### Correct Option: C. The induced voltage increases. ### Detailed Explanation: To understand why the induced voltage increases when the magnetic field passing through a closed loop of wire is increased at a constant rate, we need to refer to **Faraday's Law of Electromagnetic Induction**. This law states that the induced electromotive force (emf) in any closed circuit is equal to the rate of change of the magnetic flux through the circuit. The formula for Faraday's Law can be expressed as: \[ \text{emf} = -\frac{d\Phi_B}{dt} \] Where: - \(\text{emf}\) is the induced voltage (in volts), - \(\Phi_B\) is the magnetic flux (in webers), - \(t\) is time (in seconds), - \(d\Phi_B/dt\) is the rate of change of magnetic flux. #### Step-by-Step Explanation: 1. **Understanding Magnetic Flux**: - Magnetic flux (\(\Phi_B\)) through a loop is defined as the product of the magnetic field strength (\(B\)) and the area (\(A\)) of the loop perpendicular to the magnetic field. Mathematically, it can be expressed as: \[ \Phi_B = B \cdot A \cdot \cos(\theta) \] where \(\theta\) is the angle between the magnetic field lines and the normal (perpendicular) to the surface of the loop. For a loop where the magnetic field is perpendicular, \(\cos(\theta) = 1\), simplifying the equation to \(\Phi_B = B \cdot A\). 2. **Increasing the Magnetic Field**: - When the magnetic field \(B\) is increased at a constant rate, the change in magnetic flux (\(d\Phi_B\)) over time (\(dt\)) also increases. Since the area \(A\) of the loop remains constant, the change in flux is directly proportional to the change in the magnetic field: \[ \frac{d\Phi_B}{dt} = A \cdot \frac{dB}{dt} \] Here, \(\frac{dB}{dt}\) is the rate at which the magnetic field is changing. 3. **Induced Voltage**: - According to Faraday's Law, as the magnetic field increases, the rate of change of magnetic flux (\(d\Phi_B/dt\)) becomes larger, leading to a larger induced voltage (\(\text{emf}\)). Therefore, if the magnetic field is increasing at a constant rate, the induced voltage will also increase. 4. **Direction of Induced Voltage**: - The negative sign in Faraday's Law indicates that the induced voltage will act in a direction to oppose the change in magnetic flux (Lenz's Law). However, this does not affect the magnitude of the induced voltage; it only indicates the direction of the induced current. ### Why Other Options Are Incorrect: - **Option A: The induced voltage remains constant.** - This option is incorrect because if the magnetic field is increasing, the rate of change of magnetic flux is also increasing, which means the induced voltage cannot remain constant. - **Option B: The induced voltage decreases.** - This option is incorrect because an increase in the magnetic field leads to an increase in the induced voltage, not a decrease. - **Option D: The induced voltage becomes zero.** - This option is incorrect because the induced voltage becomes zero only when there is no change in magnetic flux. Since the magnetic field is increasing, the induced voltage cannot be zero. ### Common Pitfalls: - Students often confuse the direction of the induced current with its magnitude. Remember that while the direction opposes the change in magnetic flux, the magnitude increases with an increasing magnetic field. - Misunderstanding the relationship between magnetic field strength and magnetic flux can lead to errors in applying Faraday's Law. ### Revision Summary: - Faraday's Law states that induced voltage is proportional to the rate of change of magnetic flux. - Increasing the magnetic field at a constant rate increases the induced voltage. - The induced voltage acts to oppose the change in magnetic flux (Lenz's Law). - Remember the relationship between magnetic field strength, area, and magnetic flux to apply these concepts correctly.