Problem 10 - A Complete Resource Book in Physics for JEE Main

Question

There are \(n\) similar resistors each of resistance \(R\). The equivalent resistance comes out to be \(x\) when connected in parallel. If they are connected in series, the resistance comes out to be (A) \(x / n^{2}\) (B) \(n^{2} x\) (C) \(x / n\) (D) \(n x\)

Step-by-Step Solution

Answer

None of the given options match the result we obtained, which is \( R_\text{series} = x \). There might be an error in the given options or it may not be a multiple choice question with a single correct answer.

Step 1: 1. Formula for Resistance in Parallel

The formula for calculating the equivalent resistance (R_parallel) when resistors are connected in parallel is: \( \frac{1}{R_\text{parallel}} = \frac{1}{R_1} + \frac{1}{R_2} + \cdots + \frac{1}{R_n} \) Since all the resistors are identical, we have: \( \frac{1}{R_\text{parallel}} = n \cdot \frac{1}{R} \) Given that the equivalent resistance when connected in parallel is x, we get: \( \frac{1}{x} = n \cdot \frac{1}{R} \)

Step 2: 2. Formula for Resistance in Series

The formula for calculating the equivalent resistance (R_series) when resistors are connected in series is: \( R_\text{series} = R_1 + R_2 + \cdots + R_n \) Since the resistors are identical, we have: \( R_\text{series} = nR \)

Step 3: 3. Substitute and Find the Relation

Now we substitute the value of R from the equation for resistors in parallel into the equation for resistors in series: \( R = \frac{x}{n} \) Therefore: \( R_\text{series} = n \cdot \frac{x}{n} = x \) Comparing this with the given expressions: (A) \( x / n^{2} \) (B) \( n^{2} x \) (C) \( x / n \) (D) \( n x \) We find that none of the options match the result we obtained, which is \( R_\text{series} = x \). So, either there is an error in the given options or it is not a multiple choice type question with a single correct answer.