Question

A conducting rod of length \( l \) and mass \( m \) is placed over a smooth horizontal plane. A magnetic field \( B \) is acting perpendicular to the rod. If a charge \( q \) is suddenly passed through the rod and the rod acquires an initial velocity \( v \) on the plane surface, then the charge \( q \) is:

• A. \( \frac{m v}{B l} \)
• B. \( \frac{m}{v B l} \)
• C. \( \frac{m v}{B l} \)
• D. \( \frac{1 B v}{m} \)

Hint:

Remember, when a rod moves in a magnetic field, the induced emf and the magnetic force depend on the velocity of the rod, the magnetic field, and the length of the rod.

Answer 1

The Correct Option is A

Using the concept of induced electromotive force (emf) in a moving conductor within a magnetic field, we have the following relations: - The induced emf \( \mathcal{E} \) in the rod is given by: \[ \mathcal{E} = B \cdot v \cdot l \] - The magnetic force on the rod is given by: \[ F = q \cdot \mathcal{E} = q \cdot B \cdot v \cdot l \] - From Newton’s second law, the force is also: \[ F = m \cdot a \] Equating the two expressions for force: \[ m \cdot a = q \cdot B \cdot v \cdot l \] Thus, solving for \( q \): \[ q = \frac{m \cdot v}{B \cdot l} \]