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.
Hint: Use Gauss law to find the electric field at the surface.
(a) Step 1: Find the electric field at the surface.
Let us suppose that universe is a perfect sphere of radius R and its constituent hydrogen atoms are distributed uniformly in the sphere.
As a hydrogen atom contains one proton and one electron, charge on each hydrogen atom.
If E is electric field intensity at distance R, on the surface of the sphere, then according to Gauss' theorem,
i.e.,
Step 2: Find the gravitational force on an atom at the surface.
Now, suppose, the mass of hydrogen atomMass of a proton, GR=gravitational field at distance R on the sphere.
Then,
The gravitational force on this atom is
Coulomb force on hydrogen atom at R is
Step 3: Find the value of y.
Now, to start expansion and critical value of Y to start expansion would be when
Thus, 10-18 is the required critical value of Y corresponding to which expansion of the universe would start.
(b) Step 4: Find the net force on the atom at the surface.
Net force experience by the hydrogen atom is given by:
Step 5: Find the net acceleration of the atom.
If the acceleration of hydrogen atom is represented by , then
where,
Step 6: Find the velocity of expansion.
The general solution of Eq. (iv) is given by . We are looking for expansion, here, so and .
The velocity of expansion,
Hence, i.e., the velocity of expansion is proportional to the distance from the centre.
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