a long cylindrical volume contains a uniformly distributed charge of density p(rho).find the electric field at a point inside the cylindrical volume at a distance x from its axis.

The uniformly distributed charge of density p means that the total charge in any volume element is given by

q=pxv

 

Consider an imaginary cylinder having radius x and length l.

The volume of this cylinder will be v= ?x²l

Due to Gauss law, the electric field will be mainly due to the electric charge inside the cylinder of radius x.

q=p?x²l

 

Now according to Gauss law,

flux=Q_inside/?

Flux for this cylinder will be only at the curved surface area, not the lateral surface area since the lateral surface area vector is perpendicular to the direction of the electric field.

 

So, E(2?xl)=p?x²l/?

 

Hence E=px/2?

 

This is the answer.