Moving Charges And Magnetism

The magnetic field at the centre of a current carrying loop is given by

The magnetic moment at the centre of current carrying loop is given by
Thus, (given)
When both the current and the radius are doubled, the ratio becomes

So, the correct option is (A) : .

The magnetic field at the centre of a current carrying loop is given by

The magnetic moment at the centre of current carrying loop is given by
Thus, (given)
When both the current and the radius are doubled, the ratio becomes

So, the correct option is (A) : .

Step 1: Magnetic Field and Induced Electric Field

The magnetic field inside the cylindrical region is given as: $ is time. Since the magnetic field varies with time, an induced electric field is generated according to Faraday's Law of Electromagnetic Induction: \) )

For , the induced electric field can be determined using Faraday's Law. The rate of change of the magnetic flux through a loop of radius is: \) : \) , the electric field magnitude is: \) varies linearly with for .

This confirms that option (B) is correct.

Step 3: Direction of Motion of the Charged Particle

At , the magnetic field is: \) , the electric field is: \) : \) . Thus, option (D) is incorrect.

Final Answer:

The correct option(s) is/are: \) $

Explanation:

1. Initial Motion:
- When the bar magnet is first released from rest, it starts to accelerate downward due to gravity.
- The speed of the magnet increases as it falls freely.

2. Induced Current and Magnetic Field:
- As the magnet approaches the ring, the changing magnetic flux through the ring induces a current in the wire (according to Faraday's Law).
- This induced current generates a magnetic field that opposes the motion of the falling magnet (Lenz's Law).

3. Deceleration:
- The opposing magnetic field creates a force that acts upward on the falling magnet, causing it to decelerate.
- The speed of the magnet decreases as it passes through the region where the magnetic flux is changing most rapidly.

4. Passing Through the Ring:
- As the magnet moves through the ring and starts to exit the other side, the rate of change of magnetic flux decreases.
- This reduces the induced current and thus the opposing magnetic force.
- The magnet starts to accelerate again, but the acceleration is less than the initial acceleration due to gravity.

5. After Passing Through the Ring:
- Once the magnet is far below the ring, the induced currents and opposing magnetic forces become negligible.
- The magnet resumes accelerating under gravity alone, but its speed is lower than it would be in free fall without the ring.

Graphical Representation:
- Option A shows:
- Initial increase in speed due to gravity.
- A dip in speed due to deceleration from the opposing force.
- A gradual increase in speed after passing the ring, but less steep.

Conclusion:
Option A correctly represents the speed vs. time graph for the magnet. So, the correct answer is option (A):