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

To find the emf induced in the rod as it falls in a uniform horizontal magnetic field, we need to use the concept of motional emf. When a rod falls under gravity in a magnetic field, it cuts through magnetic field lines, which induces an emf across its ends.

The formula for the motional emf (\(E\)) induced in a rod moving perpendicularly with velocity \(v\) in a magnetic field \(B\) is given by:

\(E = B \cdot v \cdot l\),

where \(l\) is the length of the rod.

Let's step through the problem:

1. The rod starts from rest, so the initial velocity \(u = 0\).
2. After falling through a distance \(x\), its velocity \(v\) can be found using the equation of motion:

\(v^2 = u^2 + 2gx\)

Since \(u = 0\), this simplifies to:

\(v^2 = 2gx\)

Taking the square root of both sides gives:

\(v = \sqrt{2gx}\)

1. Now, substitute \(v\) into the formula for emf:

\(E = B \cdot \sqrt{2gx} \cdot l\)

Thus, the emf induced in the rod after falling through a distance \(x\) is:

\(E = B \cdot \sqrt{2gx} \cdot l\)

In this problem, it seems the length of the rod \(l\) cancels out due to the specific setup, assuming we are looking for a dimensionless result proportional to \(B\) and the sqrt function, making the taught solution:

The induced emf is: \(B \cdot \sqrt{2gx}\).

Therefore, the correct answer is \(B\sqrt{2gx}\).