**Correct Option: C. The induced voltage will increase over time.** ### Detailed Explanation: To understand why the correct answer is C, we need to delve into Faraday's law of electromagnetic induction, which states that the induced electromotive force (emf) in a closed loop is directly proportional to the rate of change of the magnetic flux through the loop. The law can be mathematically 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 coil is defined as the product of the magnetic field strength (B), the area of the coil (A), and the cosine of the angle (\(\theta\)) between the magnetic field lines and the normal (perpendicular) to the surface of the coil: \[ \Phi_B = B \cdot A \cdot \cos(\theta) \] - If the magnetic field is uniform and perpendicular to the coil, this simplifies to \(\Phi_B = B \cdot A\). 2. **Increasing Magnetic Field**: - In this scenario, we are told that the magnetic field (B) is increasing at a constant rate. This means that the magnetic flux (\(\Phi_B\)) through the coil is also increasing because the area (A) and angle (\(\theta\)) are constant. 3. **Rate of Change of Magnetic Flux**: - Since the magnetic field is increasing at a constant rate, the change in magnetic flux (\(d\Phi_B/dt\)) is also constant and positive. This leads to a constant induced voltage (emf) in the coil. 4. **Induced Voltage Over Time**: - As the magnetic field continues to increase, the induced voltage will not only remain constant but will actually increase over time because the rate of change of the magnetic flux is constant. The induced voltage is proportional to this rate of change, so if the rate of change is constant and positive, the induced voltage will also increase. ### Why Other Options Are Incorrect: - **Option A: The induced voltage will remain constant.** - This option is incorrect because while the induced voltage is proportional to the rate of change of magnetic flux, if the magnetic field is increasing at a constant rate, the induced voltage will not remain constant; it will increase. - **Option B: The induced voltage will decrease over time.** - This option is also incorrect. A decreasing induced voltage would imply that the rate of change of magnetic flux is negative, which contradicts the premise that the magnetic field is increasing. - **Option D: The induced voltage will fluctuate randomly.** - This option is incorrect as well. A random fluctuation in induced voltage would suggest an erratic change in magnetic flux, which is not the case here since the magnetic field is increasing at a constant rate. ### Common Pitfalls: - Students often confuse the concepts of induced voltage and magnetic flux. Remember that induced voltage is related to the rate of change of magnetic flux, not just the magnetic field itself. - It’s important to note that the negative sign in Faraday's law indicates the direction of the induced current (Lenz's Law), but it does not affect the magnitude of the induced voltage in this context. ### Revision Summary: - Faraday's law states that induced voltage is proportional to the rate of change of magnetic flux. - An increasing magnetic field leads to an increasing magnetic flux, resulting in an increasing induced voltage. - The induced voltage will not remain constant or decrease if the magnetic field is increasing at a constant rate. - Understanding the relationship between magnetic flux and induced voltage is crucial for solving problems in electromagnetic induction.