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

When the wire falls in Earth's magnetic field, an induced emf is generated due to the motion of the wire in the magnetic field. According to Faraday's Law of Induction, the induced emf (\( \varepsilon \)) is given by: \[ \varepsilon = BvL \] where: - \( B \) is the magnetic field, - \( v \) is the velocity of the wire as it falls, and - \( L \) is the length of the wire. The velocity \( v \) of the wire can be determined using the equation for free fall: \[ v = \sqrt{2gh} \] where \( g \) is the acceleration due to gravity and \( h \) is the height from which the wire falls. Substituting this into the equation for the induced emf: \[ \varepsilon = B \sqrt{2gh} L \] The current \( I \) is given by Ohm's law: \[ I = \frac{\varepsilon}{R} = \frac{B \sqrt{2gh} L}{R} \] Thus, the current is proportional to the magnetic field \( B \), the height \( h \), and the length \( L \), and inversely proportional to the resistance \( R \). The correct expression for the current through the wire is \( \frac{hB}{R} \).