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The problem is:

A sphere with radius R is centered at the origin, an infinite cylinder with radius R has its axis along the z axis, and an infinite slab with thickness 2R lies between the planes z=−R and z=R. The uniform volume densities of these objects are ρ1, ρ2, and ρ3, respectively. The objects are superposed on top of each other; the densities add where the objects overlap.

How should the three densities be related so that the electric field is zero everywhere throughout the volume of the sphere? What do you think the ratios $\frac{\rho_{1}}{\rho_{2}}$, $\frac{\rho_{1}}{\rho_{2}}$ and $\frac{\rho_{1}}{\rho_{2}}$ are equal to?

Hint: Find a vector expression for the field inside each object, and then use superposition.

enter image description here

The way I approached the problem:

Find individual Electric fields and add them up to 0. Find the relation between the charge densities.

To do that, I used Gauss' law. I bounded the region (the sphere) and that gives me $E_{sphere} = \frac{\rho_{1}R}{3\epsilon_{o}}$. However, because my region of interest is the sphere, I used the same bounding surface for the cylinder and the slab as well thereby getting the equations $E_{cylinder} = \frac{\rho_{2}R}{3\epsilon_{o}}$ and $E_{slab} = \frac{\rho_{3}R}{3\epsilon_{o}}$

Then, since the $E_{net} = 0$, $$E_{sphere} + E_{cylinder} + E_{slab} = 0$$ which, when simplified, gives $$\rho_{1} + \rho_{2} + \rho_{3} = 0$$

However, I'm not sure if a) what I've done so far is correct, b) how to proceed from here to calculate the ratios of the charge densities.

Any help would be really appreciated. Thank you.

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closed as off-topic by John Rennie, Brandon Enright, Dimensio1n0, Waffle's Crazy Peanut, Kyle Kanos Feb 14 at 14:22

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1 Answer 1

What you have done is calculating the densities for which the net charge density is zero. That does not mean the field is zero.

You may be confused with the the principle that there is no field inside a conductor (and hence no net charge). This is not the case here.

What the hint implies is that you need to calculate the field each object produces inside itself (this may not be the same as outside the object!), and then add up these fields and pick the charge densities so that there is no net field inside the sphere.

First sketch a graph for what you think the field strength is as you go through the objects. Do this for each object separately.

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