Superposition principle for potential?

[timnswede]

Homework Statement

A solid, conducting sphere of radius "a" and charge -Q is concentric with a spherical conducting shell of inner radius "b" and outer radius "c". The net charge on the shell is +3Q. Take the zero of electric potential to be at some point at infinity.
a.) Use Gauss's law to find the charge on the inner and outer surface of the shell.
b.) Use the superposition principle for potential to find the potential at all points in space.
c.) Use the results of part b) to find the electric field at all points in space.

Homework Equations

E= -dv/dx

The Attempt at a Solution

For part a) I took a gaussian surface between the radius b and c, so the E field is zero. Therefore Qinner-Q=0, so Qinner=Q. So outer charge must be 2Q then.
I am completely stuck on part b). What I think I am supposed to do is get the net charge, 2Q, and multiply that by Ke, and divide it by the radius. So basically the answer will be 2QK/r, where r is the radius of the point in space I am trying to find, but that seems really wrong, especially when I am trying to find the E fields from that, as taking the negative derivative will just give me the same result. How should I do this?

 

[mfb]

What I think I am supposed to do is get the net charge, 2Q, and multiply that by Ke, and divide it by the radius. So basically the answer will be 2QK/r, where r is the radius of the point in space I am trying to find

That is correct outside the outer shell, it is not correct everywhere.

as taking the negative derivative will just give me the same result

What do you mean?

 

[timnswede]

That is correct outside the outer shell, it is not correct everywhere.
What do you mean?

Well what my professor said to me as an example find the potential between the outer radius c and the inner radius b. He said that the answer to that would be the net charge of the whole system, 2Q, divided by the radius r, from the center to a point inbetween the inner and outer radius multiplied by Ke. So if I am going based off that then I will always have 2KQ over whatever the radius is, which seems wrong to me. It also makes it so if I try to get the E-field between, say, the inner and outer radius by the formula E= -dv/dx, I will not get zero, which the E-field should be.. I think I am just really not understanding how to use the superposition principle.

 

[mfb]

Well what my professor said to me as an example find the potential between the outer radius c and the inner radius b. He said that the answer to that would be the net charge of the whole system, 2Q, divided by the radius r, from the center to a point inbetween the inner and outer radius multiplied by Ke.

For a specific radius, this happens to be right, but this is a weird approach.

Potentials are continuous - they don't make jumps. You know the potential for all points outside the outer surface, so you know the potential at the surface.
Now you have two alternative ways to proceed, both will lead to the potential between b and c:
a) What is the electric field in the conducting shell? What does that tell you about the potential?
b) Find a more general expression for the potential of a charged sphere where the potential at infinity can be some other value. Try to apply it to the region between b and c.

 

[timnswede]

For a specific radius, this happens to be right, but this is a weird approach.

Potentials are continuous - they don't make jumps. You know the potential for all points outside the outer surface, so you know the potential at the surface.
Now you have two alternative ways to proceed, both will lead to the potential between b and c:
a) What is the electric field in the conducting shell? What does that tell you about the potential?
b) Find a more general expression for the potential of a charged sphere where the potential at infinity can be some other value. Try to apply it to the region between b and c.

The electric field in the conducting shell would be zero, so V would be the same as the V on the surface correct? So 2QK/c?
I'm not sure how to go about doing it the way you describe in b) though.

 

[mfb]

The electric field in the conducting shell would be zero, so V would be the same as the V on the surface correct? So 2QK/c?

Right.

I'm not sure how to go about doing it the way you describe in b) though.

This will be needed for the region between b and a.

Potentials describe the same physics even if you add a constant to it because the potentials itself are not physical reality, only potential differences are real. You can use this constant to find a solution for the range between a and b. It has to follow the law for the field around a charge, but it also has to match the potential at a radius of b.

 

[timnswede]

Right.
This will be needed for the region between b and a.

Potentials describe the same physics even if you add a constant to it because the potentials itself are not physical reality, only potential differences are real. You can use this constant to find a solution for the range between a and b. It has to follow the law for the field around a charge, but it also has to match the potential at a radius of b.

I'm really not sure how to do that at all. How would I get this constant?

 

[mfb]

It comes from the part where you get the potential at r=b right (=continuous).

 

[timnswede]

It comes from the part where you get the potential at r=b right (=continuous).

What do you mean with "right"? And the potential at r=b, that would be 2QK/b right?

 

[mfb]

"right" as in "not wrong".

And the potential at r=b, that would be 2QK/b right?

Yes, and that potential should not make a "jump" at r=b if you go to the formula for the next region.

 

[timnswede]

"right" as in "not wrong".
Yes, and that potential should not make a "jump" at r=b if you go to the formula for the next region.

So then when r=a the potential would be 2QK/a which is the same when r<a. If that is right all I have left is a<r<b which seems to be the trickiest one. I calculated the electric field using a gaussian surface and got the e-field to be -QK/r^2. So the potential between a and b should be -QK/r right? Now how to find that using the superposition principle I still don't get. And what do you mean by the formula for the next region?

 

[mfb]

So then when r=a the potential would be 2QK/a

No. How did you get that result?
You'll need the potential between b and a to find the potential for r<a.

So the potential between a and b should be -QK/r right?

That does not work, the potential at r=b would not be continuous. -QK/r reproduces all potential differences, but the potential has to be something else.

And what do you mean by the formula for the next region?

Exactly the point you are working on.

 

[timnswede]

No. How did you get that result?
You'll need the potential between b and a to find the potential for r<a.

That does not work, the potential at r=b would not be continuous. -QK/r reproduces all potential differences, but the potential has to be something else.
Exactly the point you are working on.

For potential between a and b, would it work if I made the problem into 3 separate spheres, so one sphere of radius c, one of radius b, and one of radius a? so when r<c the potential would be 2QK/c, when r<b the potential would be QK/b, and when r>a the potential would be -QK/r? So the adding all those together the potential between a and b would be QK(2/c + 1/b - 1/r)

 

[mfb]

when r<b the potential would be QK/b

You already calculated that there is an electric field - the potential cannot be constant. Also, your solution would be not continuous.

Adding them would give a constant potential everywhere, which is certainly not true either.

You found the potential for $r\geq c$ and for $c \geq r \geq b$. You found the derivative of the potential for $b \geq r \geq a$. If you know the potential at one point (b=r) and its derivative, how can you find the potential (between a and b)?

 

[timnswede]

You already calculated that there is an electric field - the potential cannot be constant. Also, your solution would be not continuous.

Adding them would give a constant potential everywhere, which is certainly not true either.

You found the potential for $r\geq c$ and for $c \geq r \geq b$. You found the derivative of the potential for $b \geq r \geq a$. If you know the potential at one point (b=r) and its derivative, how can you find the potential (between a and b)?

Would Vb-Va= -∫-QK/r^2 work? Starting to feel even more lost.
Also earlier when I said that the potential at r=b is 2QK/b, are you certain that is right? Because if I do Vc-Vb= -∫Edr, the right side will equal zero, so Vc=Vb which would mean Vb=2QK/c

 

[mfb]

Integrating the electric field is a good idea, yes.

Also earlier when I said that the potential at r=b is 2QK/b, are you certain that is right?

Oops, that should be c in the denominator, as you wrote in post 5.

 

[timnswede]

OK for the integral the limits are 'a' to 'b' so -∫-QK/r^2 = QK/a - QK/b. But then wouldn't I still need the derivative of that to be equal to -QK/r^2? Since both of those are constants that wouldn't work.

 

[mfb]

You found the potential at "a" by setting one border to "a" - but you can choose a different integration range.

 

[TSny]

The question asks you to use superposition to get V in all the regions of your problem. The charge in your system is located on three separate spherical surfaces. Start by considering V due to a charge q uniformly spread over a single spherical surface of radius R. What are the expressions for V outside and inside the surface? Once you have that, it should be easy to use superposition to get the answer for V in the different regions of your problem.

 

[timnswede]

You found the potential at "a" by setting one border to "a" - but you can choose a different integration range.

Well if I took it from r' to b, then I would end up with QK/r' - QK/b. The negative derivative of that would not give me the correct answer for the electric field. If I set it from a to r' it would work, but that doesn't seem right either does it?

 

[timnswede]

The question asks you to use superposition to get V in all the regions of your problem. The charge in your system is located on three separate spherical surfaces. Start by considering V due to a charge q uniformly spread over a single spherical surface of radius R. What are the expressions for V outside and inside the surface? Once you have that, it should be easy to use superposition to get the answer for V in the different regions of your problem.

Well V outside would be Kq/r and when it is on the surface it would be Kq/R and inside it would also be Kq/R. In a post earlier I thought of using three different spheres, with radius a, b, and c. But maybe that is not what you meant? This problem is making me feel really dumb

 

[TSny]

Well V outside would be Kq/r and when it is on the surface it would be Kq/R and inside it would also be Kq/R. In a post earlier I thought of using three different spheres, with radius a, b, and c. But maybe that is not what you meant?

Yes, That's what I meant. Maybe I'm not seeing why this is causing a problem. Since you know the general result for a single spherical surface, it seems to me that it should be easy to get the answer for 3 spherical surfaces using superposition.

 

[timnswede]

Yes, That's what I meant. Maybe I'm not seeing why this is causing a problem. Since you know the general result for a single spherical surface, it seems to me that it should be easy to get the answer for 3 spherical surfaces using superposition.

Would this be the right way of doing it for when we are looking for the potential outside the radius c? 2KQ/r (for the sphere with radius c) + QK/r (sphere with radius b) + -QK/r (sphere with radius a). So I would get 2KQ/r which is correct.
So then for when the radius is in between a and b: The potential due to c is 2QK/c, b: QK/b, and a: -QK/r. So all added together -QK/r + 2QK/c + QK/b. That is what I put before, but I was told it was wrong.

 

[TSny]

I think your answer is correct.

 

[TSny]

Earlier you wrote: "so when r<c the potential would be 2QK/c, when r<b the potential would be QK/b, and when r>a the potential would be -QK/r? So the adding all those together the potential between a and b would be QK(2/c + 1/b - 1/r)".

This is the correct answer, but the way you stated it was perhaps not entirely clear. For example, when you wrote "so when r<c the potential would be 2QK/c", it's maybe not clear whether you were referring to the total potential or just the potential due to the charge of 2Q. Similarly when you said "when r<b the potential would be QK/b", etc.

Anyway, I think you have a correct answer.

 

[timnswede]

OK thank you for the help you guys.