A sphere of radius R, uniformly charged with the surface charge density `sigma` rotates around the axis passing through its centre at an angular velocity. (a) Find the magnetic induction at the centre of the rotating sphere. (b) Also, find its magnetic moment.

Options

A. `(2)/(3) mu_(0)omega sigmaR(hat(k)),(4)/(3)pisigmaR^(4)omega`
B. `(2)/(3) mu_(0)omega sigmaR^(2)(hat(k)),(4)/(3)pisigmaR^(4)omega`
C. `(2)/(3) mu_(0)omega sigmaR^(3)(hat(k)),(4)/(3)pisigmaR^(4)omega`
D. `(2)/(3) mu_(0)omega sigmaR^(4)(hat(k)),(4)/(3)pisigmaR^(4)omega`

Answer

A

Explanation

Charge on the differential circular strip is

`dq=(2pi Rsin theta)(R d theta)sigma`
or `" "dq=2pi sigma R^(2)sin theta d theta` ltbgt Then `dl=(omegadq)/(2pi)=omegasigmaR^(2)sin theta d theta`
`(a)" "`Magnetic field at the centre is
`dB=(mu_(0)dI(R^(2)sin^(2)theta))/(2R^(3))=(mu_(0)omegasigmaRsin ^(3)thetad theta)/(2)`
`B=(mu_(0)dI(R^(2)sin^(2)theta))/(2R^(3))=(mu_(0)omegasigmaRsin^(3)theta d theta)/(2)`
`B=(mu_(0)omegasigmaR)/(2)int_(0)^(pi)sin^(3)thetad theta=mu_(0)omegasigmRint_(0)^(pi//2)sin^(3)thetad theta`
or `" "vec(B)=(2)/(3)mu_(0)omega sigmaR(hat(k))`
`(b)` Magnetic moment due to elementary ring.
`d mu=(dI)pi r^(2)" ":." "mu=int(dI)pir^(2)`
`=2int_(0)^(pi//2)(omega sigmaR^(2)sin theta ) pi R^(2)sin^(2)theta d theta` ,brgt `=2pi R^(4)omega sigma int _(0)^(pi//2)sin^(3)theta d theta`
`=(4)/(3)pi sigma R^(4)omega`