How is the net charge distributed on the inner surface of the outer sphere?

[SparkimusPrime]

Excerpt from:

Beicher and Serway, "Physics for Scientists and Engineers with
Modern Physics, 5th edition"


Chapter 24, Problem 53, part e:

A solid insulating sphere of radius a carries a net positive charge 3Q, uniformly distributed throughout its volume. Concentric with this sphere is a conducting spherical shell with inner radius b and outer radius c, and having a net charge -Q, as shown...

(e) construct a spherical Gaussian surface of radius r, where
b < r < c, and find the net charge enclosed by this surface.

So far, I can safely assume that both charged surfaces can be treated as point charges independently. But together their net electric field is 2Qke / r^2 (where r is a radius of a Gaussian surface greater than length c) and their net charge is 2Q. The outer sphere (since its a conductor) has an internal electric field equal and opposite to the inner electric field (due to conservation of energy). The net electric field is base on a net charge of 2Q, but I'm not sure how to approach that charge distribution on the inside of the outer sphere. It seems that the excess charge is pulled to the inner surface of the outer sphere due to induction (caused by the inner electric force), so should the total charge for part e, be 2Q as well? Or is the net charge inside the outer sphere zero and the total net charge equal to the inner sphere's charge?

 

[cookiemonster]

Charge rests on the surface of a conductor.

cookiemonster

 

[gnome]

So far, I can safely assume that both charged surfaces can be treated as point charges independently.

Be careful: this only true for both of them TOGETHER in the region r > c, and for the inner (solid) sphere ALONE in the region a < r < b.

The outer sphere (since its a conductor) has an internal electric field equal and opposite to the inner electric field (due to conservation of energy).

No. The outer sphere has a CHARGE on its inner surface equal and opposite to the charge on the inner sphere. But the outer sphere does not produce an electric field inside itself to cancel the field from the inner sphere. So in the region a < r < b there is just the electric field produced by the inner sphere, and it is this field that causes negative charges to migrate to the inner surface of the outer sphere. (I'm pretty sure that it is incorrect to attribute this to conservation of energy, but I won't swear to it.)

 

[SparkimusPrime]

gnome:

No. The outer sphere has a CHARGE on its inner surface equal and opposite to the charge on the inner sphere. But the outer sphere does not produce an electric field inside itself to cancel the field from the inner sphere. So in the region a < r < b there is just the electric field produced by the inner sphere, and it is this field that causes negative charges to migrate to the inner surface of the outer sphere. (I'm pretty sure that it is incorrect to attribute this to conservation of energy, but I won't swear to it.)

So charge effects charge internal to the outer sphere but the electric fields extend beyond either.

This is a almost a direct contradiction of what my teacher told me about a previous problem.

Why isn't there an electric field produced internally to the out sphere? There's an electric force between the internal charges (inner and outer surface, negative and positive charges respectively) drawing the opposing charges together.

 

[gnome]

First of all, I'm not a physics professor - far from it - so I offer no guarantees, but I think what I'm saying is right. If not, I hope someone wiser will correct me.

Now ...

So charge effects charge internal to the outer sphere but the electric fields extend beyond either.

I don't understand what you're saying here. Please explain.

This is a almost a direct contradiction of what my teacher told me about a previous problem.

As to that, can you tell me what the problem was & what the teacher said. Maybe we're just misunderstanding each other here.

Why isn't there an electric field produced internally to the out sphere? There's an electric force between the internal charges (inner and outer surface, negative and positive charges respectively) drawing the opposing charges together.

By Gauss's law the electric field at any point on a Gaussian surface is proportional to the total charge inside/enclosed by that surface. If you use a Gaussian surface INSIDE the outer sphere, it is not enclosing any charge related to the outer sphere, so it has no electric field from that sphere.

Here's something interesting to think about: Suppose your outer conducting sphere had NO net charge of its own, but had, inside but not touching it, a sphere with a charge of +Q. Would you expect that outer sphere (with a net charge of 0) to exert a force on a charged particle placed inside it? Even though there would be an induced total charge of (almost*) -Q on the inner surface of the outer sphere? (And, of course, an induced charge of almost +Q on the outer surface).

*almost, except for the effect that the extra charged particle would have on it.

 

[SparkimusPrime]

quote:
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So charge effects charge internal to the outer sphere but the electric fields extend beyond either.
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I don't understand what you're saying here. Please explain.

Looking at that sentence, I'm not even sure what I meant, ^_^. I'll try to reconstruct my thought patterns as best I can. i suppose I was saying that the charges create an equal and opposite relationship, the charge on the inner sphere is balanced by the negative net charge on the inside of the outer sphere. But that the electric fields do not behave this way, they're additive, like vectors.

quote:
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This is a almost a direct contradiction of what my teacher told me about a previous problem.
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As to that, can you tell me what the problem was & what the teacher said. Maybe we're just misunderstanding each other here.

I'll quote it for you:

In the figure a very tiny spherical plastic shell of radius 15cm carries a uniformly-distributed negative charge of -8E-9C on its surface. An uncharged solid metal block is placed nearby. The block is 10cm thick, and its is 10cm away from the surface of the sphere.

(part b) Draw the electric field vector at the center of the metal block that is due solely to induction.

My teacher said that there is an internal electric field, created by induction, inside the metal block and that it was equal and opposite the electric field produced by the sphere. Does this only apply to neutral objects?

 

[gnome]

AHA!

" ... an internal electric field, created by induction, inside the metal block ... "

Inside; do you see that now?


" ... electric fields do not behave this way, they're additive, like vectors."

Electric fields are not like vectors. They are
vectors.
The negative charges that accumulate on the inner surface of the outer sphere have migrated there because of the electric field from the positive charges on the inner sphere. Consequently, an electric field is induced within the outer sphere (I don't mean enclosed by it; I mean in the material of the shell itself). An excess of negative charges now exists on the inner surface of the shell. And therefore, an excess of positive charges now exists on the OUTER surface of the shell due to the "absence" of the negative charges that migrated inwards. Equilibrium occurs when the strength of the field exactly at the inner surface of the shell coming from the inner (enclosed) sphere is exactly balance by the strength of the induced field coming from the outer surface of the shell.

 

[SparkimusPrime]

Equilibrium occurs when the strength of the field exactly at the inner surface of the shell coming from the inner (enclosed) sphere is exactly balance by the strength of the induced field coming from the outer surface of the shell.

So the if I'm interpreting this correctly, the charge on the inner surface of the outer sphere should be -3q and the surface charge on the outer sphere is 2q. The "internal" electric field (inside the material of the outer sphere) should be equal and opposite the field due to the inner sphere?

 

[gnome]

This part:

the charge on the inner surface of the outer sphere should be -3q and the surface charge on the outer (surface of the outer) sphere is 2q

is definitely correct.

But I have a problem with this part

The "internal" electric field (inside the material of the outer sphere) should be equal and opposite the field due to the inner sphere?

I know that's pretty much what I said, but now it doesn't sound quite right. First of all, there should be no electric field inside a conductor unless a current is flowing. Second, if we allow that, then Gauss's law doesn't work either. I think we can get out of trouble this way:

Let's say that there is a field in the body of the outer sphere only while the charges are migrating. Once equilibrium is established and the surface charges are -3q and +2q, there is no longer a field in the body of the conductor. There is a field in the entire region enclosed by the inner surface of the outer sphere (but not ON the surface itself). And of course there is a field outside the outer surface of the shell.

Now Gauss's law works. Imagine a Gaussian surface that is a sphere within the body of the outer shell, midway between the inner and outer surfaces. What is the total charge enclosed by this Gaussian surface? 0 Because it encloses the -3q charge on the inner solid sphere and the +3q charge on the inner surface of the outer sphere. Net enclosed charge: 0. Field: 0.

Why don't you run this by your teacher & see if she agrees?