A metallic rod of 'L' length is rotated with angular frequency of 'w' with one end hinged at the centre and the other end at the circumference of a circular metallic ring of radius L, about an axis passing through the centre and perpendicular to the plane of the ring. A constant and uniform magnetic field B parallel to the axis is present everywhere. Deduce the expression for the emf between the centre and the metallic ring.

Explanation

To derive the expression for the electromotive force (emf) between the center and the metallic ring, we follow these steps:

### Step 1: Understand the Setup
We have a metallic rod of length \( L \) that is rotating about an axis through its center with an angular frequency \( \omega \). One end of the rod is hinged at the center of a circular metallic ring of radius \( L \), and the other end is at the circumference of the ring. A uniform magnetic field \( B \) is present parallel to the axis of rotation.

### Step 2: Define the Elemental Length
Consider an elemental length \( dx \) of the rod at a distance \( x \) from the center. This elemental length will be used to calculate the induced emf.

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