AIIMS 2007 Physics Impedance MCQ Question

To solve the question regarding the total potential difference in an AC circuit where an inductance and resistance are connected in series, we need to understand the concept of impedance, which combines the effects of resistance (R) and inductive reactance (X_L).

Given Data:

• Potential difference across the resistor, $V_R = 20 \, \text{V}$
• Potential difference across the inductor, $V_L = 16 \, \text{V}$

Total Potential Difference:

In a series circuit, the total potential difference supplied by the source $V_{total}$ can be calculated using the following relationship:

$V_{total} = \sqrt{V_R^2 + V_L^2}$

This formula arises from the fact that the voltage across the resistor and the voltage across the inductor are not in phase. The voltage across the resistor $V_R$ is in phase with the current, while the voltage across the inductor $V_L$ leads the current by $90^\circ$.

Step-by-Step Calculation:

1. Square the individual voltages:

   $$V_R^2 = (20 \, \text{V})^2 = 400 \, \text{V}^2$$ $$V_L^2 = (16 \, \text{V})^2 = 256 \, \text{V}^2$$

2. Add the squared values:

   $$V_R^2 + V_L^2 = 400 \, \text{V}^2 + 256 \, \text{V}^2 = 656 \, \text{V}^2$$

3. Take the square root to find the total potential difference:

   $$V_{total} = \sqrt{656 \, \text{V}^2} \approx 25.6 \, \text{V}$$

Thus, the total potential difference of the source is approximately $25.6 \, \text{V}$, which corresponds to option B.

Explanation of Other Options:

• A) 20.0 V: This option only considers the potential difference across the resistor and ignores the contribution from the inductor. It is incorrect because it does not account for the phase difference between the two voltages.

• C) 31.9 V: This value does not arise from any correct calculation involving the given voltages. It is likely a random number and is not close to the calculated total.

• D) 53.5 V: Similar to option C, this value does not correspond to any legitimate combination of the given voltages. It is incorrect as it overestimates the total potential difference without a basis in the relevant physics of the circuit.

Conclusion:

The correct answer is B) 25.6 V, derived from the relationship between the potential differences across components in an AC circuit where inductance and resistance are in series. The use of the Pythagorean theorem is essential due to the non-phase alignment of the voltages.