Question

An infinitely long right circular cylinder has radius ?. There is a non-constant cylindrically symmetric volume charge density ?(?), where ? is the (radial) distance from the axis of the cylinder, given by ?(?) = ((?0*?)/?)sin((2??)/?), where ?0 is a constant.

1. Consider a concentric cylinder with radius ? and length ?. Compute the total charge ?(?) inside the cylinder for 0 < ? < ? and for ? > ?.

2. Go back to the infinite cylinder setup and compute the electric field ?(?) for all ? > 0. Explain why the electric field does not have an azimuthal component, i.e., ̂? ⋅ ? = 0. In this problem ,̂? ̂? and ̂? are the unit vectors associated with cylindrical coordinates (?, ?, ?).

Can you please write out a detailed solution. Thank you.

Answer 1

The direction of the electric field is radial. Hence in vector form,

if r<R

and 0 if r>R