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 find the resultant resistance when a wire of resistance \(R\) and length \(L\) is cut into 5 equal parts and then joined in parallel, we need to follow these steps: 

1. When the wire is cut into 5 equal parts, each part will have a resistance of \(\frac{R}{5}\). This is because resistance is directly proportional to length.
2. If each part has a resistance of \(\frac{R}{5}\) and all parts are connected in parallel, the formula for equivalent resistance \(R_{\text{eq}}\) of resistors in parallel 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 resistances:

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

1. This simplifies to:

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

1. Taking the reciprocal gives the equivalent resistance:

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

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

Hence, the correct answer is \(\frac{R}{25}\).

Answer 2

Each part will have resistance:

\[ R' = \frac{R}{5} \]

When connected in parallel, the total resistance \(R_{eq}\) is:

\[ \frac{1}{R_{eq}} = 5 \times \frac{1}{R'} = 5 \times \frac{1}{R/5} = 5 \times \frac{5}{R} = \frac{1}{R/25} \]

Thus,

\[ R_{eq} = \frac{R}{25} \]