A rod of length l revolves at a small, uniform angular velocity around its perpendicular bisector. A uniform magnetic field B is present parallel to the rotation axis. What is the potential difference between the rod's centre and one of its ends?

Step-by-step solution

Let the centre of the rod be the origin and the axis of rotation pass through it. The distance from the centre to one of the ends is l/2 . Consider a small element of length dx at a distance x from the centre. The velocity of this element is v = x . The motional emf induced in this small element is given by dE = B v dx = B x dx . The total potential difference between the centre and one end is obtained by integrating dE from x = 0 to x = l/2 : E = ₀^ l/2 B x dx E = B [ x^2 2 ]₀^ l/2 E = B ( (l/2)^2 2 ) = 1 8 B l^2