Find the interaction potential energy

[ajaysabarish]

Homework Statement

2 concentric shells are placed with inner shell having charge q and outer charge -q, with radii a and b respectively.

Homework Equations

don't know

The Attempt at a Solution

the question asked me to find self energy of the 2 shells and interaction potential energy,i found the self energy but i didn't know what is in interaction energy and how to find it,please help

 

[TSny]

The meaning of "interaction potential energy" might be open to some interpretation. Did you give the complete statement of the problem "word for word"?

The system of two shells has a total electrostatic energy U_tot.

Each shell has a self energy: U_self,a for the inner shell and U_self,b for the outer shell.

I would think that the interaction energy would be the difference between the total energy of the system and the self energies. That is, U_interaction = U_tot - (U_self,a +U_self,b).

Equivalently, this interaction energy is the sum of the electrostatic energy of every pair of charge elements where one element of the pair is on the inner sphere and the other element is on the outer sphere.

 

[ajaysabarish]

The meaning of "interaction potential energy" might be open to some interpretation. Did you give the complete statement of the problem "word for word"?

The system of two shells has a total electrostatic energy U_tot.

Each shell has a self energy: U_self,a for the inner shell and U_self,b for the outer shell.

I would think that the interaction energy would be the difference between the total energy of the system and the self energies. That is, U_interaction = U_tot - (U_self,a +U_self,b).

Equivalently, this interaction energy is the sum of the electrostatic energy of every pair of charge elements where one element of the pair is on the inner sphere and the other element is on the outer sphere.

i don't think so,sir,this is the answer given(k(-q)/b)(a/b)q
and it is also given that the total potential energy stored in these shells is sum of self energy of shells and interaction energy between the shells

 

[TSny]

i don't think so,sir,this is the answer given(k(-q)/b)(a/b)q

Is this the answer for the "interaction energy"? It looks strange. It goes to zero as the radius a goes to zero.
I get a different expression for the interaction energy that is actually independent of a (as long as a is less than b).

and it is also given that the total potential energy stored in these shells is sum of self energy of shells and interaction energy between the shells

OK, that's equivalent to my interpretation in post #2.

 

[Vibhor]

i don't think so,sir,this is the answer given(k(-q)/b)(a/b)q

This is the interaction potential energy when outer shell has charge -q and inner shell is earthed .

 

[TSny]

This is the interaction potential energy when outer shell has charge -q and inner shell is earthed .

Vibhor: Yes, I agree. That's interesting. Thanks!
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ajaysabarish, may I ask you to please write out the question word for word as it was given to you?

 

[Vibhor]

@TSny , the self energy of a point charge is infinite . Could you please explain ?

And why do we not consider it while finding the self energy of any continuous charge distribution ( for eg . charged shell in the OP ) ?

 

[TSny]

@TSny , the self energy of a point charge is infinite . Could you please explain ?

Yes, the self energy of a point charge is infinite. Are you asking for a reason why the self energy is infinite for the point charge?

And why do we not consider it while finding the self energy of any continuous charge distribution ( for eg . charged shell in the OP ) ?

In classical electromagnetism, a continuous distribution of charge is a convenient fiction where we treat the charge as spread out continuously in a mathematical sense. We should not think of the continuous distribution as made up of a bunch of point charges.

The self energy of a finite charge distribution is the (hypothetical) work required to assemble the distribution starting with the finite charge spread out as an "infinitely diffuse cloud" of charge such that the electric field is initially zero everywhere in space. A point charge is the extreme limit of squeezing the initial cloud to a mathematical point of zero volume. Calculation shows that this would take an infinite amount of work to accomplish.

If you squeeze the cloud to make a line of charge of finite length and zero cross-sectional area, it also takes an infinite amount of work. So again the self energy is infinite.

But if you squeeze the cloud to make a finite surface charge of zero thickness, it only takes a finite amount of work.

So, for the surface charge on a sphere, we can calculate the self energy and find that it is finite.

 

[Vibhor]

Are you asking for a reason why the self energy is infinite for the point charge

Yes . Please explain .

 

[Vibhor]

The teacher mentioned it without giving any explanation . Could you please explain why the self energy of point charge is considered infinite .

 

[BvU]

Well, take two half charges and try to bring them together to form one point charge. How much work is needed ?

 

[Vibhor]

Well, take two half charges and try to bring them together to form one point charge. How much work is needed ?

I somehow feel this is not correct reasoning . By this reasoning self energy of continuous charge distribution like shell would also be infinite .

 

[BvU]

No: in that case dq goes to zero if dr goes to zero.

 

[TSny]

Another approach is to consider the self energy of a sphere of radius R with a charge Q spread uniformly over the surface. Or, you could take the charge Q as spread uniformly throughout the volume of the sphere. Either way, you will find that the self energy is finite and inversely proportional to R. So, if you let R approach zero while Q remains fixed, you can see what happens if you try to create a point charge.

 

[Vibhor]

Another approach is to consider the self energy of a sphere of radius R with a charge Q spread uniformly over the surface. Or, you could take the charge Q as spread uniformly throughout the volume of the sphere. Either way, you will find that the self energy is finite and inversely proportional to R. So, if you let R approach zero while Q remains fixed, you can see what happens if you try to create a point charge.

Great ! You said "Another approach" .What is the first approach you are referring to ? Are you referring to BvU's Post 11?

 

[TSny]

You said "Another approach" .What is the first approach you are referring to ? Are you referring to BvU's Post 11?

Yes, I was referring to BvU's line of argument.