Question

A wire of length \( L \) having Resistance \( R \) falls from a height \( h \) in Earth's horizontal magnetic field. What is the current through the wire?

• A. \( \frac{hB}{R} \)
• B. \( \frac{hB^2}{R} \)
• C. \( \frac{hB^2}{2R} \)
• D. \( \frac{hB}{2R} \)

Hint:

When dealing with induced emf in a falling wire in a magnetic field, remember to apply Faraday's law and use the appropriate motion equations for falling objects to determine the velocity.

Answer 1

The Correct Option is A

An induced electromotive force (emf) is produced when a wire moves through Earth's magnetic field. Faraday's Law of Induction states that the induced emf (\( \varepsilon \)) is calculated as: \[ \varepsilon = BvL \] with \( B \) representing the magnetic field strength, \( v \) the wire's velocity, and \( L \) its length. The wire's velocity \( v \) during free fall is given by: \[ v = \sqrt{2gh} \] where \( g \) is the acceleration due to gravity and \( h \) is the fall height. Substituting this velocity into the emf equation yields: \[ \varepsilon = B \sqrt{2gh} L \] Ohm's law defines the current \( I \) as: \[ I = \frac{\varepsilon}{R} = \frac{B \sqrt{2gh} L}{R} \] Consequently, the current is directly proportional to \( B \), \( h \), and \( L \), and inversely proportional to \( R \). The accurate formula for the current flowing through the wire is \( \frac{hB}{R} \).