You are given three pieces of wire that have different shapes (dimensions). You

Question

You are given three pieces of wire that have different shapes (dimensions). You connect each piece of wire separately to a battery. The first piece has a length L and cross-sectional area A. The second is twice as long as the first, but has the same thickness. The third is the same length as the first, but has twice the cross-sectional area. Rank the wires in order of which carries the most current (has the lowest resistance) when connected to batteries with the same voltage difference. Rank the wires from most current (least resistance) to least current (most resistance).

Explanation / Answer

Resistance of a wire is given by

R = L/A, where is the resistivity of the wire, L being length and A being area of the wire

If connected to a batter of voltage V, the current passing through the wire is given by I=V/R

For wire of length L and area A

R1 = L/A = R       and I1 = V/R1 = I

For wire of length 2L and area A

R2 = 2L/A = 2R       and I2 = V/R2 = V/2R = I/2

For wire of length L and area 2A

R3 = L/2A = R/2      and I3 = V/R3 = V/(R/2) = 2I

We can see that I3 > I1 > I2

so ranking of the wires will be

wire of length L and area 2A > For wire of length L and area A > For wire of length 2L and area A

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