A non-conducting sphere of radius `R = 50 mm` charged uniformly with surface density `sigma = 10.0 muC//m^(2)` rotates with an angular velocity `omega = 70` rad/s about the asxis passign thorugh its centre. Find the magnetic induction at the centre of the sphere.

Explanation

As only the outer surface of the sphere is charged, consider the element as a ring, as shown in figure.
The equivalent current due to the ring element,
`di = (omega)/(2pi) (2pi r sin theta rd theta) sigma` ....(1)
and magentic indruction due to this loop element at the centre of the sphere, `O`,
`dB = (mu_(0))/(4pi) di (2pi r sin theta r sin theta)/(r^(3)) = (mu_(0))/(4pi) di (sin^(2) theta)/(r)`
Hence, the total magentic induction due to the sphere at the centre, `O`,
`B = int dB = int_(0)^(pi//2) (mu_(0))/(4pi) (omega)/(2pi) (2pi r^(2) sin theta d theta sin^(2) theta sigma)/(r)`
Hence, `B = int_(0)^(pi//2) (mu_(0) sigma omega r)/(4pi) sin^(3) theta d theta = (2)/(3) mu_(0) sigma omega r = 29 pT`