#### alexmahone

(Any number of options may be correct.)

Three concentric spherical shells have radii r, 2r and 3r with charges q1, q2 and q3 respectively. Innermost and outermost shells are earthed. Then,

(a) q1+q3=-q2
(b) q1=-q2/4
(c) q3/q1=3
(d) q3/q2=-1/3

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#### alexmahone

I know that I must write the net potentials of the innermost and the outermost shells as zero.

Could someone please tell me how to write the net potential of a particular shell? Does it depend on the other shells as well?

#### borgwal

Show your work...which laws do you think you'll need?

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#### alexmahone

Show your work...which laws do you think you'll need?
$\phi=\frac{1}{4\pi\epsilon}\frac{q}{r}$

#### borgwal

That equation is correct for a single point charge (or a charge q distributed in a spherically symmetric way) once you choose your potential equal to zero at infinity.

Do you know Gauss' law?

Do you know how to calculate a potential once you know the electric field?

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#### alexmahone

That equation is correct for a point charge (or a charge q distributed in a spherically symmetric way) once you choose your potential equal to zero at infinity.

Do you know Gauss' law?
Yes I know Gauss' law.

#### alexmahone

Do you know how to calculate a potential once you know the electric field?
E=-dV/dr ?

#### borgwal

Apply Gauss' law to find E, then use E to find the potentials: not by integrating from infinity, but starting at the origin r=0.

#### borgwal

E=-dV/dr ?
Yes, kind of: you need the inverse relation since you can calculate E from Gauss' law.

#### alexmahone

Yes, kind of: you need the inverse relation since you can calculate E from Gauss' law.
I'll try your method tomorrow and tell you if I was able to solve it or not. Anyway, thanks for your help. :)

#### borgwal

Okay: you should find that exactly *one* of the answers is correct.

#### alexmahone

Okay: you should find that exactly *one* of the answers is correct.
Actually, the answer key tells me that (a), (b), (c) are correct.

#### borgwal

One condition follows from the given that the potentials on shells 3 and 1 are the same. The other two are really the same conditions then, and follow if you insist on having a potential zero at infinity (I didn't think you need that, but okay....you do).

And, did you use the approach I suggested?

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#### borgwal

Yes, but now you have to get the potential from the fields, start integrating from either r=0, or r=\infty.
Actually, in this problem r is a constant for some reason, so use a different variable for the variable distance.
(I think you mean your answer only for the constant distances given in the problem, but that is not enough).

#### alexmahone

Using Gauss' law I get

$E_14 \pi r^2=\frac{q_1}{\epsilon}$
$E_24 \pi (2r)^2=\frac{q_1+q_2}{\epsilon}$
$E_34 \pi (3r)^2=\frac{q_1+q_2+q_3}{\epsilon}$

#### alexmahone

Yes, but now you have to get the potential from the fields, start integrating from either r=0, or r=\infty.
Actually, in this problem r is a constant for some reason, so use a different variable for the variable distance.
(I think you mean your answer only for the constant distances given in the problem, but that is not enough).
Could you please complete the solution for me? I'm really stuck! :(

#### borgwal

No, I'm not gonna complete it for you.

For a variable distance r between R and 2R (I'm using R now for what is called r in the problem), the field is E(r)=q1/4\pi \epsilon r^2. You have to integrate this from R to 2R to get the potential difference between shells 2 and 1. Then you do the same for r between 2R and 3R: that gives you the potential difference between shells 3 and 1, which should be zero. That actually gives you one of the conditions a,b,c,d, but I won't tell you which one.

If you can't integrate, then you can't solve the problem.

yeah..keep going

#### alexmahone

$E=\frac{q}{4\pi\epsilon R^2}$
$E=-dV/dR$
$V_2-V_1=\int^{2r}_{r}\frac{q}{4\pi\epsilon R^2} dR$
$V_2-V_1=\frac{q}{4\pi\epsilon R} (r\ to\ 2r)$
$V_2-V_1=\frac{q}{4\pi\epsilon}(1/r-1/2r)$

#### borgwal

Now you have to put in the correct values of q...and then you're basically there!

#### borgwal

And the final point is: how do you get the outer shell to be at the same potential as at infinity: the way to do that is to have a zero field outside the third shell, and that gives you another condition.

#### borgwal

$V_2-V_1=\frac{q}{4\pi\epsilon}(1/r-1/2r)$
$V_3-V_2=\frac{q}{4\pi\epsilon}(1/2r-1/3r)$
The first line is correct, the second only if you put in the right value for q.

#### borgwal

$V_3-V_1=\frac{q}{4\pi\epsilon}(1/r-1/3r)$
So this is the wrong conclusion (see previous post)

#### borgwal

$V_2-V_1=\frac{2q}{4\pi\epsilon}(1/r-1/2r)$
$V_3-V_2=\frac{3q}{4\pi\epsilon}(1/2r-1/3r)$
It's not simply 2q and 3q, but some combinations of q1, q2, q3

#### borgwal

$V_2-V_1=\frac{q_2}{4\pi\epsilon}(1/r-1/2r)$
$V_3-V_2=\frac{q_3}{4\pi\epsilon}(1/2r-1/3r)$
almost correct!

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