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

In time t, if `theta` is angle traced by face end then area swept `A= pi l^(2) xx ((theta)/(2pi))=(1)/(2) l^(2) theta`
`phi=B A cos 0^(@)=(1)/(2) Bl^(2) theta`
`(d phi)/(dt)=(1)/(2) Bl^(2)((d phi)/(dt))=(1)/(2) Bl^(2) omega`
The positive charge of the rod will be pushed towards O be replaced by a battery of emf source `(1)/(2) B omega l^(2)` with the positive terminal towards O. The equivalent circuit diagram is shown in figure. The circular loop joins A to C by a resistance less path. The current in the resistance R is
`i=(in)/(R )=(B omega I^(2))/(2R)`