Why is the square of the diameter of a wire inversely proportional to its resistance?

Answer 1

This can be restated as: the resistance is inversely proportional to the area of the wire.

To see that this is reasonable, think of the area as being split up into N smaller areas of equal size. You then have N identical 'wires' in parallel. For parallel configurations of this type, the resistance is 1/N times the resistance of a single 'wire' (parallel combinations law) Say that the area of each 'wire' is a, then the area, A, of the original wire is N a. That is A = N a, or N = A/a

If the resistance of a 'wire' is 10 ohms, then the resistance of the original wire is 10 /N ohms. = 10 a /A

Thus if we double A, keeping a fixed, the resistance is decreased by a factor of 2. . If A is tripled, the resistance is reduced by a factor of 3. , etc.

This is equivalent to saying that the resistance is inversely proportional to A