Two wires of equal length, one of aluminium and the other of copper have the same resistance. Which of the two wire is lighter? Hence explain why aluminium wires are preferred for overhead power cables. `(rho_(Al) = 2.63 xx 10^(-8) Omega m, rho_(Cu) = 1.72 xx 10^(-8) Omega m` Relative density of Al = 2.7 of Cu = 8.9)

Explanation

Given for aluminium wire, `R_(1) = R l_(1) = l`
Relative density `d_(1) = 2.7`
Let `A_(1), A_(2)` be the area of cross section for aluminium wire and copper wire.
We know, `R_(1) = rho_(1) (l_(1))/(A_(1)) = (2.63 xx 10^(-8) xx l)/(A_(1)) `
and mass of the aluminium wire `m_(1) = A_(1) l_(1) xx d_(1) = A_(1) l xx 2.7`
`R_(2) = rho_(2) = (l_(2))/(A_(2)) = (1.72 xx 10^(-8) xx l)/(A_(2))`
Mass of copper wire `m_(2) = A_(2) l_(2) xx d_(2) = A_(2) l xx 8.9`
Since two wires are of equal resistance `R_(1) = R_(2)`
`(2.63 xx 10^(-8) xx 1)/(A_(1)) = (1.72 xx 10^(-8) xx l)/(A_(2)) "or" (A_(2))/(A_(1)) = (1.72)/(2.63)`
From (ii) and (iv) we have
`(m_(2))/(m_(1)) (A_(2) l xx 8.9)/(A_(1) l xx 2.7) = (8.9)/(2.7) xx (A_(2))/(A_(1))`
`= (8.9)/(2.7) xx (1.72)/(2.63) = 2.16`
It shows that copper wire is 2.16 times heavier than aluminium wire since for the same value of length and resistance aluminium wire has lesser mass then copper wire, therefore aluminium wire is prederred for overband power cables. A heavy cable may sag down owing to its own weight.