question_answer

An infinite cylinder of radius r with surface charge density \[\sigma \] is rotated about its central axis with angular speed co. Then the magnetic field at  any point inside the cylinder is

A) \[{{\mu }_{0}}\sigma \omega {{r}^{2}}\]                     

B) \[\frac{{{\mu }_{0}}\omega r}{\sigma }\]

C) \[{{\mu }_{0}}\sigma \omega r\]

D) \[{{\mu }_{0}}\sigma {{\omega }^{2}}r\]

Correct Answer: C

Solution :

[c]

\[i=\frac{dq}{t}=\frac{\sigma \times 2\pi rl}{2\pi /\omega }=\sigma r\ell \omega \]

Using Amphere?s law, we can write \[B\ell ={{\mu }_{0}}i\]

or \[B\ell ={{\mu }_{0}}\left( \sigma r\ell \omega  \right)\]

\[\therefore B={{\mu }_{0}}\sigma r\omega \]