Two wires of equal length, one of aluminum and the other of copper have the same resistance. Which of the two wires is lighter? Hence explain why aluminum wires are preferred for overhead power cables. (ρAl = 2.63 × 10⁻⁸ Ω m, ρCu = 1.72 × 10⁻⁸Ω m, Relative density of Al = 2.7, of Cu = 8.9.)

Resistivity of aluminium, ρAl = 2.63 x 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 x 10^-8 Ωm

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 =ρ₁ l₁/A₁ ——————–Eq-1

R₂ =ρ₂ l₂A₂ ——————–Eq-2

It is given that ,R₁=R_{2 =>} ρ₁ l₁/A₁= ρ₂ l₂A₂

=> ρ₁ /A₁ = ρ₂ /A₂   ( as   ,l₁=l₂)

=> A₁ /A₂  =ρ₁/ρ₂ = (2.63 x 10⁻⁸ )/(1.72 x 10⁻⁸ )=2.63/1.72

Mass of the copper wire ,m₁= Volume x Density

=A₁L₁ x d₁ =A₁l₁d_₁———————-Eq -3

Mass of the copper wire ,m2= Volume x Density

=A2L2 x d2 =A2l2d2———————-Eq -4,from which we obtain

m₁/m2 = (A1l1d1)/(A2l2d2)

For l1=l2

m1/m2 = A1d1/A2d2

For A1/A2 =2.63/1.72,

Therefore

m1/m2= 2.63/1.72 x 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.