A metallic rod of length 'l' is tied to a string of length '2l' and made to rotate with angular speed '2 ' on a horizontal table with one end of the string fixed. If there exists a vertical magnetic field 'B' in the region, the emf 'e' induced across the ends of rod is

Step-by-step solution

Let the fixed end of the string be the origin. The rod is situated between the distances r₁ = 2l and r₂ = 2l + l = 3l from the fixed end. Consider a small element of the rod of length dx at a distance x from the fixed end. The velocity of this element is v = x(2 ) . The motional emf induced in this small element is de = Bv ,dx = B(x 2 ) ,dx . The total induced emf across the ends of the rod is obtained by integrating de from x = 2l to x = 3l : e = _ 2l ^ 3l 2B x ,dx e = 2B [ x^2 2 ]_ 2l ^ 3l e = B ((3l)^2 - (2l)^2)