Question

In how many parts (equal) a wire of $100\Omega$ be cut so that a resistance of $1\Omega$ is obtained by connecting them in parallel?

• A. $10$
• B. $5$
• C. $100$
• D. $50$

Answer 1

The correct option is A $10$

The resistance of the conductor is proportional to length of the conductor.
That is, $R=\rho \frac{l}{A}$.
The resistance of one wire of length l, R1 = 100 ohms.
If we cut it in a half: $\frac{R_1}{l}=\frac{R_2}{\frac{l}{2}}\to R_2=\frac{R_1}{2}$.
If we cut it in n parts the resistance will be $R=\frac{R_1}{n}$.
Because we shorten the length n times so it became $\frac{l}{n}$.
When we connect these parts in parallel the output resistance must be 1 ohm.
So, $1=\frac{1}{R}+\frac{1}{R}+...=n\frac{1}{R}=n\frac{1}{\frac{R_1}{n}}=\frac{n^2}{R_1}=\frac{n^2}{100}\to n=10$.
Hence, a resistance of 1 ohm is obtained by connecting 10 equal parts of the wire and connecting them in parallel.