Gauss' Law with Superposition Principle

[Quantum1990]

Homework Statement

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.

Homework Equations

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

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.

 

[SammyS]

Homework Statement

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.

Homework Equations

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

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.

Homework Statement

Homework Equations

The Attempt at a Solution

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.

 

[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.

 

[TSny]

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,

The above statement is correct

and thus the E field is zero withing the cavity.

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