Question

Charge is distributed uniformly throughout the volume of an infinitely long solid Cylinder of radius R what is the electric field when r < Select one : O a . Zero O b . E = / 2 € d . E = pr / 2 € O e . E = / 2 €

Answer 1

Solution :

Let us consider the Gaussian surface that is in the form of a cylinder having a radius of r and a length of A which is $\text{coaxial with the charged cylinder}$.

The charged enclosed by the cylinder is given by,

$q=\rho V$       (here, V = $\pi r^2l$  is the volume of the cylinder)

  $=\pi r^2lp$    

If $\rho$ is positive, then the electric field lines moves in the radial outward direction and is normal to Gaussian surface which is distributed uniformly.

Therefore, total flux through Gaussian cylinder is :

$\phi=EA_{cyl}$

   $=E(2\pi rl)$

Now using Gauss' law, we get

$2\pi \epsilon_0rlE = \pi r^2lp$

or $E=\frac{\rho r}{2 \epsilon_0}$

Therefore, the electric field is $E=\frac{\rho r}{2 \epsilon_0}$

Hence, option (d) is correct.