Question

Nine identical wires, each of diameter d and length L, are connected in parallel. The combination
has the same resistance as a single similar wire of length L but whose diameter is:
A. 3d
B. 9d
C. d/3
D. d/9
E. d/81

Answer 1

Answer:

A. 3d

Explanation:

N resistors, all identical, with a resistance equal to R, when connected in parallel, have a equivalent resistance of R/N.

The general expression of the resistance of a wire (assumed of circular section) is as follows:

R =ρ*(L/A) = ρ*(L/(π*r²)) = ρ*  $\frac{L}{\pi*\frac{d^{2}}{4}}$ (1)

If the equivalent resistance of the nine wires connected in parallel, is equal to R/9, the resistance of this wire will be as follows:

Req=ρ*(L/Aeq)/ 9 = ρ*(L/9*(π*req²)) = ρ* $\frac{L}{\ 9*pi*\frac{d^{2}}{4}}= rho* \frac{L}{\pi*\frac{(3*d)^{2}}{4}}$ (2)

So, from (1) and (2), we can see that the combination has the same resistance as a single wire whose diameter is 3 times the diameter of each  individual wire.