AIIMS 2005 Physics Resistors MCQ Question

To understand how to ensure that the same energy is dissipated in all three resistors $R_1$, $R_2$, and $R_3$, we need to analyze the circuit configuration (assumed to be a series or parallel arrangement). In this case, we will discuss the parallel connection as it is commonly implied in such questions concerning equal energy dissipation.

Correct Answer Explanation

The correct answer provided is C) $R_1 = R_2$ and $R_3 = \frac{1}{4}R_2$.

In a parallel circuit, the voltage across each resistor is the same. The power dissipated by each resistor can be calculated using the formula:

$P = \frac{V^2}{R}$

Where:

• $P$ is the power (or energy per unit time) dissipated by the resistor.
• $V$ is the voltage across the resistor.
• $R$ is the resistance.

If we denote the common voltage across the resistors as $V$:

1. The power dissipated in $R_1$ is: $P_1 = \frac{V^2}{R_1}$

2. The power dissipated in $R_2$ is: $P_2 = \frac{V^2}{R_2}$

3. The power dissipated in $R_3$ is: $P_3 = \frac{V^2}{R_3}$

For the energy dissipated to be the same in all three resistors, we need:

$P_1 = P_2 = P_3$

Substituting the power equations:

$\frac{V^2}{R_1} = \frac{V^2}{R_2} = \frac{V^2}{R_3}$

Since $V^2$ is common and non-zero, we can simplify these equations to:

$R_1 = R_2 \quad \text{and} \quad R_2 = R_3$

Given the relationship from option C, if we set $R_2 = x$, then:

• $R_1 = x$
• $R_3 = \frac{1}{4}x$

This implies that $R_1$ and $R_2$ are equal while $R_3$ is one-fourth of their value, thus ensuring that the power dissipated in $R_3$ is equal to that in $R_1$ and $R_2$ when the voltage is the same across all three resistors.

Incorrect Options Explanation

Option A: $R_1 = R_2 = R_3$

This option implies that all three resistances are equal. While this would result in equal power dissipation, it does not satisfy the specific condition that $R_3$ should be one-fourth of either $R_1$ or $R_2$ as per the correct answer. Hence, it is not valid in this context.

Option B: $R_2 = R_3$ and $R_1 = 4R_2$

In this case, if $R_1 = 4R_2$, then substituting $R_2$ as $x$ gives $R_1 = 4x$ and $R_3 = x$. The power would be:

• $P_1 = \frac{V^2}{4x}$
• $P_2 = \frac{V^2}{x}$
• $P_3 = \frac{V^2}{x}$

Here, $P_1$ is not equal to $P_2$ and $P_3$, proving that this option does not ensure equal energy dissipation.

Option D: $R_1 = R_2 + R_3$

This option suggests a relationship typically found in series circuits. In a series connection, the power dissipated is not equal across resistors due to different resistances leading to different current through each. Thus, this statement does not apply to our requirement of equal energy dissipation in a parallel connection.

Conclusion

Thus, the correct answer is indeed:

C) $R_1 = R_2$ and $R_3 = \frac{1}{4}R_2$

This relationship ensures that the energy dissipated across all three resistors is equal, adhering to the principles of electrical circuits in parallel arrangements.