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I figure that the resistance of multiple wires would work just like adding in parallel, a bunch of wires that is. But couldn't any wire just be viewed as a bunch of wires, albeit small ones? I know that it isn't added like that, the area is put in the denominator, but why can't a single wire be viewed as a lot of small wires side by side?

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2 Answers 2

Yes, that's exactly how it works. As you know, the resistances for multiple resistors (such as wires) in parallel are "added" up as

$$\frac{1}{R} = \frac{1}{R_1} + \frac{1}{R_2} + \cdots$$

And the formula for resistance of a thin wire of uniform cross section and composition is $R = \frac{\rho L}{A}$. So if you have a bunch of wires with areas $A_1$, $A_2$, etc., and you combine their resistances, you get

$$\begin{align} \frac{1}{R} &= \frac{A_1}{\rho L} + \frac{A_2}{\rho L} + \cdots \\ &= \frac{A_1 + A_2 + \cdots}{\rho L} \\ &= \frac{A_\text{total}}{\rho L} \end{align}$$

which matches the resistance of one wire of area $A_\text{total}$.

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This is a good, insightful question.

If you're only considering a constant current (DC), where resistance is effectively the only consideration, then your intuition is correct.

However, for alternating currents (AC), a single wire is not necessarily equivalent to a "bunch of wires" due to, for example, the skin effect.

Consider Litz wire as a counter example.

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