A thin circular disk of radius `R` is uniformly charged with density `sigma gt 0` per unit area.The disk rotates about its axis with a uniform angular speed `omega`.The magnetic moment of the disk is :

Options

A. `pi R^(4) sigma omega`
B. `(pi R^(4))/2 sigma omega`
C. `(pi R^(4))/4 sigma omega`
D. `2 pi R^(4) sigma omega`

Answer

c

Explanation

Let us select a ring of radius x and thickness dx on the surface of disc. Charge on this ring segment can be written as follows:
`dq = sigma 2pi x dx `
` di = (sigma xx 2 pi x d x)/((2pi)/omega) = sigma omega x d x `
Magnetic dipole moment for this ring can be written as follows:
` d mu = (di) xx A = sigma x d x xx pi x^(2) = sigma omega pi x^(3) dx`
`mu = underset(0) overset(R) int sigma omega pi x^(3) dx = 1/4 sigmaomegapi R^(4) `
Hence , option (c ) is correct.