Question

Equivalent resistance \( R \) is obtained when \( n \) wires having the same length and same thickness of the same material are joined in parallel. The equivalent resistance on joining them in series will be:

• A. \( nR \)
• B. \( n^2 R \)
• C. \( R/n \)
• D. \( R/n^2 \)

Hint:

Let each wire be \( r \); parallel gives \( r/n = R \) so \( r = nR \), and series gives \( nr \).

Answer 1

The Correct Option is B

Step 1: Since all \( n \) wires have the same length, thickness and material, they have the same resistance. Let each wire have resistance \( r \).

Step 2 (parallel): For \( n \) equal resistances in parallel, \[ \frac{1}{R_{parallel}} = \frac{n}{r} \ \Rightarrow\ R_{parallel} = \frac{r}{n}. \] We are told this equals \( R \): \[ \frac{r}{n} = R \ \Rightarrow\ r = nR. \]

Step 3 (series): For \( n \) equal resistances in series, \[ R_{series} = n\,r. \]

Step 4: Substitute \( r = nR \): \[ R_{series} = n \times (nR) = n^2 R. \] Hence option 2.

\[\boxed{R_{series} = n^2 R}\]