To solve the problem regarding the power dissipated by resistors \( R_1 \) and \( R_2 \) in a circuit, we need to understand how power is calculated in resistive circuits and how resistors behave in series and parallel configurations. ### Step-by-Step Explanation 1. **Understanding Power in Resistors**: The power \( P \) dissipated by a resistor can be calculated using the formula: \[ P = I^2 R \] where \( I \) is the current flowing through the resistor and \( R \) is the resistance. 2. **Analyzing the Circuit**: In the given circuit, we have two resistors \( R_1 \) and \( R_2 \). Since the internal resistance of the cell is zero, we can assume that the entire voltage from the cell is applied across the resistors. 3. **Current Distribution**: If \( R_1 \) and \( R_2 \) are connected in parallel, the voltage across both resistors is the same. Let's denote the voltage across the resistors as \( V \). The current through each resistor can be expressed as: \[ I_1 = \frac{V}{R_1} \quad \text{and} \quad I_2 = \frac{V}{R_2} \] 4. **Calculating Power for Each Resistor**: Using the power formula for each resistor: - For \( R_1 \): \[ P_1 = I_1^2 R_1 = \left(\frac{V}{R_1}\right)^2 R_1 = \frac{V^2}{R_1} \] - For \( R_2 \): \[ P_2 = I_2^2 R_2 = \left(\frac{V}{R_2}\right)^2 R_2 = \frac{V^2}{R_2} \] 5. **Finding the Ratio of Powers**: Now, we can find the ratio of the powers \( P_1 \) and \( P_2 \): \[ \frac{P_1}{P_2} = \frac{\frac{V^2}{R_1}}{\frac{V^2}{R_2}} = \frac{R_2}{R_1} \] This means that the ratio of the powers dissipated by \( R_1 \) and \( R_2 \) is: \[ \frac{P_1}{P_2} = \frac{R_2}{R_1} \] 6. **Identifying the Correct Option**: The question asks for the ratio of the powers \( P_1 \) and \( P_2 \). Since we found that: \[ \frac{P_1}{P_2} = \frac{R_2}{R_1} \] This corresponds to option **A**: \( \frac{R_2}{R_1} \). ### Why Other Options Are Incorrect - **Option B: \( \frac{R_1}{R_2} \)**: This option suggests that the power in \( R_1 \) is greater than that in \( R_2 \) when the opposite is true based on our calculations. - **Option C: \( \frac{R_1 + R_2}{R_1} \)**: This option does not represent the ratio of powers but rather a different relationship that does not apply to the power dissipation in parallel resistors. - **Option D: \( \frac{R_1 + R_2}{R_2} \)**: Similar to option C, this does not reflect the correct relationship for power dissipation in the context of the given circuit. ### Common Pitfalls - **Misunderstanding Circuit Configuration**: Ensure you know whether resistors are in series or parallel, as this affects current and voltage distribution. - **Forgetting to Square the Current**: When calculating power, remember that power is proportional to the square of the current, which can lead to incorrect ratios if overlooked. ### Revision Summary - Power in a resistor is given by \( P = I^2 R \). - For resistors in parallel, the voltage across each is the same, leading to different currents. - The ratio of powers for resistors in parallel is given by \( \frac{P_1}{P_2} = \frac{R_2}{R_1} \). - The correct answer is option **A**: \( \frac{R_2}{R_1} \).