To determine the amplitude of the induced voltage in the given circuit, we can use Faraday's law of electromagnetic induction. This law states that the induced electromotive force (emf) in a closed loop is equal to the rate of change of magnetic flux through the loop. The formula for the induced emf (ε) in a rotating loop in a magnetic field is given by: \[ \epsilon = N \cdot B \cdot A \cdot \omega \cdot \sin(\theta) \] Where: - \( \epsilon \) = induced emf (voltage) - \( N \) = number of loops (50 loops in this case) - \( B \) = magnetic field strength (0.4 T) - \( A \) = area of the loop (0.4 m²) - \( \omega \) = angular velocity (10 rad/s) - \( \theta \) = angle between the magnetic field and the normal to the plane of the loop ### Step-by-Step Calculation 1. **Identify the Variables**: - Number of loops, \( N = 50 \) - Magnetic field strength, \( B = 0.4 \, \text{T} \) - Area of the loop, \( A = 0.4 \, \text{m}^2 \) - Angular velocity, \( \omega = 10 \, \text{rad/s} \) 2. **Determine the Angle**: - For maximum induced voltage, we consider the angle \( \theta = 90^\circ \) (or \( \frac{\pi}{2} \) radians), where \( \sin(90^\circ) = 1 \). 3. **Substitute the Values into the Formula**: \[ \epsilon = N \cdot B \cdot A \cdot \omega \cdot \sin(90^\circ) \] \[ \epsilon = 50 \cdot 0.4 \cdot 0.4 \cdot 10 \cdot 1 \] 4. **Calculate Step-by-Step**: - First, calculate \( N \cdot B \cdot A \cdot \omega \): \[ N \cdot B = 50 \cdot 0.4 = 20 \] \[ 20 \cdot A = 20 \cdot 0.4 = 8 \] \[ 8 \cdot \omega = 8 \cdot 10 = 80 \] 5. **Final Result**: \[ \epsilon = 80 \, \text{V} \] ### Conclusion The amplitude of the induced voltage is **80 V**, which corresponds to option **D**. ### Explanation of Other Options - **Option A (8V)**: This value is incorrect because it does not account for the total number of loops and the angular velocity correctly. It seems to be a miscalculation of the parameters. - **Option B (16V)**: This option is also incorrect. It may arise from incorrectly calculating the product of the area and magnetic field without considering the angular velocity and the number of loops. - **Option C (20V)**: This value is incorrect as well. It likely results from miscalculating the contributions of the area and magnetic field without the full multiplication by the angular velocity and number of loops. ### Revision Summary - Use Faraday's law to calculate induced emf: \( \epsilon = N \cdot B \cdot A \cdot \omega \cdot \sin(\theta) \). - For maximum induced voltage, set \( \theta = 90^\circ \) (or \( \sin(90^\circ) = 1 \)). - Carefully substitute all values and perform calculations step-by-step. - Always check the units and ensure they are consistent throughout the calculation.