A uniform magnetic field fills a cylindrical volume of radius R. A metal rod of length L is placed as shown in the figure. If B is changing at the rate , we have to show that the emf that is produced by the changing magnetic field and that acts between the ends of the rod is given by

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Because of the cylindrical symmetry and the direction of the magnetic field as shown in the figure, the electric field that will appear because of the change in flux due to change in the magnetic field will be circular. The induced electric field will not have radial component.

For computing the induced emf across the ends of the rod, which is placed in the magnetic field as shown in the figure, we join its ends A and B with the point O on the axis as shown.

We compute along the closed contour AOBA. As the induced electric field does not have radial component, the contour integral

By Faraday’s law of induction,

,

where is the flux enclosed.

The flux enclosed by the loop AOBA will be B times the area of the triangle AOB. It, therefore, will be

Therefore, the emf produced by the changing magnetic field across the ends of the rod will be