Question

A wire of resistance \( R \) and length \( L \) is cut into 5 equal parts. If these parts are joined in parallel, then the resultant resistance will be:

• A. 5 R
• B. 25 R
• C. \( \frac{R}{25} \)
• D. \( \frac{R}{5} \)

Answer 1

The Correct Option is C

To determine the equivalent resistance when a wire of resistance \(R\) and length \(L\) is divided into 5 equal segments and connected in parallel, the following procedure is applied:

1. Dividing the wire into 5 equal segments results in each segment having a resistance of \(\frac{R}{5}\), as resistance is directly proportional to length.
2. When these 5 segments, each with resistance \(\frac{R}{5}\), are connected in parallel, the formula for equivalent resistance \(R_{\text{eq}}\) is:

\(\frac{1}{R_{\text{eq}}} = \frac{1}{R_1} + \frac{1}{R_2} + \frac{1}{R_3} + \frac{1}{R_4} + \frac{1}{R_5}\), where \(R_1 = R_2 = R_3 = R_4 = R_5 = \frac{R}{5}\).

1. Substituting the individual resistances into the formula yields:

\(\frac{1}{R_{\text{eq}}} = \frac{1}{\frac{R}{5}} + \frac{1}{\frac{R}{5}} + \frac{1}{\frac{R}{5}} + \frac{1}{\frac{R}{5}} + \frac{1}{\frac{R}{5}}\)

1. This expression simplifies to:

\(\frac{1}{R_{\text{eq}}} = 5 \times \frac{1}{\frac{R}{5}} = \frac{25}{R}\)

1. Inverting this equation provides the equivalent resistance:

\(R_{\text{eq}} = \frac{R}{25}\)

Consequently, the resultant resistance when the wire is cut into 5 equal parts and connected in parallel is \(\frac{R}{25}\).

The correct value is therefore \(\frac{R}{25}\).