# A spherical shell of radius #R# and uniformly charged with charge #Q# is rotating about its axis with frequency #f#. Find the magnetic moment of the sphere?

##### 1 Answer

Uniformly charged spherical shell of radius

Hence it has surface charge density

#sigma=Q/(4πR^2)#

It rotates about its axis with frequency

Suppose the angular velocity

To find the magnetic moment of spinning shell we can divide

it into infinitesimal charges.

Using spherical polar coordinates

Here

#dq=sigmadS=sigmaR^2sinthetad thetadphi#

Current in the ring is given by

#dI=(dq)xxf=(sigmaR^2sinthetad thetadphi)xxomega/(2pi)#

#=>dI=(omegasigmaR^2sinthetad thetadphi)/(2pi)#

Now the magnetic dipole moment of ring is given by the expression in terms of position vector

#dvecm=1/2int_"ring"vecrxxvecJ#

#=>dvecm=1/2dIint_"ring"vecrxxdvecl#

where#dvecl# is element length of the ring.

Line integral becomes equal to the circumference of ring

#:.dvecm=dI(piR^2sin^2theta)hatz#

Inserting value of

#dvecm=(omegasigmaR^2sinthetad thetadphi)/(2pi)(piR^2sin^2theta)hatz#

#=>dvecm=(vecomegasigmaR^4sin^3thetad thetadphi)/2#

Total magnetic moment is integral with respect to both variables within respective limits

#vecm=(vecomegasigmaR^4)/2int_0^pisin^3thetad thetaint_0^(2pi)dphi#

#=>vecm=(vecomegasigmaR^4)/2xx4/3xx2pi#

#=>vecm=4/3pisigmaR^4vecomega#

Rewriting in terms of charge

#vecm=4/3pi(Q/(4πR^2))R^4vecomega#

#vecm=Q/3R^2vecomega#

.-.-.-.-.-.-.-.-

Reference figure for

Using online integral calculator

#int_0^pisin^3thetad theta=|cos^3theta/3-costheta|_0^pi=4/3#