Question

A piece of wire having resistance 'R' is cut into four equal parts.

a) How does the resistance of each part compare with the original resistance?

b) If the four are placed in parallel, how will the resistance of combination compare with the resistance of the original wire.

Answer 1

(a) We know that total resistance = R

So, when the wire is cut into four equal parts, we actually have divided the total resistance into 4 equal parts. Hence, the resistance of each part is now $\frac{1}{4}$th of the original Resistance, i.e. $\frac{R}{4}$.

(b) When these 4 parts are joined in parallel, the equivalent resistance $R_p$ will be

$\frac{1}{R_p}=\frac{4}{R}+\frac{4}{R}+\frac{4}{R}+\frac{4}{R}=4\times \left(\frac{4}{R}\right)=\frac{16}{R}$
i.e. $R_p=\frac{R}{16}$.

So, the total resistance = $\frac{R}{16}$, i.e., $\frac{1}{16}$th times of R.