Derive the expression for the electric field

[amiv4]

Homework Statement

A very long, solid cylinder with radius R has positive charge uniformly distributed throughout it, with charge per unit volume \rho

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 \rho.

The Attempt at a Solution

\rho/(r*2*pi*\epsilon_0)

 

[leright]

Do you know Gauss's law? For an appropriate gaussian surface (D-field perpendicular to the guassian surface and constant everywhere on the surface) DA = Q or D = Q/A, where Q is the total charge enclosed by the Gaussian surface and A is the surface area of the Gaussian surface.

Choose a cylindrical Gaussian surface and assume that a negligible amount of flux passes through the end caps, since it is a 'very long' cylinder (meaning the end cap area is small compared to the total area of the cylinder).

Q is the charge density multiplied by the volume of the Gaussian surface. V = h*pi*r^2, where 'r' is the distance from the axis of the cylinder. So Q = rho*h*pi*r^2.

A is the area of the gaussian surface (neglecting endcap area). A = 2*pi*r*h.

So, D = Q/A = (rho*r)/2

The relationship between D and E is E = D/ep_0, so E = (rho*r)/(2*ep_0)