Understanding of Mathematical tools is important before starting this chapter
1.1 Introduction
All of us have the experience of seeing a spark or hearing a crackle when we take off our synthetic clothes or sweater, particularly in dry weather. This is almost inevitable with ladies garments like a polyester saree. Have you ever tried to find any explanation for this phenomenon? Another common example of electric discharge is the lightning that we see in the sky during thunderstorms. We also experience a sensation of an electric shock either while opening the door of a car or holding the iron bar of a bus after sliding from our seat. The reason for these experiences is discharge of electric charges through our body, which were accumulated due to rubbing of insulating surfaces. You might have also heard that this is due to generation of static electricity. This is precisely the topic we are going to discuss in this and the next chapter. Static means anything that does not move or change with time.Electrostatics deals with the study of forces, fields and potentials arising from static charges.
1.2 Electric Charge
Historically the credit of discovery of the fact that amber rubbed with wool or silk cloth attracts light objects goes to Thales of Miletus, Greece, around 600 BC. The name electricity is coined from the Greek wordelektron meaningamber. Many such pairs of materials were known which on rubbing could attract light objects like straw, pith balls and bits of papers. You can perform the following activity at home to experience such an effect. Cut out long thin strips of white paper and lightly iron them. Take them near a TV screen or computer monitor. You will see that the strips get attracted to the screen. In fact they remain stuck to the screen for a while.
Figure 1.1 Rods and pith balls: like charges repel and unlike charges attract each other.
It was observed that if two glass rods rubbed with wool or silk cloth are brought close to each other, they repel each other [Fig. 1.1(a)]. The two strands of wool or two pieces of silk cloth, with which the rods were rubbed, also repel each other. However, the glass rod and wool attracted each other. Similarly, two plastic rods rubbed with cat’s fur repelled each other [Fig. 1.1(b)] but attracted the fur. On the other hand, the plastic rod attracts the glass rod [Fig. 1.1(c)] and repel the silk or wool with which the glass rod is rubbed. The glass rod repels the fur.
If a plastic rod rubbed with fur is made to touch two small pith balls (now-a-days we can use polystyrene balls) suspended by silk or nylon thread, then the balls repel each other [Fig. 1.1(d)] and are also repelled by the rod. A similar effect is found if the pith balls are touched with a glass rod rubbed with silk [Fig. 1.1(e)]. A dramatic observation is that a pith ball touched with glass rod attracts another pith ball touched with plastic rod [Fig. 1.1(f)].
These seemingly simple facts were established from years of efforts and careful experiments and their analyses. It was concluded, after many careful studies by different scientists, that there were only two kinds of an entity which is called theelectric charge. We say that the bodies like glass or plastic rods, silk, fur and pith balls are electrified. They acquire an electric charge on rubbing. The experiments on pith balls suggested that there are two kinds of electrification and we find that (i)like charges repel and (ii)unlike chargesattract each other. The experiments also demonstrated that the charges are transferred from the rods to the pith balls on contact. It is said that the pith balls are electrified or are charged by contact. The property which differentiates the two kinds of charges is called thepolarity of charge.
When a glass rod is rubbed with silk, the rod acquires one kind of charge and the silk acquires the second kind of charge. This is true for any pair of objects that are rubbed to be electrified. Now if the electrified glass rod is brought in contact with silk, with which it was rubbed, they no longer attract each other. They also do not attract or repel other light objects as they did on being electrified.
Thus, the charges acquired after rubbing are lost when the charged bodies are brought in contact. What can you conclude from these observations? It just tells us that unlike charges acquired by the objects neutralise or nullify each other’s effect. Therefore, the charges were named aspositive andnegative by the American scientist Benjamin Franklin. We know that when we add a positive number to a negative number of the same magnitude, the sum is zero. This might have been the philosophy in naming the charges as positive and negative. By convention, the charge on glass rod or cat’s fur is called positive and that on plastic rod or silk is termed negative. If an object possesses an electric charge, it is said to be electrified or charged. When it has no charge it is said to be electrically neutral.
A simple apparatus to detect charge on a body is thegold-leaf electroscope[Fig. 1.2(a)]. It consists of a vertical metal rod housed in a box, with two thin gold leaves attached to its bottom end. When a charged object touches the metal knob at the top of the rod, charge flows on to the leaves and they diverge. The degree of divergance is an indicator of the amount of charge.
Unification of electricity and magnetism
In olden days, electricity and magnetism were treated as separate subjects. Electricity dealt with charges on glass rods, cat’s fur, batteries, lightning, etc., while magnetism described interactions of magnets, iron filings, compass needles, etc. In 1820 Danish scientist Oersted found that a compass needle is deflected by passing an electric current through a wire placed near the needle. Ampere and Faraday supported this observation by saying that electric charges in motion produce magnetic fields and moving magnets generate electricity. The unification was achieved when the Scottish physicist Maxwell and the Dutch physicist Lorentz put forward a theory where they showed the interdependence of these two subjects. This field is calledelectromagnetism. Most of the phenomena occurring around us can be described under electromagnetism. Virtually every force that we can think of like friction, chemical force between atoms holding the matter together, and even the forces describing processes occurring in cells of living organisms, have its origin in electromagnetic force. Electromagnetic force is one of the fundamental forces of nature.
Maxwell put forth four equations that play the same role in classical electromagnetism as Newton’s equations of motion and gravitation law play in mechanics. He also argued that light is electromagnetic in nature and its speed can be found by making purely electric and magnetic measurements. He claimed that the science of optics is intimately related to that of electricity and magnetism.
The science of electricity and magnetism is the foundation for the modern technological civilisation. Electric power, telecommunication, radio and television, and a wide variety of the practical appliances used in daily life are based on the principles of this science. Although charged particles in motion exert both electric and magnetic forces, in the frame of reference where all the charges are at rest, the forces are purely electrical. You know that gravitational force is a long-range force. Its effect is felt even when the distance between the interacting particles is very large because the force decreases inversely as the square of the distance between the interacting bodies. We will learn in this chapter that electric force is also as pervasive and is in fact stronger than the gravitational force by several orders of magnitude (refer to Chapter 1 of Class XI Physics Textbook).
Students can make a simple electroscope as follows [Fig. 1.2(b)]: Take a thin aluminium curtain rod with ball ends fitted for hanging the curtain. Cut out a piece of length about 20 cm with the ball at one end and flatten the cut end. Take a large bottle that can hold this rod and a cork which will fit in the opening of the bottle. Make a hole in the cork sufficient to hold the curtain rod snugly. Slide the rod through the hole in the cork with the cut end on the lower side and ball end projecting above the cork. Fold a small, thin aluminium foil (about 6 cm in length) in the middle and attach it to the flattened end of the rod by cellulose tape. This forms the leaves of your electroscope. Fit the cork in the bottle with about 5 cm of the ball end projecting above the cork. A paper scale may be put inside the bottle in advance to measure the separation of leaves. The separation is a rough measure of the amount of charge on the electroscope.
Figure 1.2Electroscopes: (a) The gold leaf electroscope, (b) Schematics of a simple electroscope.
To understand how the electroscope works, use the white paper strips we used for seeing the attraction of charged bodies. Fold the strips into half so that you make a mark of fold. Open the strip and iron it lightly with the mountain fold up, as shown in Fig. 1.3. Hold the strip by pinching it at the fold. You would notice that the two halves move apart.This shows that the strip has acquired charge on ironing. When you fold it into half, both the halves have the same charge. Hence they repel each other. The same effect is seen in the leaf electroscope. On charging the curtain rod by touching the ball end with an electrified body, charge is transferred to the curtain rod and the attached aluminium foil. Both the halves of the foil get similar charge and therefore repel each other. The divergence in the leaves depends on the amount of charge on them. Let us first try to understand why material bodies acquire charge.
You know that all matter is made up of atoms and/or molecules. Although normally the materials are electrically neutral, they do contain charges; but their charges are exactly balanced. Forces that hold the molecules together, forces that hold atoms together in a solid, the adhesive force of glue, forces associated with surface tension, all are basically electrical in nature, arising from the forces between charged particles. Thus the electric force is all pervasive and it encompasses almost each and every field associated with our life. It is therefore essential that we learn more about such a force.
Figure 1.3Paper strip experiment.
To electrify a neutral body, we need to add or remove one kind of charge. When we say that a body is charged, we always refer to this excess charge or deficit of charge. In solids, some of the electrons, being less tightly bound in the atom, are the charges which are transferred from one body to the other. A body can thus be charged positively by losing some of its electrons. Similarly, a body can be charged negatively by gaining electrons. When we rub a glass rod with silk, some of the electrons from the rod are transferred to the silk cloth. Thus the rod gets positively charged and the silk gets negatively charged. No new charge is created in the process of rubbing. Also the number of electrons, that are transferred, is a very small fraction of the total number of electrons in the material body. Also only the less tightly bound electrons in a material body can be transferred from it to another by rubbing. Therefore, when a body is rubbed with another, the bodies get charged and that is why we have to stick to certain pairs of materials to notice charging on rubbing the bodies.
1.3 Conductors and Insulators
A metal rod held in hand and rubbed with wool will not show any sign of being charged. However, if a metal rod with a wooden or plastic handle is rubbed without touching its metal part, it shows signs of charging. Suppose we connect one end of a copper wire to a neutral pith ball and the other end to a negatively charged plastic rod. We will find that the pith ball acquires a negative charge. If a similar experiment is repeated with a nylon thread or a rubber band, no transfer of charge will take place from the plastic rod to the pith ball. Why does the transfer of charge not take place from the rod to the ball?
Some substances readily allow passage of electricity through them, others do not.Those which allow electricity to pass through them easily are calledconductors. They have electric charges (electrons) that are comparatively free to move inside the material. Metals, human and animal bodies and earth are conductors. Most of the non-metals like glass, porcelain, plastic, nylon, wood offer high resistance to the passage of electricity through them. They are calledinsulators.Most substances fall into one of the two classes stated above*.
When some charge is transferred to a conductor, it readily gets distributed over the entire surface of the conductor. In contrast, if some charge is put on an insulator, it stays at the same place. You will learn why this happens in the next chapter.
This property of the materials tells you why a nylon or plastic comb gets electrified on combing dry hair or on rubbing, but a metal article like spoon does not. The charges on metal leak through our body to the ground as both are conductors of electricity.
When we bring a charged body in contact with the earth, all the excess charge on the body disappears by causing a momentary current to pass to the ground through the connecting conductor (such as our body). This process of sharing the charges with the earth is calledgrounding or earthing. Earthing provides a safety measure for electrical circuits and appliances. A thick metal plate is buried deep into the earth and thick wires are drawn from this plate; these are used in buildings for the purpose of earthing near the mains supply. The electric wiring in our houses has three wires: live, neutral and earth. The first two carry electric current from the power station and the third is earthed by connecting it to the buried metal plate. Metallic bodies of the electric appliances such as electric iron, refrigerator, TV are connected to the earth wire. When any fault occurs or live wire touches the metallic body, the charge flows to the earth without damaging the appliance and without causing any injury to the humans; this would have otherwise been unavoidable since the human body is a conductor of electricity.
* There is a third category called semiconductors, which offer resistance to the movement of charges which is intermediate between the conductors and insulators.
1.4 Charging by Induction
When we touch a pith ball with an electrified plastic rod, some of the negative charges on the rod are transferred to the pith ball and it also gets charged. Thus the pith ball ischarged by contact. It is then repelled by the plastic rod but is attracted by a glass rod which is oppositely charged. However, why a electrified rod attracts light objects, is a question we have still left unanswered. Let us try to understand what could be happening by performing the following experiment.
Figure 1.4Charging by induction.
(i) Bring two metal spheres, A and B, supported on insulating stands, in contact as shown in Fig. 1.4(a).
(ii) Bring a positively charged rod near one of the spheres, say A, taking care that it does not touch the sphere. The free electrons in the spheres are attracted towards the rod. This leaves an excess of positive charge on the rear surface of sphere B. Both kinds of charges are bound in the metal spheres and cannot escape. They, therefore, reside on the surfaces, as shown in Fig. 1.4(b). The left surface of sphere A, has an excess of negative charge and the right surface of sphere B, has an excess of positive charge. However, not all of the electrons in the spheres have accumulated on the left surface of A. As the negative charge starts building up at the left surface of A, other electrons are repelled by these. In a short time, equilibrium is reached under the action of force of attraction of the rod and the force of repulsion due to the accumulated charges. Fig. 1.4(b) shows the equilibrium situation. The process is calledinduction of charge and happens almost instantly. The accumulated charges remain on the surface, as shown, till the glass rod is held near the sphere. If the rod is removed, the charges are not acted by any outside force and they redistribute to their original neutral state.
(iii) Separate the spheres by a small distance while the glass rod is still held near sphere A, as shown in Fig. 1.4(c). The two spheres are found to be oppositely charged and attract each other.
(iv) Remove the rod. The charges on spheres rearrange themselves as shown in Fig. 1.4(d). Now, separate the spheres quite apart. The charges on them get uniformly distributed over them, as shown in Fig. 1.4(e).
In this process, the metal spheres will each be equal and oppositely charged. This ischarging by induction. The positively charged glass rod does not lose any of its charge, contrary to the process of charging by contact.
When electrified rods are brought near light objects, a similar effect takes place. The rods induce opposite charges on the near surfaces of the objects and similar charges move to the farther side of the object. [This happens even when the light object is not a conductor. The mechanism for how this happens is explained later in Sections 1.10 and 2.10.] The centres of the two types of charges are slightly separated. We know that opposite charges attract while similar charges repel. However, the magnitude of force depends on the distance between the charges and in this case the force of attraction overweighs the force of repulsion. As a result the particles like bits of paper or pith balls, being light, are pulled towards the rods.
Example 1.1How can you charge a metal sphere positively without touching it?
SolutionFigure 1.5(a) shows an uncharged metallic sphere on an insulating metal stand. Bring a negatively charged rod close to the metallic sphere, as shown in Fig. 1.5(b). As the rod is brought close to the sphere, the free electrons in the sphere move away due to repulsion and start piling up at the farther end. The near end becomes positively charged due to deficit of electrons. This process of charge distribution stops when the net force on the free electrons inside the metal is zero. Connect the sphere to the ground by a conducting wire. The electrons will flow to the ground while the positive charges at the near end will remain held there due to the attractive force of the negative charges on the rod, as shown in Fig. 1.5(c). Disconnect the sphere from the ground. The positive charge continues to be held at the near end [Fig. 1.5(d)]. Remove the electrified rod. The positive charge will spread uniformly over the sphere as shown in Fig. 1.5(e).
Figure 1.5
In this experiment, the metal sphere gets charged by the process of induction and the rod does not lose any of its charge.
Similar steps are involved in charging a metal sphere negatively by induction, by bringing a positively charged rod near it. In this case the electrons will flow from the ground to the sphere when thesphere is connected to the ground with a wire. Can you explain why?
1.5 Basic Properties of Electric Charge
We have seen that there are two types of charges, namely positive and negative and their effects tend to cancel each other. Here, we shall now describe some other properties of the electric charge.
If the sizes of charged bodies are very small as compared to the distances between them, we treat them aspoint charges. All the charge content of the body is assumed to be concentrated at one point in space.
1.5.1 Additivity of charges
We have not as yet given a quantitative definition of a charge; we shall follow it up in the next section. We shall tentatively assume that this can be done and proceed. If a system contains two point chargesq1 andq2, the total charge of the system is obtained simply by adding algebraicallyq1 andq2,i.e., charges add up like real numbers or they are scalars like the mass of a body. If a system containsn chargesq1, q2, q3, …, qn, then the total charge of the system isq1 + q2 + q3 + … + qn . Charge has magnitude but no direction, similar to mass. However, there is one difference between mass and charge. Mass of a body is always positive whereas a charge can be either positive or negative. Proper signs have to be used while adding the charges in a system. For example, the total charge of a system containing five charges +1, +2, –3, +4 and –5, in some arbitrary unit, is (+1) + (+2) + (–3) + (+4) + (–5) = –1 in the same unit.
1.5.2 Charge is conserved
We have already hinted to the fact that when bodies are charged by rubbing, there is transfer of electrons from one body to the other; no new charges are either created or destroyed. A picture of particles of electric charge enables us to understand the idea of conservation of charge. When we rub two bodies, what one body gains in charge the other body loses. Within an isolated system consisting of many charged bodies, due to interactions among the bodies, charges may get redistributed but it is found thatthe total charge of the isolated system is always conserved. Conservation of charge has been established experimentally.
It is not possible to create or destroy net charge carried by any isolated system although the charge carrying particles may be created or destroyed in a process. Sometimes nature creates charged particles: a neutron turns into a proton and an electron. The proton and electron thus created have equal and opposite charges and the total charge is zero before and after the creation.
1.5.3 Quantisation of charge
Experimentally it is established that all free charges are integral multiples of a basic unit of charge denoted bye. Thus chargeq on a body is always given by
q = ne
where n is any integer, positive or negative. This basic unit of charge is the charge that an electron or proton carries. By convention, the charge on an electron is taken to be negative; therefore charge on an electron is written as–e and that on a proton as+e.
The fact that electric charge is always an integral multiple ofe is termed asquantisation of charge. There are a large number of situations in physics where certain physical quantities are quantised. The quantisation of charge was first suggested by the experimental laws of electrolysis discovered by English experimentalist Faraday. It was experimentally demonstrated by Millikan in 1912.
In the International System (SI) of Units, a unit of charge is called acoulomb and is denoted by the symbol C. A coulomb is defined in terms the unit of the electric current which you are going to learn in a subsequent chapter. In terms of this definition, one coulomb is the charge flowing through a wire in 1 s if the current is 1 A (ampere), (see Chapter 2 of Class XI, Physics Textbook , Part I). In this system, the value of the basic unit of charge is
e =1.602192 × 10–19C
Thus, there are about 6 × 1018 electrons in a charge of –1C. In electrostatics, charges of this large magnitude are seldomencountered and hence we use smaller units 1µC(micro coulomb) = 10–6 C or 1 mC(milli coulomb) = 10–3 C.
If the protons and electrons are the only basic charges in the universe, all the observable charges have to be integral multiples ofe. Thus, if a body containsn1 electrons andn2 protons, the total amount of charge on the body isn2 × e + n1 × (–e) = (n2 – n1) e. Sincen1 andn2 are integers, their difference is also an integer. Thus the charge on any body is always an integral multiple ofe and can be increased or decreased also in steps ofe.
The step sizeeis, however, very small because at the macroscopic level, we deal with charges of a fewµC. At this scale the fact that charge of a body can increase or decrease in units ofe is not visible. In this respect, the grainy nature of the charge is lost and it appears to be continuous.
This situation can be compared with the geometrical concepts of points and lines. A dotted line viewed from a distance appears continuous to us but is not continuous in reality. As many points very close to each other normally give an impression of a continuous line, many small charges taken together appear as a continuous charge distribution.
At the macroscopic level, one deals with charges that are enormous compared to the magnitude of chargee. Sincee = 1.6 × 10–19 C, a charge of magnitude, say 1µC, contains something like 1013 times the electronic charge. At this scale, the fact that charge can increase or decrease only in units ofe is not very different from saying that charge can take continuous values. Thus, at the macroscopic level, the quantisation of charge has no practical consequence and can be ignored. However, at the microscopic level, where the charges involved are of the order of a few tens or hundreds ofe, i.e., they can be counted, they appear in discrete lumps and quantisation of charge cannot be ignored. It is the magnitude of scale involved that is very important.
Example 1.2 If 109 electrons move out of a body to another body every second, how much time is required to get a total charge of 1 Con the other body?
Solution In one second 109 electrons move out of the body. Therefore the charge given out in one second is 1.6 × 10–19 × 109 C =1.6 × 10–10 C.
The time required to accumulate a charge of 1 C can then be estimated to be 1 C ÷ (1.6 × 10–10 C/s) = 6.25 × 109 s =6.25 × 109 ÷ (365 × 24 × 3600) years = 198 years. Thus to collect a charge of one coulomb, from a body from which 109 electrons move out every second, we will need approximately 200 years. One coulomb is, therefore, a very large unit for many practical purposes.
It is, however, also important to know what is roughly the number of electrons contained in a piece of one cubic centimetre of a material. A cubic piece of copper of side 1 cm contains about 2.5 × 1024 electrons.
Example 1.3How much positive and negative charge is there in a cup of water?
Solution Let us assume that the mass of one cup of water is 250 g. The molecular mass of water is 18g. Thus, one mole (= 6.02 × 1023 molecules) of water is 18 g. Therefore the number of molecules in one cup of water is (250/18) × 6.02 × 1023.
Each molecule of water contains two hydrogen atoms and one oxygen atom, i.e., 10 electrons and 10 protons. Hence the total positive and total negative charge has the same magnitude. It isequal to(250/18) × 6.02 × 1023 × 10 × 1.6 × 10–19 C= 1.34 × 107 C.
1.6 Coulomb’s Law
Coulomb’s law is a quantitative statement about the force between two point charges. When the linear size of charged bodies are much smaller than the distance separating them, the size may be ignored and the charged bodies are treated aspoint charges. Coulomb measured the force between two point charges and found thatit varied inversely as the square of the distance between the charges and was directly proportional to the product of the magnitude of the two charges and acted along the line joining the two charges. Thus, if two point chargesq1, q2 are separated by a distancer in vacuum, the magnitude of the force (F) between them is given by
(1.1)
How did Coulomb arrive at this law from his experiments? Coulomb used a torsion balance* for measuring the force between two charged metallicspheres. When the separation between two spheres is much larger than the radius of each sphere, the charged spheres may be regarded as point charges. However, the charges on the spheres were unknown, to begin with. How then could he discover a relation like Eq. (1.1)? Coulomb thought of the following simple way: Suppose the charge on a metallic sphere isq. If the sphere is put in contact with an identical uncharged sphere, the charge will spread over the two spheres. By symmetry, the charge on each sphere will beq/2*. Repeating this process, we can get chargesq/2, q/4, etc. Coulomb varied the distance for a fixed pair of charges and measured the force for different separations. He then varied the charges in pairs, keeping the distance fixed for each pair. Comparing forces for different pairs of charges at different distances, Coulomb arrived at the relation, Eq. (1.1).
*A torsion balance is a sensitive device to measure force. It was also used later by Cavendish to measure the very feeble gravitational force between two objects, to verify Newton’s Law of Gravitation.
Coulomb’s law, a simple mathematical statement, was initially experimentally arrived at in the manner described above. While the original experiments established it at a macroscopic scale, it has also been established down to subatomic level (r ~ 10–10 m).
Coulomb discovered his law without knowing theexplicit magnitude of the charge. In fact, it is the other way round: Coulomb’s law cannow be employed to furnish a definition for a unit of charge. In the relation, Eq. (1.1),k is so far arbitrary. We can choose any positive value ofk. The choice ofk determines the size of the unit of charge. In SI units, the value ofk is about 9 × 109. The unit of charge that results from this choice is called a coulomb which we defined earlier in Section 1.4. Putting this value ofk in Eq. (1.1), we see that forq1 =q2 = 1 C,r = 1 m
F = 9 × 109 N
Charles Augustin de Coulomb (1736 – 1806)Coulomb, a French physicist, began his career as a military engineer in the West Indies. In 1776, he returned to Paris and retired to a small estate to do his scientific research. He invented a torsion balance to measure the quantity of a force and used it for determination of forces of electric attraction or repulsion between small charged spheres. He thus arrived in 1785 at the inverse square law relation, now known as Coulomb’s law. The law had been anticipated by Priestley and also by Cavendish earlier, though Cavendish never published his results. Coulomb also found the inverse square law of force between unlike and like magnetic poles.
That is, 1 C is the charge that when placed at a distance of 1 m from another charge of the same magnitudein vacuum experiences an electrical force of repulsion of magnitude 9 × 109 N. One coulomb is evidently too big a unit to be used. In practice, in electrostatics, one uses smaller units like 1 mC or 1µC.
The constantk in Eq. (1.1) is usually put as k = 1/4πε0 for later convenience, so that Coulomb’s law is written as
(1.2)
ε0 is called thepermittivity of free space. The value ofε0 in SI units is
= 8.854 × 10–12 C2 N–1m–2
*Implicit in this is the assumption of additivity of charges and conservation: two charges (q/2 each) add up to make a total charge q.
Since force is a vector, it is better to write Coulomb’s law in the vector notation. Let the position vectors of chargesq1 andq2 ber1 andr2 respectively [see Fig.1.6(a)]. We denote force onq1 due toq2 byF12 and force onq2due toq1 byF21. The two point chargesq1 andq2 have been numbered 1 and 2 for convenience and the vector leading from 1 to 2 is denoted byr21:
r21 =r2 –r1
In the same way, the vector leading from 2 to 1 is denoted byr12:
r12 =r1 –r2 = –r21
The magnitude of the vectorsr21 andr12 is denoted byr21 and r12,respectively (r12 = r21). The direction of a vector is specified by a unit vector along the vector. To denote the direction from 1 to 2 (or from 2 to 1), we define the unit vectors:
,
Coulomb’s force law between two point chargesq1 andq2 located atr1 andr2 is then expressed as
(1.3)
Some remarks on Eq. (1.3) are relevant:
Figure 1.6(a)Geometry and (b) Forces between charges.
• Equation (1.3) is valid for any sign ofq1 andq2 whether positive or negative. Ifq1 andq2 are of the same sign (either both positive or both negative),F21 is along21, which denotes repulsion, as it should be for like charges. Ifq1 andq2 are of opposite signs,F21 is along –21(=12), which denotes attraction, as expected for unlike charges. Thus, we do not have to write separate equations for the cases of like and unlikecharges. Equation (1.3) takes care of both cases correctly [Fig. 1.6(b)].
• The forceF12 on chargeq1 due to chargeq2, is obtained from Eq. (1.3), by simply interchanging 1 and 2, i.e.,
Thus, Coulomb’s law agrees with the Newton’s third law.
• Coulomb’s law [Eq. (1.3)] gives the force between two chargesq1 andq2 in vacuum. If the charges are placed in matter or the intervening space has matter, the situation gets complicated due to the presence of charged constituents of matter. We shall consider electrostatics in matter in the next chapter.
Example 1.4 Coulomb’s law for electrostatic force between two point charges and Newton’s law for gravitational force between two stationary point masses, both have inverse-square dependence on the distance between the charges and masses respectively. (a) Compare the strength of these forces by determining the ratio of their magnitudes (i) for an electron and a proton and (ii) for two protons. (b) Estimate the accelerations of electron and proton due to the electrical force of their mutual attraction when they are 1 Å (= 10-10 m) apart? (mp = 1.67 × 10–27 kg, me = 9.11 × 10–31 kg)
Solution
(a) (i) The electric force between an electron and a proton at a distancer apart is:
where the negative sign indicates that the force is attractive. The corresponding gravitational force (always attractive) is:
where mp andme are the masses of a proton and an electron respectively.
(ii) On similar lines, the ratio of the magnitudes of electric force to the gravitational force between two protons at a distancer apart is:
1.3 × 1036
However, it may be mentioned here that the signs of the two forces are different. For two protons, the gravitational force is attractive in nature and the Coulomb force is repulsive. The actual values of these forces between two protons inside a nucleus (distance between two protons is ~ 10-15 m inside a nucleus) areFe ~ 230 N, whereas,FG ~ 1.9 × 10–34 N.
The (dimensionless) ratio of the two forces shows that electrical forces are enormously stronger than the gravitational forces.
(b) The electric forceF exerted by a proton on an electron is same in magnitude to the force exerted by an electron on a proton; however, the masses of an electron and a proton are different. Thus, the magnitude of force is
Using Newton’s second law of motion,F =ma,the acceleration that an electron will undergo is
a= 2.3×10–8N / 9.11 ×10–31 kg = 2.5 × 1022 m/s2
Comparing this with the value of acceleration due to gravity, we can conclude that the effect of gravitational field is negligible on the motion of electron and it undergoes very large accelerations under the action of Coulomb force due to a proton.
The value for acceleration of the proton is
2.3 × 10–8N / 1.67 × 10–27kg = 1.4 × 1019m/s2
Example 1.5 A charged metallic sphere A is suspended by a nylon thread. Another charged metallic sphere B held by an insulating handle is brought close to A such that the distance between their centres is 10 cm, as shown in Fig. 1.7(a). The resulting repulsion of A is noted (for example, by shining a beam of light and measuring the deflection of its shadow on a screen). Spheres A and B are touched by uncharged spheres C and D respectively, as shown in Fig. 1.7(b). C and D are then removed and B is brought closer to A to a distance of 5.0 cm between their centres, as shown in Fig. 1.7(c).What is the expected repulsion of A on the basis of Coulomb’s law? Spheres A and C and spheres B and D have identical sizes. Ignore the sizes of A and B in comparison to the separation between their centres.
Figure 1.7
Solution Let the original charge on sphere A beqand that on B beq′. At a distancer between their centres, the magnitude of the electrostatic force on each is given by
neglecting the sizes of spheres A and B in comparison tor. When an identical but uncharged sphere C touches A, the charges redistribute on A and C and, by symmetry, each sphere carries a chargeq/2. Similarly, after D touches B, the redistributed charge on each is q′/2. Now, if the separation between A and B is halved, the magnitude of the electrostatic force on each is
Thus the electrostatic force on A, due to B, remains unaltered.
1.7 Forces between Multiple Charges
The mutual electric force between two charges is givenby Coulomb’s law. How to calculate the force on a charge where there are not one but several charges around? Consider a system ofnstationary chargesq1, q2, q3, ..., qn in vacuum.What is the force onq1 due toq2, q3, ...,qn? Coulomb’s law is not enough to answer this question. Recall that forces of mechanical origin add according to the parallelogram law of addition. Is the same true for forces of electrostatic origin?
Figure 1.8A system of (a)three charges (b) multiple charges.
Experimentally, it is verified thatforce on any chargedue to a number of other charges is the vector sum of all the forces on that charge due to theother charges, taken one at a time. The individual forces are unaffected due to the presence of other charges. This is termed as the principle of superposition.
To better understand the concept, consider a system of three chargesq1, q2 and q3, as shown in Fig. 1.8(a). The force on one charge, sayq1, due to two other chargesq2, q3 can therefore be obtained by performing a vector addition of the forces due to each one of these charges. Thus, if the force onq1 due toq2 is denoted byF12, F12 is given by Eq. (1.3) even though other charges are present.Thus, F12
In the same way, the force onq1 due toq3, denoted byF13, is given by
which again is the Coulomb force onq1 due toq3, even though other chargeq2 is present.
Thus the total forceF1 on q1 due to the two chargesq2 andq3 is given as
(1.4)
The above calculation of force can be generalised to a system of charges more than three, as shown in Fig. 1.8(b).
The principle of superposition says that in a system of chargesq1, q2, ..., qn, the force onq1 due toq2 is the same as given by Coulomb’s law, i.e., it is unaffected by the presence of the other chargesq3, q4, ..., qn. The total forceF1 on the chargeq1, due to all other charges, is then given by the vector sum of the forcesF12, F13, ...,F1n:
i.e.,
(1.5)
The vector sum is obtained as usual by the parallelogram law of addition of vectors. All of electrostatics is basically a consequence of Coulomb’s law and the superposition principle.
Example 1.6 Consider three chargesq1, q2, q3 each equal toq at the vertices of an equilateral triangle of sidel. What is the force on a chargeQ (with the same sign asq) placed at the centroid of the triangle, as shown in Fig. 1.9?
Figure 1.9
SolutionIn the given equilateral triangle ABC of sides of lengthl, if we draw a perpendicular AD to the side BC,
AD = AC cos 30º = () l and the distance AO of the centroid O from A is (2/3) AD = () l. By symmatry AO = BO = CO.
Thus,
Force F1on Q due to chargeq at A =along AO
Force F2 on Q due to chargeq at B =along BO
Force F3 on Q due to chargeq at C =along CO
The resultant of forcesF2 andF3 isalong OA, by the parallelogram law. Therefore, the total force onQ = = 0, whereis theunit vector along OA.
It is clear also by symmetry that the three forces will sum to zero. Suppose that the resultant force was non-zero but in some direction. Consider what would happen if the system was rotated through 60° about O.
Example 1.7Consider the chargesq, q, and –q placed at the vertices of an equilateral triangle, as shown in Fig. 1.10. What is the force on each charge?
Figure 1.10
Solution The forces acting on chargeq at A due to chargesq at B and–q at C areF12 along BA andF13 along AC respectively, as shown in Fig. 1.10. By the parallelogram law, the total forceF1onthecharge q at A is given by
F1 =Fwhereis a unit vector along BC.
The force of attraction or repulsion for each pair of charges has the same magnitude
The total forceF2 on chargeq at B is thusF2 =F2, where2 is a unit vector along AC.
Similarly the total force on charge –q at Cis F3 =F, whereis the unit vector along the direction bisecting the∠BCA.
It is interesting to see that the sum of the forces on the three charges is zero, i.e.,
F1 + F2 + F3 = 0
The result is not at all surprising. It follows straight from the fact that Coulomb’s law is consistent with Newton’s third law. The proof is left to you as an exercise.
1.8 Electric Field
Let us consider a point chargeQ placed in vacuum, at the origin O. If we place another point chargeq at a point P, whereOP =r, then the chargeQ will exert a force onq as per Coulomb’s law. We may ask the question: If chargeq is removed, then what is left in the surrounding? Is there nothing? If there is nothing at the point P, then how does a force act when we place the chargeq at P. In order to answer such questions, the early scientists introduced the concept of field. According to this, we say that the chargeQ produces an electric field everywhere in the surrounding. When another chargeq is brought at some point P, the field there acts on it and produces a force. The electric field produced by the chargeQ at a pointr is given as
(1.6)
wherer/r, is a unit vector from the origin to the pointr. Thus, Eq.(1.6) specifies the value of the electric field for each value of the position vectorr. The word “field” signifies how some distributed quantity (which could be a scalar or a vector) varies with position. The effect of the charge has been incorporated in the existence of the electric field. We obtain the forceFexerted by acharge Q on a chargeq, as
(1.7)
Note that the chargeq also exerts an equal and opposite force on the chargeQ. The electrostatic force between the chargesQ and q can be looked upon as an interaction between chargeq and the electric field ofQ and vice versa.If we denote the position of chargeq by the vectorr,it experiences a forceF equal to the chargeq multiplied by the electric fieldE at the location ofq. Thus,
F(r) =q E(r) (1.8)
Equation (1.8) defines the SI unit of electric field as N/C*.
Some important remarks may be made here:
(i)From Eq. (1.8), we can infer that ifq is unity, the electric field due to a chargeQ is numerically equal to the force exerted by it. Thus, theelectric field due to a chargeQ at a point in space may be defined as the force that a unit positive charge would experience if placedat that point. The chargeQ, which is producing the electric field, is called asource chargeand the chargeq, which tests the effect of a source charge, is called atest charge. Note that the source chargeQ must remain at its original location. However, if a chargeq is brought at any point around Q, Q itself is bound to experience an electrical force due toq and will tend to move. A way out of this difficulty is to makeq negligibly small. The forceF is then negligibly small but the ratioF/q is finite and defines the electric field:
(1.9)
Figure 1.11Electric field (a) due to a chargeQ, (b) due to a charge–Q.
A practical way to get around the problem (of keepingQ undisturbed in the presence ofq) is to holdQ to its location by unspecified forces! This may look strange but actually this is what happens in practice. When we are considering the electric force on a test chargeq due to a charged planar sheet (Section 1.15), the charges on the sheet are held to their locations by the forces due to the unspecified charged constituents inside the sheet.
(ii)Note that the electric fieldE due toQ, though defined operationally in terms of some test chargeq, is independent ofq. This is becauseF is proportional toq, so the ratioF/q does not depend onq. The forceF on the chargeq due to the chargeQ depends on the particular location of chargeq which may take any value in the space around the chargeQ. Thus, the electric fieldE due toQ is also dependent on the space coordinater.For different positions of the chargeq all over the space, we get different values of electric fieldE. The field exists at every point in three-dimensional space.
(iii)For a positive charge, the electric field will be directed radially outwards from the charge. On the other hand, if the source charge is negative, the electric field vector, at each point, points radially inwards.
(iv)Since the magnitude of the forceF on chargeq due to chargeQ depends only on the distancer of the chargeq from chargeQ, the magnitude of the electric fieldE will also depend only on the distancer. Thus at equal distances from the chargeQ, the magnitude of its electric fieldE is same.The magnitude of electric fieldEdue to a point charge is thus same on a sphere with the point charge at its centre; in other words, it has a spherical symmetry.
1.8.1 Electric field due to a system of charges
Consider a system of chargesq1, q2, ..., qn with position vectorsr1, r2, ...,rn relative to some origin O. Like the electric field at a point in space due to a single charge, electric field at a point in space due to the system of charges is defined to be the force experienced by a unit test charge placed at that point, without disturbing the original positions of chargesq1, q2,..., qn. We can use Coulomb’s law and the superposition principle to determine this field at a point P denoted by position vectorr.
Electric fieldE1 atr due toq1 atr1 is given by
E1 =
whereis a unit vector in the direction fromq1 to P, andr1P is the distance betweenq1 and P.
In the same manner, electric fieldE2 atr due toq2at r2is E2 =
whereis a unit vector in the direction fromq2 to P andr2P is the distance betweenq2 and P. Similar expressions hold good for fieldsE3, E4, ..., En due to chargesq3, q4, ...,qn.
By the superposition principle, the electric fieldE atr due to the system of charges is (as shown in Fig. 1.12)
Figure 1.12Electric field at a point due to a system of charges is the vector sum of the electric fields at the point due to individual charges.
E(r) =E1 (r) +E2 (r) + … +En(r)
=
E(r)(1.10)
E is a vector quantity that varies from one point to another point in space and is determined from the positions of the source charges.
1.8.2 Physical significance of electric field
You may wonder why the notion of electric field has been introduced here at all. After all, for any system of charges, the measurable quantity is the force on a charge which can be directly determined using Coulomb’s law and the superposition principle [Eq. (1.5)]. Why then introduce this intermediate quantity called the electric field?
For electrostatics, the concept of electric field is convenient, but not really necessary. Electric field is an elegant way of characterising the electrical environment of a system of charges. Electric field at a point in the space around a system of charges tells you the force a unit positive test charge would experience if placed at that point (without disturbing the system). Electric field is a characteristic of the system of charges and is independent of the test charge that you place at a point to determine the field. The termfield in physics generally refers to a quantity that is defined at every point in space and may vary from point to point. Electric field is a vector field, since force is a vector quantity.
The true physical significance of the concept of electric field, however, emerges only when we go beyond electrostatics and deal with time-dependent electromagnetic phenomena. Suppose we consider the force between two distant chargesq1, q2 in accelerated motion. Now the greatest speed with which a signal or information can go from one point to another isc, the speed of light. Thus, the effect of any motion ofq1 onq2 cannot arise instantaneously. There will be some time delay between the effect (force onq2) and the cause (motion ofq1). It is precisely here that the notion of electric field (strictly, electromagnetic field) is natural and very useful.The field picture is this: the accelerated motion of charge q1 produces electromagnetic waves, which then propagate with the speed c, reach q2 and cause a force on q2. The notion of field elegantly accounts for the time delay. Thus, even though electric and magnetic fields can be detected only by their effects (forces) on charges, they are regarded as physical entities, not merely mathematical constructs. They have anindependent dynamicsof their own, i.e., they evolve according to laws of theirown. They can also transport energy. Thus, a source of time-dependent electromagnetic fields, turned on for a short interval of time and then switched off, leaves behind propagating electromagnetic fields transporting energy. The concept of field was first introduced by Faraday and is now among the central concepts in physics.
Example 1.8 An electron falls through a distance of 1.5 cm in a uniform electric field of magnitude 2.0 × 104 N C–1 [Fig. 1.13(a)]. The direction of the field is reversed keeping its magnitude unchanged and a proton falls through the same distance [Fig. 1.13(b)]. Compute the time of fall in each case. Contrast the situation with that of ‘free fall under gravity’.
Figure 1.13
Solution In Fig. 1.13(a) the field is upward, so the negatively charged electron experiences a downward force of magnitudeeE whereE is the magnitude of the electric field. The acceleration of the electron is
ae =eE/me
where me is the mass of the electron.
Starting from rest, the time required by the electron to fall through a distanceh is given by
For e = 1.6 × 10–19C,me = 9.11 × 10–31 kg,
E = 2.0 × 104 N C–1, h = 1.5 × 10–2 m,
te = 2.9 × 10–9s
In Fig. 1.13 (b), the field is downward, and the positively charged proton experiences a downward force of magnitudeeE. The acceleration of the proton is
ap =eE/mp
where mp is the mass of the proton;mp = 1.67 × 10–27 kg. The time of fall for the proton is
Thus, the heavier particle (proton) takes a greater time to fall through the same distance. This is in basic contrast to the situation of ‘free fall under gravity’ where the time of fall is independent of the mass of the body. Note that in this example we have ignored the acceleration due to gravity in calculating the time of fall. To see if this is justified, let us calculate the acceleration of the proton in the given electric field:
which is enormous compared to the value ofg (9.8 m s–2), the acceleration due to gravity. The acceleration of the electron is even greater. Thus, the effect of acceleration due to gravity can be ignored in this example.
Example 1.9Two point chargesq1 andq2, of magnitude +10–8 C and –10–8 C, respectively, are placed 0.1 m apart. Calculate the electric fields at points A, B and C shown in Fig. 1.14.
Figure 1.14
SolutionThe electric field vectorE1A at A due to the positive chargeq1 points towards the right and has a magnitude
= 3.6 × 104 N C–1
The electric field vectorE2A at A due to the negative chargeq2 points towards the right and has the same magnitude. Hence the magnitude of the total electric fieldEA at A is
EA =E1A +E2A = 7.2 × 104 N C–1
EA is directed toward the right.
The electric field vectorE1B at B due to the positive chargeq1 points towards the left and has a magnitude
= 3.6 × 104 N C–1
The electric field vectorE2B at B due to the negative chargeq2 points towards the right and has a magnitude
= 4 × 103 N C–1
The magnitude of the total electric field at B is
EB= E1B– E2B = 3.2 × 104 N C–1
EB is directed towards the left.
The magnitude of each electric field vector at point C, due to chargeq1 andq2 is
= 9 × 103 N C–1
The directions in which these two vectors point are indicated in Fig. 1.14. The resultant of these two vectors is
= 9 × 103 N C–1
ECpoints towards the right.
1.9 Electric Field Lines
We have studied electric field in the last section. It is a vector quantity and can be represented as we represent vectors. Let us try to representE due to a point charge pictorially. Let the point charge be placed at the origin. Draw vectors pointing along the direction of the electric field with their lengths proportional to the strength of the field at each point. Since the magnitude of electric field at a point decreases inversely as the square of the distance of that point from the charge, the vector gets shorter as one goes away from the origin, always pointing radially outward. Figure 1.15 shows such a picture. In this figure, each arrow indicates the electric field, i.e., the force acting on a unit positive charge, placed at the tail of that arrow. Connect the arrows pointing in one direction and the resulting figure represents a field line. We thus get manyfield lines, all pointing outwards from the point charge. Have we lost the information about the strength or magnitude of the field now, because it was contained in the length of the arrow? No. Now the magnitude of the field is represented by the density of field lines.E is strong near the charge, so the density of field lines is more near the charge and the lines are closer. Away from the charge, the field gets weaker and the density of field lines is less, resulting in well-separated lines.
Another person may draw more lines. But the number of lines is not important. In fact, an infinite number of lines can be drawn in any region. It is the relative density of lines in different regions which is important.
We draw the figure on the plane of paper,i.e., in two-dimensions but we live in three-dimensions. So if one wishes to estimate the density of field lines, one has to consider the number of lines per unit cross-sectional area, perpendicular to the lines. Since the electric field decreases as the square of the distance from a point charge and the area enclosing the charge increases as the square of the distance, the number of field lines crossing the enclosing area remains constant, whatever may be the distance of the area from the charge.
We started by saying that the field lines carry information about the direction of electric field at different points in space. Having drawn a certain set of field lines, the relative density (i.e., closeness) of the field lines at different points indicates the relative strength of electric field at those points. The field lines crowd where the field is strong and are spaced apart where it is weak. Figure 1.16 shows a set of field lines. We can imagine two equal and small elements of area placed at points R and S normal to the field lines there. The number of field lines in our picture cutting the area elements is proportional to the magnitude of field at these points. The picture shows that the field at R is stronger than at S.
Figure 1.15Field of a point charge.
To understand the dependence of the field lines on the area, or rather thesolid angle subtended by an area element, let us try to relate the area with the solid angle, a generalisation of angle to three dimensions. Recall how a (plane) angle is defined in two-dimensions. Let a small transverse line element∆l be placed at a distancer from a point O. Then the angle subtended by∆l at O can be approximated as∆θ= ∆l/r. Likewise, in three-dimensions the solid angle* subtended by a small perpendicular plane area∆S, at a distancer, can be written as ∆Ω= ∆S/r2. We know that in a given solid angle the number of radial field lines is the same. In Fig. 1.16, for two points P1 and P2 at distancesr1 andr2from the charge, the element of area subtending the solid angle∆Ω is∆Ω at P1 and an element of area
Figure 1.16Dependence of electric field strength on the distance and its relation to the number of field lines.
∆Ω at P2, respectively. The number of lines (sayn) cutting these area elements are the same. The number of field lines, cutting unit area element is thereforen/(∆Ω) at P1 andn/(∆Ω) at P2, respectively. Sincen and∆Ω are common, the strength of the field clearly has a 1/r2dependence.
The picture of field lines was invented by Faraday to develop an intuitive non-mathematical way of visualising electric fields around charged configurations. Faraday called themlines of force. This term is somewhat misleading, especially in case of magnetic fields. The more appropriate term is field lines(electric or magnetic) that we have adopted in this book.
Electric field lines are thus a way of pictorially mapping the electric field around a configuration of charges. An electric field line is, in general,a curve drawn in such a way that the tangent to it at each point is in the direction of the net field at that point. An arrow on the curve is obviously necessary to specify the direction of electric field from the two possible directions indicated by a tangent to the curve. A field line is a space curve, i.e., a curve in three dimensions.
Figure 1.17Field lines due to some simple charge configurations.
Figure 1.17 shows the field lines around some simple charge configurations. As mentioned earlier, the field lines are in 3-dimensional space, though the figure shows them only in a plane. The field lines of a single positive charge are radially outward while those of a single negative charge are radially inward. The field lines around a system of two positive charges (q, q) give a vivid pictorial description of their mutual repulsion, while those around the configuration of two equal and opposite charges (q, –q), a dipole, show clearly the mutual attraction between the charges. The field lines follow some important general properties:
(i) Field lines start from positive charges and end at negative charges. If there is a single charge, they may start or end at infinity.
(ii) In a charge-free region, electric field lines can be taken to be continuous curves without any breaks.
(iii) Two field lines can never cross each other. (If they did, the field at the point of intersection will not have a unique direction, which is absurd.)
(iv) Electrostatic field lines do not form any closed loops. This follows from the conservative nature of electric field (Chapter 2).
1.10 Electric Flux
Consider flow of a liquid with velocityv, through a small flat surface dS, in a direction normal to the surface. The rate of flow of liquid is given by the volume crossing the area per unit timev dS and represents the flux of liquid flowing across the plane. If the normal to the surface is not parallel to the direction of flow of liquid,i.e., tov, but makes an angleθ with it, the projected area in a plane perpendicular tov isv dS cosθ. Therefore, the flux going out of the surface dS is v.dS. For the case of the electric field, we define an analogous quantity and call itelectric flux. We should, however, note that there is noflow of a physically observable quantity unlike the case of liquid flow.
In the picture of electric field lines described above, we saw that the number of field lines crossing a unit area, placed normal to the field at a point is a measure of the strength of electric field at that point. This means that if we place a small planar element of area∆S normal toE at a point, the number of field lines crossing it is proportional* toE ∆S. Now suppose we tilt the area element by angleθ. Clearly, the number of field lines crossing the area element will be smaller. The projection of the area element normal toE is∆S cosθ. Thus, the number of field lines crossing∆S is proportional toE ∆S cosθ. Whenθ = 90°, field lines will be parallel to∆Sand will not cross it at all (Fig. 1.18).
Figure 1.18 Dependence of flux on the inclinationθ betweenE and n
The orientation of area element and not merely its magnitude is important in many contexts. For example, in a stream, the amount of water flowing through a ring will naturally depend on how you hold the ring. If you hold it normal to the flow, maximum water will flowthrough it than if you hold it with some other orientation. This shows that an area element should be treated as a vector. It has a magnitude and also a direction. How to specify the direction of a planar area? Clearly, the normal to the plane specifies the orientation of the plane. Thus the direction of a planar area vector is along its normal.
How to associate a vector to the area of a curved surface? We imagine dividing the surface into a large number of very small area elements. Each small area element may be treated as planar and a vector associated with it, as explained before.
Notice one ambiguity here. The direction of an area element is along its normal. But a normal can point in two directions. Which direction do we choose as the direction of the vector associated with the area element? This problem is resolved by some convention appropriate to the given context. For the case of a closed surface, this convention is very simple. The vector associated with every area element of a closed surface is taken to be in the direction of the outward normal. This is the convention used in Fig. 1.19. Thus, the area element vector∆S at a point on a closed surface equals∆Sn where ∆S is the magnitude of the area element and nis a unit vector in the direction of outward normal at that point.
We now come to the definition of electric flux. Electric flux∆φ through an area element∆S is defined by
∆φ =E.∆S = E ∆S cosθ (1.11)
which, as seen before, is proportional to the number of field lines cutting the area element. The angleθ here is the angle betweenE and∆S. For a closed surface, with the convention stated already,θ is the angle betweenE and the outward normal to the area element. Notice we could look at the expressionE∆S cosθ in two ways: E (∆S cosθ) i.e.,E times theprojection of area normal toE, orE⊥∆S, i.e., component ofE along the normal to the area element times the magnitude of the area element. The unit of electric flux is N C–1 m2.
The basic definition of electric flux given by Eq. (1.11) can be used, in principle, to calculate the total flux through any given surface. All we have to do is to divide the surface into small area elements, calculate the flux at each element and add them up. Thus, the total fluxφ through a surfaceS is
φ~ΣE.∆S (1.12)
The approximation sign is put because the electric fieldE is taken to be constant over the small area element. This is mathematically exact only when you take the limit∆S → 0 and the sum in Eq. (1.12) is written as an integral.
Figure 1.19Convention for defining normal
1.11 Electric Dipole
An electric dipole is a pair of equal and opposite point chargesq and –q, separated by a distance 2a. The line connecting the two charges defines a direction in space. By convention, the direction from –q toq is said to be the direction of the dipole. The mid-point of locations of –q andq is called the centre of the dipole.
The total charge of the electric dipole is obviously zero. This does not mean that the field of the electric dipole is zero. Since the chargeq and –q are separated by some distance, the electric fields due to them, when added, do not exactly cancel out. However, at distances much larger than the separation of the two charges forming a dipole (r >> 2a), the fields due toq and –q nearly cancel out. The electric field due to a dipole therefore falls off, at large distance, faster than like 1/r2 (the dependence onr of the field due to a single chargeq). These qualitative ideas are borne out by the explicit calculation as follows:
1.11.1 The field of an electric dipole
The electric field of the pair of charges (–q and q) at any point in space can be found out from Coulomb’s law and the superposition principle. The results are simple for the following two cases: (i) when the point is on the dipole axis, and (ii) when it is in theequatorial plane of the dipole, i.e., on a plane perpendicular to the dipole axis through its centre. The electric field at any general point P is obtained by adding the electric fieldsE–q due to the charge –q andE+qdue to the chargeq, by the parallelogram law of vectors.
Let the point P be at distancer from the centre of the dipole on the side of the chargeq, as shown in Fig. 1.20(a). Then
[1.13(a)]
whereis the unit vector along the dipole axis (from –q to q). Also
[1.13(b)] The total field at P is
(1.14) Forr >>a
(r >>a) (1.15)
Figure 1.20Electric field of a dipole at (a) a point on the axis, (b) a point on the equatorial plane of the dipole.
p is the dipole moment vector of magnitudep =q × 2a and
directed from –q toq.
The magnitudes of the electric fields due to the two charges +q and –q are given by
[1.16(a)]
[1.16(b)]
and are equal.
The directions ofE+q andE–q are as shown in Fig. 1.20(b). Clearly, the components normal to the dipole axis cancel away. The components along the dipole axis add up. The total electric field is opposite to. We have
E = –(E +q +E –q) cosθ
(1.17)
At large distances (r >>a), this reduces to
(1.18)
From Eqs. (1.15) and (1.18), it is clear that the dipole field at large distances does not involveq anda separately; it depends on the productqa. This suggests the definition of dipole moment. Thedipole momentvectorp of an electric dipole is defined by
p =q × 2a(1.19)
that is, it is a vector whose magnitude is chargeq times the separation 2a (between the pair of chargesq, –q) and the direction is along the line from –q toq. In terms ofp, the electric field of a dipole at large distances takes simple forms:
At a point on the dipole axis
(r >> a)(1.20)
At a point on the equatorial plane
(r >> a) (1.21)
Notice the important point that the dipole field at large distances falls off not as 1/r2 but as1/r3. Further, the magnitude and the direction of the dipole field depends not only on the distancerbut also on theangle between the position vectorr and the dipole momentp.
We can think of the limit when the dipole size 2a approaches zero, the chargeq approaches infinity in such a way that the productp =q × 2a is finite. Such a dipole is referred to as apoint dipole. For a point dipole, Eqs. (1.20) and (1.21) are exact, true for anyr.
1.11.2 Physical significance of dipoles
In most molecules, the centres of positive charges and of negative charges* lie at the same place. Therefore, their dipole moment is zero. CO2 and CH4 are of this type of molecules. However, they develop a dipole moment when an electric field is applied. But in some molecules, the centres of negative charges and of positive charges do not coincide. Therefore they have a permanent electric dipole moment, even in the absence of an electric field. Such molecules are called polar molecules. Water molecules, H2O, is an example of this type. Various materials give rise to interesting properties and important applications in the presence or absence of electric field.
Example 1.10Two charges ±10µC are placed 5.0 mm apart. Determine the electric field at (a) a point P on the axis of the dipole 15 cm away from its centre O on the side of the positive charge, as shown in Fig. 1.21(a), and (b) a point Q, 15 cm away from O on a line passing through O and normal to the axis of the dipole, as shown in Fig. 1.21(b).
figure 1.21
Solution(a) Field at P due to charge +10µC
=
= 4.13 × 106 N C–1 along BP
Field at P due to charge –10µC
= 3.86 × 106 N C–1 along PA
The resultant electric field at P due to the two charges at A and B is = 2.7 × 105 N C–1 along BP.
In this example, the ratio OP/OB is quite large (= 60). Thus, we can expect to get approximately the same result as above by directly using the formula for electric field at a far-away point on the axis of a dipole. For a dipole consisting of charges ±q, 2a distance apart, the electric field at a distancer from the centre on the axis of the dipole has a magnitude
(r/a >> 1)
where p = 2a q is the magnitude of the dipole moment.
The direction of electric field on the dipole axis is always along the direction of the dipole moment vector (i.e., from –q toq). Here, p =10–5 C × 5 × 10–3 m = 5 × 10–8 C m
Therefore,
E == 2.6 × 105 N C–1
along the dipole moment direction AB, which is close to the result obtained earlier.
(b) Field at Q due to charge + 10µC at B
=
= 3.99 × 106 N C–1 along BQ
Field at Q due to charge –10µC at A
=
= 3.99 × 106 N C–1 along QA.
Clearly, the components of these two forces with equal magnitudes cancel along the direction OQ but add up along the direction parallel to BA. Therefore, the resultant electric field at Q due to the two charges at A and B is
.
= 2 ×along BA
= 1.33 × 105 N C–1 along BA.
As in (a), we can expect to get approximately the same result by directly using the formula for dipole field at a point on the normal to the axis of the dipole:
(r/a >> 1)
= 1.33 × 105 N C–1.
The direction of electric field in this case is opposite to the direction of the dipole moment vector. Again, the result agrees with that obtained before.
1.12 Dipole in a Uniform External Field
Consider a permanent dipole of dipole momentp in a uniform external fieldE, as shown in Fig. 1.22. (By permanent dipole, we mean thatp exists irrespective ofE; it has not been induced byE.)
There is a forceqE onq and a force –qE on –q. The net force on the dipole is zero, sinceE is uniform. However, the charges are separated, so the forces act at different points, resulting in a torque on the dipole. When the net force is zero, the torque (couple) is independent of the origin. Its magnitude equals the magnitude of each force multiplied by the arm of the couple (perpendicular distance between the two antiparallel forces).
Figure 1.22Dipole in a uniform electric field.
Magnitude of torque =q E × 2a sinθ
= 2q a E sinθ
Its direction is normal to the plane of the paper, coming out of it.
The magnitude ofp ×E is alsopE sinθ and its direction is normal to the paper, coming out of it. Thus,
τ =p ×E (1.22)
This torque will tend to align the dipole with the fieldE. Whenp is aligned withE, the torque is zero.
What happens if the field is not uniform? In that case, the net force will evidently be non-zero. In addition there will, in general, be a torque on the system as before. The general case is involved, so let us consider the simpler situations whenp is parallel toE or antiparallel toE. In either case, the net torque is zero, but there is a net force on the dipole ifE is not uniform.
Figure 1.23 is self-explanatory. It is easily seen that whenp is parallel toE, the dipole has a net force in the direction of increasing field. Whenp is antiparallel toE, the net force on the dipole is in the direction of decreasing field. In general, the force depends on the orientation ofp with respect toE.
This brings us to a common observation in frictional electricity. A comb run through dry hair attracts pieces of paper. The comb, as we know, acquires charge through friction. But the paper is not charged. What then explains the attractive force? Taking the clue from the preceding discussion, the charged comb ‘polarises’ the piece of paper, i.e., induces a net dipole moment in the direction of field. Further, the electric field due to the comb is not uniform. In this situation, it is easily seen that the paper should move in the direction of the comb!
1.13 Continuous Charge Distribution
We have so far dealt with charge configurations involving discrete chargesq1, q2, ...,qn. One reason why we restricted to discrete charges is that the mathematical treatment is simpler and does not involve calculus. For many purposes, however, it is impractical to work in terms of discrete charges and we need to work with continuous charge distributions. For example, on the surface of a charged conductor, it is impractical to specify the charge distribution in terms of the locations of the microscopic charged constituents. It is more feasible to consider an area element∆S (Fig. 1.24) on the surface of the conductor (which is very small on the macroscopic scale but big enough to include a very large number of electrons) and specify the charge∆Q on that element. We then define asurface chargedensityσ at the area element by
(1.23)
We can do this at different points on the conductor and thus arrive at a continuous functionσ, called the surface charge density. The surface charge densityσ so defined ignores the quantisation of charge and the discontinuity in charge distribution at the microscopic level*. σrepresents macroscopic surface charge density, which in a sense, is a smoothed out average of the microscopic charge density over an area element∆S which, as said before, is large microscopically but small macroscopically. The units forσ are C/m2.
Similar considerations apply for a line charge distribution and a volume charge distribution. Thelinear charge densityλ of a wire is defined by
Figure 1.23Electric force on a dipole: (a)E parallel top,(b) E antiparallel top.
(1.24)
where ∆l is a small line element of wire on the macroscopic scale that, however, includes a large number of microscopic charged constituents, and∆Q is the charge contained in that line element. The units forλ are C/m. Thevolume charge density (sometimes simply called charge density) is defined in a similar manner:
(1.25)
where ∆Q is the charge included in the macroscopically small volume element∆V that includes a large number of microscopic charged constituents. The units forρ are C/m3.
The notion of continuous charge distribution is similar to that we adopt for continuous mass distribution in mechanics. When we refer tothe density of a liquid, we are referring to its macroscopic density. We regard it as a continuous fluid and ignore its discrete molecular constitution.
Figure 1.24 Definition of linear, surface and volume charge densities.
In each case, the element (∆l, ∆S, ∆V) chosen is small on the macroscopic scale but containsa very large number of microscopic constituents.
The field due to a continuous charge distribution can be obtained in much the same way as for a system of discrete charges, Eq. (1.10). Suppose a continuous charge distribution in space has a charge densityρ. Choose any convenient origin O and let the position vector of any point in the charge distribution ber. The charge densityρ may vary from point to point, i.e., it is a function ofr. Divide the charge distribution into small volume elements of size∆V. The charge in a volume element∆V isρ∆V.
Now, consider any general point P (inside or outside the distribution) with position vectorR(Fig. 1.24). Electric field due to the chargeρ∆V is given by Coulomb’s law:
(1.26) where r′ is the distance between the charge element and P, and′ is a unit vector in the direction from the charge element to P. By the superposition principle, the total electric field due to the charge distribution is obtained by summing over electric fields due to different volume elements:
(1.27)
Note thatρ, r′,all can vary from point to point. In a strict mathematical method, we should let∆V→0 and the sum then becomes an integral; but we omit that discussion here, for simplicity. In short, using Coulomb’s law and the superposition principle, electric field can be determined for any charge distribution, discrete or continuous or part discrete and part continuous.
1.14 Gauss’s Law
As a simple application of the notion of electric flux, let us consider the total flux through a sphere of radiusr, which encloses a point chargeq at its centre. Divide the sphere into small area elements, as shown in Fig. 1.25.
Figure 1.25 Flux through a sphere enclosing a point chargeq at its centre.
The flux through an area element∆S is
(1.28)
where we have used Coulomb’s law for the electric field due to a single chargeq. The unit vectoris along the radius vector from the centre to the area element. Now, since the normal to a sphere at every point is along the radius vector at that point, the area element∆S andhave the same direction. Therefore,
(1.29)
since the magnitude of a unit vector is 1.
The total flux through the sphere is obtained by adding up flux through all the different area elements:
Since each area element of the sphere is at the same distancer from the charge,
Now S, the total area of the sphere, equals 4πr2. Thus,
(1.30)
Figure 1.26 Calculation of the flux of uniform electric field through the surface of a cylinder.
Equation (1.30) is a simple illustration of a general result of electrostatics called Gauss’s law.
We stateGauss’s law without proof:
Electric flux through a closed surface S
= q/ε0(1.31)
q = total charge enclosed by S.
The law implies that the total electric flux through a closed surface is zero if no charge is enclosed by the surface. We can see that explicitly in the simple situation of Fig. 1.26.
Here the electric field is uniform and we are considering a closed cylindrical surface, with its axis parallel to the uniform fieldE. The total fluxφ through the surface isφ =φ1 +φ2 +φ3, whereφ1 andφ2 represent the flux through the surfaces 1 and 2 (of circular cross-section) of the cylinder andφ3 is the flux through the curved cylindrical part of the closed surface. Now the normal to the surface 3 at every point is perpendicular toE, so by definition of flux,φ3 = 0. Further, the outward normal to 2 is alongE while the outward normal to 1 is opposite toE. Therefore,
φ1 = –ES1, φ2 = +ES2
S1 =S2 =S
where S is the area of circular cross-section. Thus, the total flux is zero, as expected by Gauss’s law. Thus, whenever you find that the net electric flux through a closed surface is zero, we conclude that the total charge contained in the closed surface is zero.
The great significance of Gauss’s law Eq. (1.31), is that it is true in general, and not only for the simple cases we have considered above. Let us note some important points regarding this law:
(i) Gauss’s law is true for any closed surface, no matter what its shape or size.
(ii) The termq on the right side of Gauss’s law, Eq. (1.31), includes the sum of all charges enclosed by the surface. The charges may be located anywhere inside the surface.
(iii) In the situation when the surface is so chosen that there are some charges inside and some outside, the electric field [whose flux appears on the left side of Eq. (1.31)] is due to all the charges, both inside and outsideS. The termq on the right side of Gauss’s law, however, represents only the total charge insideS.
(iv) The surface that we choose for the application of Gauss’s law is called the Gaussian surface. You may choose any Gaussian surface and apply Gauss’s law. However, take care not to let the Gaussian surface pass through any discrete charge. This is because electric field due to a system of discrete charges is not well defined at the location of any charge. (As you go close to the charge, the field grows without any bound.) However, the Gaussian surface can pass through a continuous charge distribution.
(v) Gauss’s law is often useful towards a much easier calculation of the electrostatic fieldwhen the system has some symmetry. This is facilitated by the choice of a suitable Gaussian surface.
(vi) Finally, Gauss’s law is based on the inverse square dependence on distance contained in the Coulomb’s law. Any violation of Gauss’s law will indicate departure from the inverse square law.
Example 1.11 The electric field components in Fig. 1.27 areEx =αx1/2, Ey =Ez = 0, in whichα = 800 N/C m1/2. Calculate (a) the flux through the cube, and (b) the charge within the cube. Assume thata = 0.1 m.
Figure 1.27
Solution
(a) Since the electric field has only anx component, for faces perpendicular tox direction, the angle betweenE and∆S is ± π/2. Therefore, the fluxφ =E.∆S is separately zero for each face of the cube except the two shaded ones. Now the magnitude of the electric field at the left face is
EL =αx1/2 =αa1/2
(x =a at the left face).
The magnitude of electric field at the right face is
(b) We can use Gauss’s law to find the total chargeq inside the cube. We haveφ = q/ε0 orq =φε0. Therefore,
q = 1.05 × 8.854 × 10–12 C = 9.27 × 10–12 C.
Example 1.12 An electric field is uniform, and in the positivex direction for positivex, and uniform with the same magnitude but in the negativex direction for negativex. It is given thatE= 200N/C forx > 0 andE = –200N/C forx < 0. A right circular cylinder of length 20 cm and radius 5 cm has its centre at the origin and its axis along thex-axis so that one face is atx = +10 cm and the other is atx = –10 cm (Fig. 1.28). (a) What is the net outward flux through each flat face? (b) What is the flux through the side of the cylinder? (c) What is the net outward flux through the cylinder? (d) What is the net charge inside the cylinder?
Solution
(a) We can see from the figure that on the left faceE and∆S are parallel. Therefore, the outward flux is
φL= E.∆S = – 200
= + 200∆S, since= –∆S
= + 200 ×π (0.05)2 = + 1.57 N m2 C–1
On the right face,E and∆S are parallel and therefore
φR = E.∆S = + 1.57 N m2 C–1.
(b) For any point on the side of the cylinderE is perpendicular to∆S and henceE.∆S = 0. Therefore, the flux out of the side of the cylinder is zero.
(c) Net outward flux through the cylinder
φ= 1.57 + 1.57 + 0 = 3.14 N m2 C–1
Figure 1.28
(d) The net charge within the cylinder can be found by using Gauss’s law which gives
q= ε0φ
= 3.14 × 8.854 × 10–12 C
= 2.78 × 10–11 C
1.15 Applications of Gauss’s Law
The electric field due to a general charge distribution is, as seen above, given by Eq. (1.27). In practice, except for some special cases, the summation (or integration) involved in this equation cannot be carried out to give electric field at every point in space. For some symmetric charge configurations, however, it is possible to obtain the electric field in a simple way using the Gauss’s law. This is best understood by some examples.
1.15.1 Field due to an infinitely long straight uniformly charged wire
Consider an infinitely long thin straight wire with uniform linear charge densityλ. The wire is obviously an axis of symmetry. Suppose we take the radial vector from O to P and rotate it around the wire. The points P, P′, P′′ so obtained are completely equivalent with respect to the charged wire. This implies that the electric field must have the same magnitude at these points. The direction of electric field at every point must be radial (outward ifλ > 0, inward ifλ < 0). This is clear from Fig. 1.29.
Figure 1.29 (a) Electric field due to an infinitely long thin straight wire is radial, (b) The Gaussian surface for a long thin wire of uniform linear charge density.
Consider a pair of line elements P1 and P2 of the wire, as shown. The electric fields produced by the two elements of the pair when summed give a resultant electric field which is radial (the components normal to the radial vector cancel). This is true for any such pair and hence the total field at any point P is radial. Finally, since the wire is infinite, electric field does not depend on the position of P along the length of the wire. In short, the electric field is everywhere radial in the plane cutting the wire normally, and its magnitude depends only on the radial distancer.
To calculate the field, imagine a cylindrical Gaussian surface, as shown in the Fig. 1.29(b).Since the field is everywhere radial, flux through the two ends of the cylindrical Gaussian surface is zero. At the cylindrical part of the surface,E is normal to the surface at every point, and its magnitude is constant, since it depends only onr. The surface area of the curved part is 2πrl, wherel is the length of the cylinder.
Flux through the Gaussian surface
= flux through the curved cylindrical part of the surface
= E × 2πrl
The surface includes charge equal toλ l. Gauss’s law then gives
E × 2πrl =λl/ε0
i.e., E =
Vectorially,E at any point is given by
(1.32)
whereis the radial unit vector in the plane normal to the wire passing through the point.E is directed outward ifλ is positive and inward ifλ is negative.
Note that when we write a vectorA as a scalar multiplied by a unit vector, i.e., asA =A, the scalarA is an algebraic number. It can be negative or positive. The direction ofA will be the same as that of the unit vectorif A > 0 and opposite toif A < 0. When we want to restrict to non-negative values, we use the symboland call it the modulus ofA. Thus,.
Also note that though only the charge enclosed by the surface (λl) was included above, the electric fieldE is due to the charge on the entire wire. Further, the assumption that the wire is infinitely long is crucial. Without this assumption, we cannot takeE to be normal to the curved part of the cylindrical Gaussian surface. However, Eq. (1.32) is approximately true for electric field around the central portions of a long wire, where the end effects may be ignored.
1.15.2 Field due to a uniformly charged infinite plane sheet
Letσ be the uniform surface charge density of an infinite plane sheet (Fig. 1.30). We take thex-axis normal to the given plane. By symmetry, the electric field will not depend ony andzcoordinates and its direction at every point must be parallel to thex-direction.
We can take the Gaussian surface to be a rectangular parallelepiped of cross-sectional areaA, as shown. (A cylindrical surface will also do.) As seen from the figure, only the two faces 1 and 2 will contribute to the flux; electric field lines are parallel to the other faces and they, therefore, do not contribute to the total flux.
Figure 1.30 Gaussian surface for a uniformly charged infinite plane sheet.
The unit vector normal to surface 1 is in –x direction while the unit vector normal to surface 2 is in the +x direction. Therefore, fluxE.∆S through both the surfaces are equal and add up. Therefore the net flux through the Gaussian surface is 2EA. The charge enclosed by the closed surface isσA. Therefore by Gauss’s law,
2 EA =σA/ε0
or, E =σ/2ε0
Vectorically,
(1.33)
where n is a unit vector normal to the plane and going away from it.
E is directed away from the plate ifσ is positive and toward the plate ifσ is negative. Note that the above application of the Gauss’ law has brought out an additional fact:E is independent ofx also.
For a finite large planar sheet, Eq. (1.33) is approximately true in the middle regions of the planar sheet, away from the ends.
1.15.3 Field due to a uniformly charged thin spherical shell
Letσ be the uniform surface charge density of a thin spherical shell of radiusR (Fig. 1.31). The situation has obvious spherical symmetry. The field at any point P, outside or inside, can depend only onr (the radial distance from the centre of the shell to the point) and must be radial (i.e., along the radius vector).
(i) Field outside the shell: Consider a point P outside the shell with radius vectorr. To calculateE at P, we take the Gaussian surface to be a sphere of radiusr and with centre O, passing through P. All points on this sphere are equivalent relative to the given charged configuration. (That is what we mean by spherical symmetry.) The electric field at each point of the Gaussian surface, therefore, has the same magnitudeE and is along the radius vector at each point. Thus,E and∆Sat every point are parallel and the flux through each element isE∆S. Summing over all∆S, the flux through the Gaussian surface isE × 4π r2. The charge enclosed isσ × 4π R2. By Gauss’s lawE × 4π r2=
Or,
where q = 4π R2σ is the total charge on the spherical shell.
Vectorially,
(1.34)
Figure 1.31 Gaussian surfaces for a point with (a) r> R, (b)r < R.
The electric field is directed outward ifq > 0 and inward ifq < 0.This, however, is exactly the field produced by a chargeq placed at the centre O. Thus for points outside the shell, the field due to a uniformly charged shell is as if the entire charge of the shell is concentrated at its centre.
E × 4π r2 = 0
i.e., E = 0 (r <R ) (1.35)
that is, the field due to a uniformly charged thin shell is zero at all points inside the shell*. This important result is a direct consequence of Gauss’s law which follows from Coulomb’s law. The experimental verification of this result confirms the 1/r2 dependence in Coulomb’s law.
Example 1.13 An early model for an atom considered it to have a positively charged point nucleus of chargeZe, surrounded by a uniform density of negative charge up to a radiusR. The atom as a whole is neutral. For this model, what is the electric field at a distancer from the nucleus?
Figure 1.32
Solution The charge distribution for this model of the atom is as shown in Fig. 1.32. The total negative charge in the uniform spherical charge distribution of radiusR must be –Z e, since the atom (nucleus of chargeZ e+ negative charge) is neutral. This immediately gives us the negative charge densityρ, since we must have
or
To find the electric fieldE(r) at a point P which is a distancer away from the nucleus, we use Gauss’s law. Because of the spherical symmetry of the charge distribution, the magnitude of the electric fieldE(r) depends only on the radial distance, no matter what the direction ofr. Its direction is along (or opposite to) the radius vectorr from the origin to the point P. The obvious Gaussian surface is a spherical surface centred at the nucleus. We consider two situations, namely,r <R andr >R.
(i) r <R : The electric fluxφenclosed by the spherical surface is
φ =E (r) × 4πr2
where E (r) is the magnitude of the electric field atr. This is because the field at any point on the spherical Gaussian surface has the same direction as the normal to the surface there, and has the same magnitude at all points on the surface.
The chargeq enclosed by the Gaussian surface is the positive nuclear charge and the negative charge within the sphere of radiusr,
i.e.,
Substituting for the charge densityρobtained earlier, we have
Gauss’s law then gives,
The electric field is directed radially outward.
(ii) r >R: In this case, the total charge enclosed by the Gaussian spherical surface is zero since the atom is neutral. Thus, from Gauss’s law,
E (r) × 4πr2 = 0 orE (r) = 0;r >R
At r =R, both cases give the same result:E = 0.
On symmetry operations
In Physics, we often encounter systems with various symmetries. Consideration of these symmetries helps one arrive at results much faster than otherwise by a straightforward calculation. Consider, for example an infinite uniform sheet of charge (surface charge densityσ) along they-z plane. This system is unchanged if (a) translated parallel to they-z plane in any direction, (b) rotated about thex-axis through any angle. As the system is unchanged under such symmetry operation, so must its properties be. In particular, in this example, the electric fieldE must be unchanged.
Translation symmetry along they-axis shows that the electric field must be the same at a point (0,y1, 0) as at (0,y2, 0). Similarly translational symmetry along thez-axis shows that the electric field at two point (0, 0,z1) and (0, 0,z2) must be the same. By using rotation symmetry around thex-axis, we can conclude thatE must be perpendicular to they-z plane, that is, it must be parallel to thex-direction.
Try to think of a symmetry now which will tell you that the magnitude of the electric field is a constant, independent of thex-coordinate. It thus turns out that the magnitude of the electric field due to a uniformly charged infinite conducting sheet is the same at all points in space. The direction, however, is opposite of each other on either side of the sheet.
Compare this with the effort needed to arrive at this result by a direct calculation using Coulomb’s law.
Summary
1. Electric and magnetic forces determine the properties of atoms, molecules and bulk matter.
2. From simple experiments on frictional electricity, one can infer that there are two types of charges in nature; and that like charges repel and unlike charges attract. By convention, the charge on a glass rod rubbed with silk is positive; that on a plastic rod rubbed with fur is then negative.
3. Conductors allow movement of electric charge through them, insulators do not. In metals, the mobile charges are electrons; in electrolytes both positive and negative ions are mobile.
4. Electric charge has three basic properties: quantisation, additivity and conservation.
Quantisation of electric charge means that total charge (q) of a body is always an integral multiple of a basic quantum of charge (e) i.e., q =n e, wheren = 0, ±1, ±2, ±3, .... Proton and electron have charges +e, –e, respectively. For macroscopic charges for whichn is a very large number, quantisation of charge can be ignored.
Additivity of electric charges means that the total charge of a system is the algebraic sum (i.e., the sum taking into account proper signs) of all individual charges in the system.
Conservation of electric charges means that the total charge of an isolated system remains unchanged with time. This means that when bodies are charged through friction, there is a transfer of electric charge from one body to another, but no creation or destruction of charge.
5. Coulomb’s Law: The mutual electrostatic force between two point chargesq1 andq2 is proportional to the productq1q2 and inversely proportional to the square of the distancer21separating them. Mathematically,
F21 = force onq2 due to
whereis a unit vector in the direction fromq1 toq2 andk =is the constant of proportionality.
In SI units, the unit of charge is coulomb. The experimental value of the constantε0 is
ε0 = 8.854 × 10–12 C2 N–1 m–2
The approximate value ofk is
k = 9 × 109 N m2 C–2
6. The ratio of electric force and gravitational force between a proton and an electron is
7. Superposition Principle:The principle is based on the property that the forces with which two charges attract or repel each other are not affected by the presence of a third (or more) additional charge(s). For an assembly of chargesq1, q2, q3, ..., the force on any charge, sayq1, is the vector sum of the force onq1 due toq2, the force on q1 due toq3, and so on. For each pair, the force is given by the Coulomb’s law for two charges stated earlier.
8. The electric fieldE at a point due to a charge configuration is the force on a small positive test chargeq placed at the point divided by the magnitude of the charge. Electric field due to a point chargeq has a magnitude |q|/4πε0r2; it is radially outwards fromq, ifq is positive, and radially inwards ifq is negative. Like Coulomb force, electric field also satisfies superposition principle.
9. An electric field line is a curve drawn in such a way that the tangent at each point on the curve gives the direction of electric field at that point. The relative closeness of field lines indicates the relative strength of electric field at different points; they crowd near each other in regions of strong electric field and are far apart where the electric field is weak. In regions of constant electric field, the field lines are uniformly spaced parallel straight lines.
10. Some of the important properties of field lines are: (i) Field lines are continuous curves without any breaks. (ii) Two field lines cannot cross each other. (iii) Electrostatic field lines start at positive charges and end at negative charges —they cannot form closed loops.
11. An electric dipole is a pair of equal and opposite chargesq and –q separated by some distance 2a. Its dipole moment vectorp has magnitude 2qa and is in the direction of the dipole axis from –q toq.
12. Field of an electric dipole in its equatorial plane (i.e., the plane perpendicular to its axis and passing through its centre) at a distancer from the centre:
Dipole electric field on the axis at a distancer from the centre:
The 1/r3 dependence of dipole electric fields should be noted in contrast to the 1/r2dependence of electric field due to a point charge.
13. In a uniform electric fieldE, a dipole experiences a torquegiven by
= p × E
but experiences no net force.
14. The flux∆φ of electric fieldE through a small area element∆S is given by
∆φ = E.∆S
The vector area element∆S is
∆S =∆S
where ∆S is the magnitude of the area element andis normal tothe area element, which can be considered planar for sufficiently small∆S. For an area element of a closed surface,is taken to be the direction ofoutward normal, by convention.
15. Gauss’s law: The flux of electric field through any closed surfaceS is 1/ε0 times the total charge enclosed byS. The law is especially useful in determining electric fieldE, when the source distribution has simple symmetry:
(i) Thin infinitely long straight wire of uniform linear charge densityλ
where r is the perpendicular distance of the point from the wire andis the radial unit vector in the plane normal to the wire passing through the point.
(ii) Infinite thin plane sheet of uniform surface charge densityσ
whereis a unit vector normal to the plane, outward on either side.
(iii) Thin spherical shell of uniform surface charge densityσ
E = 0 (r < R)
where r is the distance of the point from the centre of the shell andR the radius of the shell.q is the total charge of the shell:q = 4πR2σ.
The electric field outside the shell is as though the total charge is concentrated at the centre. The same result is true for a solid sphere of uniform volume charge density. The field is zero at all points inside the shell.
Physical quantity - Symbol - Dimensions - Unit - Remarks
Points to Ponder
1. You might wonder why the protons, all carrying positive charges, are compactly residing inside the nucleus. Why do they not fly away? You will learn that there is a third kind of a fundamental force, called the strong force which holds them together. The range of distance where this force is effective is, however, very small ~10-14 m. This is precisely the size of the nucleus. Also the electrons are not allowed to sit on top of the protons, i.e. inside the nucleus, due to the laws of quantum mechanics. This gives the atoms their structure as they exist in nature.
2. Coulomb force and gravitational force follow the same inverse-square law. But gravitational force has only one sign (always attractive), while Coulomb force can be of both signs (attractive and repulsive), allowing possibility of cancellation of electric forces. This is how gravity, despite being a much weaker force, can be a dominating and more pervasive force in nature.
3. The constant of proportionalityk in Coulomb’s law is a matter of choice if the unit of charge is to be defined using Coulomb’s law. In SI units, however, what is defined is the unit of current (A) via its magnetic effect (Ampere’s law) and the unit of charge (coulomb) is simply defined by (1C = 1 A s). In this case, the value ofk is no longer arbitrary; it is approximately 9 × 109 N m2C–2.
4. The rather large value ofk, i.e., the large size of the unit of charge (1C) from the point of view of electric effects arises because (as mentioned in point 3 already) the unit of charge is defined in terms of magnetic forces (forces on current–carrying wires) which are generally much weaker than the electric forces. Thus while 1 ampere is a unit of reasonable size for magnetic effects, 1 C = 1 A s, is too big a unit for electric effects.
5. The additive property of charge is not an ‘obvious’ property. It is related to the fact that electric charge has no direction associated with it; charge is a scalar.
6. Charge is not only a scalar (or invariant) under rotation; it is also invariant for frames of reference in relative motion. This is not always true for every scalar. For example, kinetic energy is a scalar under rotation, but is not invariant for frames of reference in relative motion.
7. Conservation of total charge of an isolated system is a property independent of the scalar nature of charge noted in point 6. Conservation refers to invariance in time in a given frame of reference. A quantity may be scalar but not conserved (like kinetic energy in an inelastic collision). On the other hand, one can have conserved vector quantity (e.g., angular momentum of an isolated system).
8. Quantisation of electric charge is a basic (unexplained) law of nature; interestingly, there is no analogous law on quantisation of mass.
9. Superposition principle should not be regarded as ‘obvious’, or equated with the law of addition of vectors. It says two things: force on one charge due to another charge is unaffected by the presence of other charges, and there are no additional three-body, four-body, etc., forces which arise only when there are more than two charges.
10. The electric field due to a discrete charge configuration is not defined at the locations of the discrete charges. For continuous volume charge distribution, it is defined at any point in the distribution. For a surfacecharge distribution, electric field is discontinuous across the surface.
11. The electric field due to a charge configuration with total charge zero is not zero; but for distances large compared to the size of the configuration, its field falls off faster than 1/r2, typical of field due to a single charge. An electric dipole is the simplest example of this fact.
Exercise
Q 1.1) What is the force between two small charged spheres having charges of 2 × 10–7C and 3 × 10–7 C placed 30 cm apart in the air?
Q 1.2) The electrostatic force on a small sphere of charge 0.4 µC due to another small sphere of charge –0.8 µC in the air is 0.2 N.(a) What is the distance between the two spheres?(b) What is the force on the second sphere due to the first?
Q 1.3) Check that the ratio ke2/G memp is dimensionless. Lookup a Table of Physical Constants and determine the value of this ratio. What does the ratio signify?
Q 1.4) (i) Explain the meaning of the statement ‘electric charge of a body is quantised’ (ii) Why can one ignore quantisation of electric charge when dealing with macroscopic i.e., large scale charges?
Q 1.5) When a glass rod is rubbed with a silk cloth, charges appear on both. A similar phenomenon is observed with many other pairs of bodies. Explain how this observation is consistent with the law of conservation of charge.
Q 1.6) Four point charges are located at corners of a square ABCD of side 10cm. What is the force on a charge of 1 µC placed at the center of the square?
Q 1.8) Two point charges qA = 3 µC and qB= –3 µC are located 20 cm apart in a vacuum.(i) What is the electric field at the midpoint O of the line AB joining the two charges?(ii) If a negative test charge of magnitude 1.5 × 10–9 C is placed at this point, what is the force experienced by the test charge?
Q 1.9) A system has two charges qa = 2.5 x C and qB -2.5 xC located at points A: (0,0,- 15 cm) and B: (0, 0, + 15 cm), respectively. What are the total charge and electric dipole moment lof the svstem?
Q 1.10) An electric dipole with dipole momentis aligned at 30° with the direction of a uniform electric field of magnitude. Calculate the magnitude of the torque acting on the dipole.
Q 1.11) A polythene piece rubbed with wool is found to have a negative charge of.(a) Estimate the number of electrons transferred (from which to which?)(b) Is there a transfer of mass from wool to polythene?
Q 1.12)(i) Two insulated charged copper spheres A and B have their centers separated by a distance of 50 cm. What is the mutual force of electrostatic repulsion if the charge on each is?The radii of A and B are negligible compared to the distance of separation.
(ii) What is the force of repulsion if each sphere is charged double the above amount, and the distance between them is halved?
Q 1.13) Suppose the spheres A and B in Exercise 1.12 have identical sizes. A third sphere of the same size but uncharged is brought in contact with the first, then brought in contact with the second, and finally removed from both. What is the new force of repulsion between A and B?
Q 1.14) The figure below shows tracks of three charged particles in a uniform electrostatic field. Give the signs of the three charges. Which particle has the highest charge to the mass ratio?
What is the net flux of the uniform electric field of Exercise 1.15 through a cube of side 20 cm oriented so that its faces are parallel to the coordinate planes?
Careful measurement of the electric field at the surface of a black box indicates that the net outward flux through the surface of the box is 8.0 × 103 Nm2/C.
(a) What is the net charge inside the box?
(b) If the net outward flux through the surface of the box were zero, could you conclude that there were no charges inside the box? Why or Why not?
A point charge +10 μC is at a distance 5 cm directly above the centre of a square of side 10 cm, as shown in Fig. What is the magnitude of the electric flux through the square?
( Hint: Think of the square as one face of a cube with edge 10 cm)
A conducting sphere of radius 10 cm has an unknown charge. If the electric field 20 cm from the center of the sphere is 1.5 × 103 N/C and points radially inward, what is the net charge on the sphere?
Two large, thin metal plates are parallel and close to each other. On their inner faces, the plates have surface charge densities of opposite signs and of magnitude 17.0 × 10–22 C /m2
1.25 An oil drop of 12 excess electrons is held stationary under a constant electric field of (Millikan’s oil drop experiment). The density of the oil is. Estimate the radius of the drop. (g = 9.81; e =).
1.27 In a certain region of space, the electric field is along the z-direction throughout. The magnitude of the electric field is, however, not constant but increases uniformly along the positive z-direction, at the rate of per meter. What are the force and torques experienced by a system having a total dipole moment equal to Cm in the negativez-direction?
(a) A conductor A with a cavity as shown in Fig. 1.36 (a) is given a chargeQ. Show that the entire charge must appear on the outer surface of the conductor.
(b) Another conductor B with chargeq is inserted into the cavity keeping B insulated from A. Show that the total charge on the outside surface of A isQ +q [Fig. 1.36(b)].
(c) A sensitive instrument is to be shielded from the strong electrostatic fields in its environment. Suggest a possible way.
1.29 A hollow charged conductor has a tiny hole cut into its surface. Show that the electric field in the hole is (, whereis the unit vector in the outward normal direction, andσ is the surface charge density near the hole.
1.30 Obtain the formula for the electric field due to a long thin wire of uniform linear charge densityE without using Gauss’s law. [Hint: Use Coulomb’s law directly and evaluate the necessary integral.]
1.31 It is now believed that protons and neutrons (which constitute nuclei of ordinary matter) are themselves built out of more elementary units called quarks. A proton and a neutron consist of three quarks each. Two types of quarks, the so-called ‘up’ quark (denoted by u) of charge + (2/3)e, and the ‘down’ quark (denoted by d) of charge (–1/3)e, together with electrons build up ordinary matter. (Quarks of other types have also been found which give rise to different unusual varieties of matter.) Suggest a possible quark composition of a proton and neutron.
1.32 (a) Consider an arbitrary electrostatic field configuration. A small test charge is placed at a null point (i.e., whereE = 0) of the configuration. Show that the equilibrium of the test charge is necessarily unstable.
(b) Verify this result for the simple configuration of two charges of the same magnitude and sign placed a certain distance apart.
1.33 A particle of massm and charge (–q) enters the region between the two charged plates initially moving along thex-axis with speedvx(like particle 1 in Fig. 1.33). The length of the plate isLand a uniform electric fieldE is maintained between the plates. Show that the vertical deflection of the particle at the far edge of the plate is.
Compare this motion with the motion of a projectile in the gravitational field discussed in Section 4.10 of the Class XI Textbook of Physics.
1.34 Suppose that the particle in Exercise in 1.33 is an electron projected with velocityvx = 2.0 × 106 m s–1. IfE between the plates separated by 0.5 cm is 9.1 × 102 N/C, where will the electron strike the upper plate? (|e|=1.6 × 10–19 C,me = 9.1 × 10–31 kg.)
1. In figure two positive charges q2 and q3 fixed along the y-axis, exert a net electric force in the +x-direction on a charge q1 fixed along the x-axis. If a positive charge Q is added at (x,0), the force on q1
(1) shall increase along the positive x-axis
(2) shall decrease along the positive x-axis
(3) shall point along the negative x-axis
(4) shall increase but the direction changes because of the intersection of Q with q2 and q3
4. Five charges q1, q2, q3, q4, and q5 are fixed at their positions as shown in figure, S is a Gaussian surface. The Gauss' law is given by. Which of the following statements is correct?
(1) E on the LHS of the above equation will have contribution from q1, q5 and q3 while q on the RHS will have a contribution from q2 and q4 only
(2) E on the LHS of the above equation will have a contribution from all charges while q on the RHS will have a contribution from q2 and q4 only
(3) E on the LHS of the above equation will have a contribution from all charges while q on the RHS will have a contribution from q1, q3 and q5 only
(4) Both E on the LHS and q on the RHS will have contributions from q2 and q4 only
13. A positive charge Q is uniformly distributed along a circular ring of radius R. A small test charge q is placed at the centre of the ring figure. Then,
(a) if q > 0 and is displaced away from the centre in the plane of the ring, it will be pushed back towards the centre
(b) if q < 0 and is displaced away from the centre in the plane of the ring, it will never return to the centre and will continue moving till it hits the ring
(c) if q < 0, it will perform SHM for small displacement along the axis
(d) q at the centre of the ring is in an unstable equitibrium within the plane of the ring for q > 0
15. A metallic spherical shell has an inner radiusR1 and outer radiusR2 A charge Q is placed at the centre of the spherical cavity. What will be surface charge density on
16. The dimensions of an atom are of the order of an Angstrom. Thus, there must be large electric fields between the protons and electrons. Why, then is the electrostatic field inside a conductor zero?
17. If the total charge enclosed by a surface is zero, does it imply that the electric field everywhere on the surface is zero? Conversely, if the electric field everywhere on a surface is zero, does it imply that the net charge inside it is zero.
20. A paisa coin is made up of Al-Mg alloy and weight 0.75 g. It has a square shape and its diagonal measures 17 mm. It is electrically neutral and contains equal amounts of positive and negative charges.
21. Consider a coin of Question 20. It is electrically neutral and contains equal amounts of positive and negative charge of magnitude 34.8 kC. Suppose that these equal charges were concentrated in two point charges separated by
(i) 1 cm (x diagonal of the one paisa coin)
(ii) 100 m ( length of a long building)
(iii) 106 m (radius of the earth). Find the force on each such point charge in each of the three cases. What do you conclude from these results?
22. Figure represents a crystal unit of cesium chloride, CsCl. The cesium atoms, represented by open circles are situated at the corners of a cube of side 0.40 nm, whereas a Cl atom is situated at the centre of the cube. The Cs atoms are deficient in one electron while the Cl atom carries an excess electron.
(i) What is the net electric field on the Cl atom due to eight Cs atoms?
(ii) Suppose that the Cs atom at the corner A is missing. What is the net force now on the Cl atom due to seven remaining Cs atoms?
23. Two charges q and -3q are placed fixed on x-axis separated by distance d. Where should a third charge 2q be placed such that it will not experience any force?
25. Five charges, q each are placed at the comers of a regular pentagon of side.
(a) (i) What will be the electric field at 0, the centre of the pentagon?
(ii) What will be the electric field at 0 if the charge from one of the corners (say A) is removed? (iii) What will be the electric field at 0 if the charge q at A is replaced by -q ?
(b) How would your answer to (a) be affected if pentagon is replaced by n-sided regular polygon with charge q at each of its corners?
26. In 1959 Lyttleton and Bondi suggested that the expansion of the universe could be explained if matter carried a net charge. Suppose that the universe is made up of hydrogen atoms with a number density N, which is maintained a constant. Let the charge on the proton be where e is the electronic charge.
(a) Find the critical value of y such that expansion may start.
(b) Show that the velocity of expansion is proportional to the distance from the centre
27. Consider a sphere of radius R with charge density distributed as p(r)=kr forfor r>R.
(a) Find the electric field as all points r.
(b) Suppose the total charge on the sphere is 2e where e is the electron charge. Where can two protons be embedded such that the force on each of them is zero. Assume that the introduction of the proton does not alter the negative charge distribution.
28. Two fixed, identical conducting plates (α and β), each of surface area S are charge to -Q and q, respectively, where, Q>q>0. A third identical plate (γ), free to move is located on the other side of the plate with charge q at a distance d (figure). The third plate is released and collides with the plate β. Assume the collision is elastic and the time of collision is sufficient to redistribute charge amongs β and γ.
(a) Find the electric field acting on the plate γ before collision.
(b) Find the charges on β and γ after the collision.
(c) Find the velocity of the plate γ after the collision and at a distance d from the plate β.
29. There is another useful system of units, besides the SI/MKS. A system, called the CGS (Centimeter-Gram-Second) system. In this system, Coulomb's law is given by
where the distance r is measured in cm (=10-2 μ), F is dynes (=10-5 N) and the charges in electrostatic units (es units), where 1 es units of chargeThe number [3] actually arises from the speed of light in vaccum which is now taken to be exactly given byAn approximate value of c, then is(i) Show that the Coulomb's law in CGS units yields 1 esu of chargeObtain the dimensions of units of charge in terms of mass M, length L and time T. Show that it is given in terms of fractional powers of M and L.
(ii) Write 1 esu of charge=xC, where x is a dimensionless number. Show that this givesWith we have
31. Total charge -Q is uniformly spread along length of a ring of radius R. A small test charge +q of mass m is kept at the centre of the ring and is given a gentle push along the axis of the ring.
(a) Show that the particle executes a simple harmonic oscillation.
