Ncert Exercises & Solutions

3.16 Two wires of equal length, one of aluminium and the other of copper have the same resistance. Which of the two wires is lighter? Hence explain why aluminium wires are preferred for overhead power cables.

$(ρ_{Al} = 2.63 \times 10^{- 8} Ω m, ρ_{Cu} = 1.72 \times 10^{- 8} Ω m, Relative density of Al = 2, 7 of Cu = 8.9)$

Resistivity of aluminium,

ρ_{Al} = 2.63 \times 10^{- 8} Ω m

Relative density of aluminium,

d_{1} = 2.7

Let l₁ be the length of aluminium wire and m₁ be its mass.

Resistance of the aluminium wire=R₁

Area of cross-section of the aluminium wire=A₁

Resistivity of copper,

ρ_{Cu} = 1.72 \times 10^{- 8} Ω m

Relative density of copper,

d_{2} = 8.9

Let l₂ be the length of copper wire and m₂ be its mass.

Resistance of the copper wire=R₂

Area of cross-section of the copper wire=A₂

The two relations can be written as

R_{1} = ρ_{1} \frac{l_{1}}{A_{1}} . . . . . . . . . (1)

R_{2} = ρ_{2} \frac{l_{2}}{A_{2}} . . . . . . . . . . . (2)

It is given that,

R_{1} = R_{2}

\Rightarrow ρ_{1} \frac{l_{1}}{A_{1}} = ρ_{2} \frac{l_{2}}{A_{2}}

\Rightarrow \frac{ρ_{1}}{A_{1}} = \frac{ρ_{2}}{A_{2}} [∵ l_{1} = l_{2}]

\Rightarrow \frac{A_{1}}{A_{2}} = \frac{ρ_{1}}{ρ_{2}} = \frac{2.63 \times 10^{- 8}}{1.72 \times 10^{- 8}} = \frac{2.63}{1.72}

Mass of the aluminium wire,

m_{1} = Volume \times Density

= A_{1} l_{1} \times d_{1} = A_{1} l_{1} d_{1} . . . . . . . . . . (3)

Mass of the copper wire,

m_{2} = Volume \times Density

= A_{2} l_{2} \times d_{2} = A_{2} l_{2} d_{2} . . . . . . . (4)

Dividing equation (3) by equation (4), we obtain

\frac{m_{1}}{m_{2}} = \frac{A_{1} l_{1} d_{1}}{A_{2} l_{2} d_{2}}

For l_{1} = l_{2},

\frac{m_{1}}{m_{2}} = \frac{A_{1} d_{1}}{A_{2} d_{2}}

For \frac{A_{1}}{A_{2}} = \frac{2.63}{1.72},

\frac{m_{1}}{m_{2}} = \frac{2.63}{1.72} \times \frac{2.7}{8.9} = 0.46

It can be inferred from this ratio that m₁ is less than m₂. Hence, aluminium is lighter than copper. Since aluminium is lighter, it is preferred for overhead power cables over copper.

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