An infinitely long nonconducting cylinder of radius R = 2.00 cm carries a uniform volume charge density of Calculate the electric field at distance r = 1.00 cm from the axis of the cylinder. (ε0 = 8.85 × 10-12 C2/N ∙ m2)

where is the volume charge density.

Explanation:

Since the volume charge density is not given, I will just call it .

Gauss's law says

To evaluate this integral, we choose a cylindrical Gaussian surface which is concentric with the nonconducting cylinder.

Let be the radius of the Gaussian cylinder, and be its length, then the right side of the Gauss's law gives

Notice that we only count the lateral area of the cylinder because the sides of the cylinder are perpendicular to , and therefore, give .

Now we turn to the right side of Gauss's law.  

Since , the charge enclosed by the Gaussian surface is

which makes the right side

Thus the Gauss's law becomes

simplifying a and solving for we get:

Putting in and , we get: