Question

An ideal solenoid is kept with its axis vertical. Current \( I_0 \) is flowing in the solenoid. A charge \( Q \) is thrown downward inside the solenoid. If acceleration of the charge particle is \( a \), then

• A. \( a>g \)
• B. \( a = g \)
• C. \( a<g \)
• D. \( a = 0 \)

Hint:

When a charged particle moves in a uniform magnetic field with velocity parallel to the field, the magnetic force does not affect its motion. Gravity is the primary force acting on it.

Answer 1

The Correct Option is B

To determine the acceleration of a charged particle falling inside an ideal solenoid, consider the following steps:

1. Understanding Solenoid Characteristics:
   • An ideal solenoid with current \( I_0 \) produces a uniform magnetic field inside it along the axis. However, the field lines are parallel to the axis of the solenoid and do not exert any force on a charged particle moving along the field lines.
2. Force on the Charged Particle:
   • The magnetic force \( \mathbf{F} \) on a charge \( Q \) moving with velocity \( \mathbf{v} \) in a magnetic field \( \mathbf{B} \) is given by the Lorentz force equation: \(\mathbf{F} = Q(\mathbf{v} \times \mathbf{B})\).
   • Since the magnetic field inside the solenoid is vertical and the charge is moving downward (also vertical), the angle between velocity \( \mathbf{v} \) and magnetic field \( \mathbf{B} \) is zero. Therefore, the cross product \( \mathbf{v} \times \mathbf{B} = 0 \), and thus the magnetic force is zero.
3. Effect on Acceleration:
   • Since the magnetic force acting on the charge is zero, the only force acting on the charged particle is gravity.
   • Hence, the acceleration of the charged particle is solely due to gravity, \( g \).
4. Conclusion:
   • Since no additional forces act on the charge except for gravity, the acceleration \( a \) of the charge is \(a = g\).

Thus, the correct answer is: \( a = g \), meaning the acceleration of the charged particle is equal to the acceleration due to gravity.