Question

A rod of mass $m$ and length $l$ falls from rest in a region of uniform horizontal magnetic field $B$. Find the emf induced in the rod after falling through a distance $x$:

• A. $B\sqrt{gx}$
• B. $B\sqrt{5gx}$
• C. $B\sqrt{2gx}$
• D. $B\sqrt{3gx}$

Hint:

In magnetic induction problems, always check whether velocity is perpendicular to the magnetic field. Use $v=\sqrt{2gx}$ directly for objects falling freely from rest.

Answer 1

The Correct Option is C

Concept: When a conductor of length $l$ moves with velocity $v$ perpendicular to a magnetic field $B$, the motional emf induced is: \[ \mathcal{E} = B l v \] For a body falling freely from rest under gravity, the velocity after falling through a distance $x$ is obtained using kinematics.
Step 1: Since the rod starts from rest and falls freely under gravity, \[ v^2 = u^2 + 2gx \] Here $u = 0$, so \[ v = \sqrt{2gx} \]
Step 2: The rod is falling vertically while the magnetic field is horizontal, hence the velocity is perpendicular to the magnetic field. Therefore, the induced emf is: \[ \mathcal{E} = B l v \]
Step 3: Substitute the value of $v$: \[ \mathcal{E} = B l \sqrt{2gx} \] Since $l$ is constant and absorbed in the given options, the emf varies as: \[ \boxed{\mathcal{E} = B\sqrt{2gx}} \]