Electric field inside non-conducting spherical shell

[Tiago3434]

Homework Statement

Figure 23-30 shows two nonconducting spherical shells fixed in place. Shell 1 has uniform surface charge density +6.0 µC/m2 on its outer surface and radius 3.0 cm. Shell 2 has uniform surface charge density -3.8 µC/m2 on its outer surface and radius 2.0 cm. The shell centers are separated by L = 14 cm. What are the magnitude and direction of the net electric field at x = 2.0 cm?

Homework Equations

The Attempt at a Solution

So this question is kinda two-fold:
First: my initial idea was that the field at x=2.0cm was just zero, since it was inside a uniformly charged sphere. I even thought about a gaussian surface inside shell 1, and because it didn't have any charges inside it, the electric field was zero. Upon seeing that the answer said that was wrong, I realized that the spheres are non conducting, so the field inside sphere 1 due to the charges of such sphere was going to be zero, but the field inside sphere 1 due to the charges of the sphere 2 was not null. Given that they are non conducting, then the charges on the surface wouldn't be able to reorganize themselves to nullify the field inside sphere 1. Is this explanation correct? It's just an idea for why my initial guess wasn't correct, and I don't know if it's true or not.

Second: why Gauss' law doesn't give the proper answer? Am I misapplying it?

 

[collinsmark]

Hello @Tiago3434,

Could you at least specify where the center of each spherical shell is located?

You mentioned that they are separated by L = 14 cm, but that's not enough information. For example, if the center of the first is at x = 100 cm, and the second at x = 114 cm, they are separated by 14 cm, but that would give a very different answer if their respective centers were at x = -7 and +7 cm.

Or another way of looking at it: with respect to the spherical shells, exactly where is x = 2.0 cm?

Maybe attaching the figure would help.

 

[Tiago3434]

Oh sorry @collinsmark, didn't realize that in the original post.
Shell 1 is centered around the origin, and shell 2 is centered around x=10cm.

 

[TSny]

Oh sorry @collinsmark, didn't realize that in the original post.
Shell 1 is centered around the origin, and shell 2 is centered around x=10cm.

Good, that helps.

I realized that the spheres are non conducting, so the field inside sphere 1 due to the charges of such sphere was going to be zero, but the field inside sphere 1 due to the charges of the sphere 2 was not null. Given that they are non conducting, then the charges on the surface wouldn't be able to reorganize themselves to nullify the field inside sphere 1. Is this explanation correct?

Yes, very good.

why Gauss' law doesn't give the proper answer? Am I misapplying it?

You are probably misapplying it. Usually, in applying Gauss' law you reach a step where you "pull E out" of the surface integral. Why is this step invalid in this problem?

 

[Tiago3434]

I think the answer to your answer is that the electric field is not constant, because it varies with distance, so I could not take it out of the integral. But then Gauss' Law tells me I only have to worry about the enclosed charges by the gaussian surface...

 

[TSny]

I think the answer to your answer is that the electric field is not constant, because it varies with distance, so I could not take it out of the integral.

Yes, that's right.

But then Gauss' Law tells me I only have to worry about the enclosed charges by the gaussian surface...

Gauss' law tells you that if you could do the integral representing the flux through the closed gaussian surface, then you would get a value equal to the $Q_{inside}/\varepsilon_0$, which would be zero for the gaussian surface that you chose. But this is not very helpful in solving the original problem, as far as I can tell.