AIPMT PRELIMS 2004 Physics Series and Parallel Circuits MCQ Question

To solve the given problem, we need to first understand how resistances behave in parallel and in series.

Step 1: Equivalent Resistance for Parallel Connection

When $n$ resistors, each of resistance $r \, \Omega$, are connected in parallel, the formula for the equivalent resistance $R$ can be expressed as:

$$\frac{1}{R} = \frac{1}{r_1} + \frac{1}{r_2} + \frac{1}{r_3} + \ldots + \frac{1}{r_n}$$

Since all resistors have the same resistance $r$:

$$\frac{1}{R} = \frac{n}{r}$$

Rearranging this gives:

$$R = \frac{r}{n}$$

This means that the equivalent resistance $R$ of $n$ resistors in parallel is $R = \frac{r}{n}$.

Step 2: Equivalent Resistance for Series Connection

When the same $n$ resistors are connected in series, the equivalent resistance $R_s$ is simply the sum of the individual resistances:

$$R_s = r_1 + r_2 + r_3 + \ldots + r_n$$

For $n$ resistors each of resistance $r$:

$$R_s = nr$$

Step 3: Relating Series Resistance to Parallel Resistance

From the parallel connection result, we have:

$$r = nR$$

When we substitute $r$ back into the series resistance formula, we get:

$$R_s = n(nR) = n^2R$$

Conclusion

Thus, when $n$ resistors, each of resistance $r \, \Omega$, are connected in series, the equivalent resistance is $R_s = n^2 R$.

Correct Answer:

The correct answer is D) $n^2 R$.

Explanation of Other Options:

• Option A) $R/n^2$: This option suggests that the series resistance decreases with an increase in $n^2$, which is incorrect since resistance adds up in series.

• Option B) $R/n$: This option is the result of resistances in parallel, not series. It implies a reduction in resistance with increasing $n$, which is not applicable for series connections.

• Option C) $nR$: This option incorrectly suggests that the series resistance is simply $n$ times the equivalent resistance of the parallel configuration. However, we have established that $r = nR$, so the correct factor should be $n^2$ times $R$.

In summary, the behavior of resistors in parallel and series circuits is fundamental to understanding the overall resistance in a circuit. The derived formulas allow us to relate the resistances in different configurations effectively.