Question

An aluminium wire 7.5 m long is connected in a parallel with a copper wire 6m long. When a current of 5A is passed through the combination, it isfound that the current in the aluminium wire is 3 A. The diameter of the aluminium wire is 1 mm. Determine the diameter of the copper wire. Resistivity of copper is 0.017μΩ-m..that of the aluminium is 0.028 μΩ-m

Answer 1

Let the subscript I represent aluminium and subscript 2 represent copper.

R₁ = ρ(l₁/a₁) and R₂ = ρ₂(l₂/a₂)

∴ R₂/R₁ = ρ₂/R₁ = ρ₂/ρ₁ x l₂/l₁ x a₁/a₂

∴ a₂ = a₁ x R₁/R₂ x ρ₂/ρ₁ x l₂/l₁ ........(i)

Now l₁ = 3A; l₂ = 5 - 3 = 2A.

If V is the common voltage across the parallel combination of aluminium and copper wires,then

V = I₁R₁ = l₂R₂

∴ R₁/R₂ = l28/l₁ = 2/3

a₁ = πd²/4 =(π x 12)/4 = π/4mm²

Substituting the given values in Eq. (i), we get

a₂ = π/4 x 2/3 x 0.017/0.028 x 6/7.5 = 0.2544m²

∴ π x d₂²/4 = 0.2544 or d₂ = 0.569mm