At one time aluminum rather than copper wires were used to carry electric current through homes. Which wire must have the larger radius if they are the same length and need to have the same electrical resistance?

Radius of the copper wire is less than the radius of aluminum wire by a factor of 0.805

Explanation:

We know that Resistance of a wire is given by

R = ρL/A

Where ρ is the resistivity of the material used, L is the length and A is the cross-sectional area of the wire.

A = πr²

R = ρL/πr²

The resistivity of copper wire is ρc = 1.72x10⁻⁸

The resistivity of aluminum wire is ρa = 2.65x10⁻⁸

If both wires has same resistance then we can compare them as

Rc = Ra

ρcL/πrc²= ρaL/πra²

Assume a 1 m length of wire

ρc/rc²= ρa/ra²

ρc/ρa = rc²/ra²

1.72x10⁻⁸/2.65x10⁻⁸ = rc²/ra²

0.649 = rc²/ra²

rc/ra = √0.649

rc/ra = 0.805

rc = 0.805ra

So that means the radius of the copper wire is less than the radius of aluminum wire by a factor of 0.805

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