Homework Help: Gauss' Law with Superposition Principle

1. Jul 22, 2012

Quantum1990

1. The problem statement, all variables and given/known data
A very long cylinder of radius R has positive charge uniformly distributed over its volume. The amount of charge is λ Coulombs per meter of length of the cylinder. A spherical cavity of radius R' < R, centered on the axis of the cylinder, has been cut out of this cylinder, and the charge in this cavity has been discarded. Find the electric field as a function of distance from the center of the sphere along the axis of the cylinder.

2. Relevant equations

Gauss' Law: Flux = q / ε
Superposition Principle: F(total) = ƩF (individual)

3. The attempt at a solution

I imagined a gaussian sphere for a r < R', which would enclose zero charge. Thus, by Gauss' law, the flux through the sphere would be zero, and thus the E field is zero withing the cavity. However, the answer is wrong, so my reasoning is flawed. Can someone help me understand? A section in the book mentions that I can imagine the cylinder without the cavity, then subtract the vector of a sphere with opposite charge density. I'm still not sure how to do this.

Any help would be great! I'm self studying the book, so there's not many people to ask.
1. The problem statement, all variables and given/known data

2. Relevant equations

3. The attempt at a solution

2. Jul 22, 2012

SammyS

Staff Emeritus
Hello Quantum1990. Welcome to PF !

A big clue here is to use superposition.

How about using a complete cylinder, with uniform positive charge density. Superimpose into that a sphere with uniform negative charge density.

3. Jul 23, 2012

Quantum1990

Thanks for the insight! But where does my argument break down? If there is no enclosed charge in the cavity, isn't there no electric field there?

Rereading, the only answer I can find is that I improperly used a Gaussian surface because the sphere has a radial inward field, but the cylinder has a radially outwards one.

4. Jul 23, 2012

TSny

The above statement is correct

This statement is not correct. What is the reasoning that you used to go from "flux is zero" to "E is zero"?