Question

What diameter must a copper wire have if its resistance is to be the same as that of an equal length of aluminium wire with diameter 2.14mm?

Short answer

The diameter of the copper wire must be 1.69mm.

Step-by-step solution

Determine the formula to find the diameter of a wire

We know,

$A=\frac{\mathrm{\pi }}{4}{d}^2$

Substituting in the equation

$R=\frac{\rho L}{A}=\frac{4\rho L}{{\mathrm{\pi d}}^2}$

Now, according to the given question, we know the resistivities of both copper and aluminium wires which thus, will make it easy to find the diameter of a copper wire.

Calculation to find the diameter of the copper wire

We are given that,

$R_{cu}=R_{AI},L_{cu}=L_{AI}$and$d_{AI}=1.72\times {10}^{-3}mm$

Also, the resistivities of aluminium and copper wire is

$p_{Al}=2.75\times {10}^{-8}\Omega m$ and$p_{cu}=1.72\times {10}^{-8}\Omega m$

So,

Therefore, the diameter of the copper wire must be 1.69 mm so as the resistance to be the same as that of an equal length of aluminium wire with diameter 2.14 mm