A metallic rod of length l is moved perpendicular to its length with velocity v in a magnetic field `vec(B)` acting perpendicular to the plane in which rod moves. Derive the expression for the inducced emf.

Explanation

To derive the expression for the induced electromotive force (emf) in a metallic rod moving in a magnetic field, we can follow these steps:

### Step 1: Understand the Setup
We have a metallic rod of length \( L \) moving with a velocity \( v \) perpendicular to its length in a magnetic field \( \vec{B} \) that is also perpendicular to the plane of motion of the rod.

### Step 2: Determine the Change in Area
As the rod moves, the area associated with the magnetic flux changes. The change in area \( dA \) as the rod moves a small distance \( dx \) can be expressed as:
\[
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