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.

(ρAl=2.63×10-8 Ω m, ρCu=1.72×10-8 Ω m, Relative density of Al=2,7 of Cu=8.9)


Resistivity of aluminium, ρAl=2.63×10-8 Ω m
Relative density of aluminium, d1=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, ρCu=1.72×10-8 Ω m
Relative density of copper, d2=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
R1=ρ1l1A1 .........(1)
R2=ρ2l2A2 ...........(2)
It is given that,
R1=R2
ρ1l1A1=ρ2l2A2
ρ1A1=ρ2A2 [l1=l2]
A1A2=ρ1ρ2=2.63×10-81.72×10-8=2.631.72
Mass of the aluminium wire, m1=Volume×Density
=A1l1×d1=A1l1d1 ..........(3)
Mass of the copper wire, m2=Volume×Density
=A2l2×d2=A2l2d2 .......(4)
Dividing equation (3) by equation (4), we obtain
m1m2=A1l1d1A2l2d2
For l1=l2,
m1m2=A1d1A2d2
For A1A2=2.631.72,
m1m2=2.631.72×2.78.9=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.