Question

An infinitely long cylinder of radius R has its axis along z-axis. It has a uniform volume charge density and it is rotating with constant angular velocity. Find the magnetic field produced by it everywhere.

Answer 1

Infinite long cylinder

$\smallint B_o.dl =\mu_0\Sigma i$

$$\smallint B_o dlcos\theta=\mu_oI=B_o(2\pi r)=\mu_0I$$

$B_o=\frac{\mu_oI}{2\pi r}$

$I=\frac{wq}{2 \pi},q=\rho v$

$I=\frac{\rho v w}{2 \pi}$

$B_0=\frac{\mu_0 }{2 \pi r}\times \frac{\rho vw}{2 \pi}=\frac{\mu_0 }{4\pi ^2r}\rho vw$

Similarly surface r=R

$B_s=\frac{\mu_0 }{2 \pi R}\times \frac{\rho vw}{2 \pi}=\frac{\mu_0 }{4\pi ^2R}\rho vw$