How Do You Apply the Differential Form of Gauss's Law to an Infinite Cylinder?

[mdxyz]

Homework Statement

Calculate E inside and outside an infinite cylinder of uniform volume charge density using the differential form of Gauss's law.

Homework Equations

$$\nabla$$ E = $$\frac{p}{e0}$$

p = charge density

Divergence in cylindrical polars:

The Attempt at a Solution

I'm aware this is much easier using the integral form. I have no problem with calculating E field of various symmetrical shapes using the integral form. However I specifically have to use the differential form. I've never seen an example of this and have looked for quite a while, and I'm not really sure what I'm doing at all.

All I can think is that by symmetry, differential of E in terms of theta and z are zero, but this still leaves an awkward derivative of E in terms of r, and I'm not sure what to do at that point.

 

[kuruman]

You have to use the awkward derivative, sorry. I am going to change symbols and use r for the radial coordinate and ρ for the volume charge density which is constant in this case. You get

$$\frac{d(E_r r)}{dr}=\frac{\rho \; r}{\epsilon_0}$$

Can you solve this differential equation?

 

[mdxyz]

That's pretty much what I did, but I didn't think to multiply through by r. After that the integration is quite straight forward, and by varying the limits accordingly I get the same result as by the integral form for E field both inside and outside the cylinder.

Thanks a lot!