Question

A wire of length \( L \) and resistance \( R \) is falling vertically through Earth's horizontal magnetic field \( B \). What is the current induced in the wire when it has fallen a height \( L \)? (Take acceleration due to gravity as \( g \))

• A. \( \frac{BL\sqrt{2gL}}{R} \)
• B. \( \frac{B\sqrt{2gL}}{R} \)
• C. \( \frac{BL^2\sqrt{2g}}{R} \)
• D. \( \frac{BLg}{R} \)

Hint:

When a wire falls through a magnetic field, the induced current depends on the velocity of the wire, which is influenced by the gravitational potential energy being converted to kinetic energy.

Answer 1

The Correct Option is A

The induced current in the wire results from its movement within the magnetic field. Faraday's law states that the induced electromotive force (\( \epsilon \)) is calculated as: \[ \epsilon = BvL \] Here, \( v \) represents the wire's velocity as it descends due to gravity. Using the kinematic equation \( v^2 = 2gL \), the velocity can be expressed as: \[ v = \sqrt{2gL} \] Substituting this velocity into the emf equation yields: \[ \epsilon = B\sqrt{2gL}L \] Ohm's law defines the current \( I \) as: \[ I = \frac{\epsilon}{R} = \frac{B\sqrt{2gL}L}{R} \] Therefore, the induced current is given by: \[ I = \frac{BL\sqrt{2gL}}{R} \]