An insulating sphere inside a conducting sphere questions

[jpruim]

I've been given a copy of my friend's midterm exam from this same class from last term, and decided to take a crack at it to help study. One question type in particular really messes me up and it looks like the following. How would I go about solving these in the future?

A solid insulating sphere with radius A is at the center of a concentric conducting spherical shell of inner radius B and outer radius C. After the sphere and shell are charged, it is found that an electric field at a distance D from the center such that A < D < B is Q1 N/C radially inward, while the field at a distance G such that G > C is Q2 radially outward.

Find

1. The charge on the insulating sphere

2. The electric field at a radius such that r < A

3. The net charge on the hollow conducting sphere

4. The electric field at a radius such that B < r < C

5. The charges on the inner and outer surfaces of the hollow conducting sphereOn the assignment given, there were actual values for A-E, Q1 and Q2. However, I’ve replaced them with variables so that I can get more conceptual explanations.

My attempt at part 1 is as follows
$$q_e = \vec{E}\cdot 4 \pi r^2 \epsilon_0$$
then set r = D, so that I get
$$q_e = \vec{E}\cdot 4 \pi D^2 \epsilon_0$$
Note that D =/= A.
Thus, you would plug in the distance D where the electric field E is to be found, and this would tell me the charge on the insulating sphere (in theory) because the sphere is enclosed within a sphere of radius D.My attempt at part 2 is
$$\vec{E} = \frac{kQr}{R^3}$$
where Q is the total charge (i.e. q_e found above), R is the sphere radius (R = A) and r is the electric field radius measurement.

Part 4 I know for sure is that it is 0 since r is within a conducting surface.However, I assume part 5 relies on part 3, which I have no idea how to calculate.
How would I go about solving part 3 and as well part 5? Are my answers for parts 1 and 2 correct?

 

[BvU]

Hello jp,

Quite an elaborate first (?) post !
PS: in PF homework sections you really do want to use the template. In fact we're not allowed to assist when it's missing ...

Could you check with the original problem statement for me:

After the sphere and shell are charged, it is found that an electric field at a distance D from the center such that A < D < B is Q1 N/C radially inward, while the field at a distance G such that G > C is Q2 radially outward.

It would be strange if Q₂ would be a given constant, unless the radius for G is also a given. Let me proceed under the assumption that Q₂ and $r_G$ are given.

I understand your attempt at making this generic, but using Q for an E field makes things hard to read ...

1) I can follow and I think it's correct. At $r_D$ the field is given as Q₁ (I don't understand your term "to be found") so you can write down the answer in terms of the givens.

2) is straight out of the book. Reason I give the link is that you don't describe how you get this answer. I'll guess: using the Gauss theorem. That same theorem helps you (twice) in part 3 ! If $|\vec E| = 0$ you know something of the charge inside a Gaussian surface. Idem If $|\vec E| = {\rm Q_2}$

and then 5) is a piece of cake.