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

Options

A. `(3B omega l^(2))/(2)`
B. `(4B omega l^(2))/(2)`
C. `(5 B omega l^(2))/(2)`
D. `(2B omega l^(2))/(2)`

Answer

C

Explanation

As rod is rotating in magnetic field thus cutting magnetic field lines, motional emf will develop a cross the rod. As each and every element of the rod is moving with different velocity. For calculating emf across rod we first consider an element of length `dx` at a distance `x` from the fixed end of the string

e.m.f induced in the element is
`d epsilon = b(omega x) d x`
Hence, the e.m.f. induced across the ends of the rod is
`epsilon = underset(2l)overset(3l)(int) B omegax dx = B omega[(x^(2))/(2)]_(2l)^(3l) = (B omega)/(2)[(3l)^(2) - (2l)^(2)]`
`= (5 B omega l^(2))/(2)`