# Finding electric field between overlapping surfaces [closed]

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.

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, centralcharge, Waffle's Crazy Peanut, Kyle KanosFeb 14 '14 at 14:22

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