AIIMS 2006 Physics Induced EMF and Current MCQ Question

To understand the behavior of the current induced in a metallic ring as it falls through a magnetic field, we need to delve into the principles of electromagnetic induction, specifically Faraday's law of electromagnetic induction.

Explanation of the Problem

When the metallic ring is dropped into a region with a constant horizontal magnetic field, it experiences a change in magnetic flux as it enters and exits this region. According to Faraday's law, the induced electromotive force (EMF) in a closed loop is given by:

$$\mathcal{E} = -\frac{d\Phi_B}{dt}$$

where $\Phi_B$ is the magnetic flux through the ring. The magnetic flux is defined as:

$$\Phi_B = B \cdot A$$

where $B$ is the magnetic field strength and $A$ is the area of the ring exposed to the magnetic field.

Analyzing the Scenario

1. Entering the Magnetic Field:

   • At $t = 0$, the ring starts to enter the magnetic field. As it enters, the area $A$ that is subject to the magnetic field increases linearly with time until the ring is fully within the field. This results in an increase in magnetic flux and thus a positive induced EMF.

2. Fully Inside the Magnetic Field:

   • Once fully inside the magnetic field, the area does not change, and thus the magnetic flux remains constant. At this point, the induced EMF becomes zero, and therefore the current in the ring also becomes zero.

3. Exiting the Magnetic Field:

   • As the ring starts to exit the magnetic field, the area exposed to the magnetic field decreases linearly. This also results in a change in magnetic flux, which induces a negative EMF, leading to a current in the opposite direction.

Current Variation Over Time

The current in the ring will thus exhibit a specific pattern over the time interval from $0$ to $T$:

• It starts from zero, increases linearly to a maximum when the ring is halfway through the magnetic field, remains at zero while fully inside the field, and then decreases linearly back to zero as it exits.

Graph of Current vs Time

The graph of current $I(t)$ versus time $t$ can be visualized as follows:

• From $t = 0$ to $t = \frac{T}{2}$: Current increases linearly.
• From $t = \frac{T}{2}$ to $t = T$: Current decreases linearly back to zero.

This pattern indicates that the induced current varies as a triangular wave, characterized by a linear relationship during each of the entering and exiting phases, leading to a peak current at $t = \frac{T}{2}$.

Correct Answer and Explanation

Given the options presented, the correct option is B, which represents the current varying in a triangular manner, indicating a linear increase followed by a linear decrease.

Why Other Options May Be Incorrect

• Option A (a): If this option suggests a constant current, it would be incorrect because the current cannot remain constant as the ring moves in and out of the magnetic field.
• Option C (c): If it suggests a sinusoidal variation, this is also incorrect as the behavior is not oscillatory but rather linear during the entry and exit.
• Option D (d): If it implies an exponential decay or rise, this does not represent the induced current, which is linear in nature during the entry and exit phases.

Conclusion

In summary, the metallic ring experiences induced current due to the changing magnetic flux as it moves through the magnetic field. The correct interpretation of the current's variation leads us to choose option B, which accurately reflects the linear increase and decrease of current in the ring as it interacts with the magnetic field.

Thus, understanding the relationship between magnetic fields and induced currents through Faraday's law is crucial in solving problems related to electromagnetic induction efficiently.