A very long, solid cylinder with radius R has positive charge uniformly distributed throughout it, with charge per unit volume r. (a) Derive the expression for the electric field inside the volume at a distance r from the axis of the cylinder in terms of the charge density r. (b) What is the electric field at a point outside the volume in terms of the charge per unit length l in the cylinder

Explanation:

From the given information.

The charge density of the cylinder can be computed as:

where:

= total charge on cylinder

= density of the cylinder

net volume on cylinder

Considering the charge enclosed by the Gaussian surface; we have:

Q = ρV

Now, determining the volume of the cylinder at a distance r from the axis of the cylinder as follows:

Here;

are in the same direction;

If we replace with and q with ; Then:

By rearrangement;

(b)

Using the same formula:

where;

are in the same direction;

If we replace with and with q;

Then:

Replacing for .

From above = the charge per unit length

∴

a. are