How to find electrostatic interaction energy?

[gracy]

How to find electrostatic interaction energy between two uniformly charged conducting spheres /uniformly charged non conducting spheres or between a charge and uniformly charged spherical shell I mean what is general method of finding electrostatic energy in a given system.I don't have any specific problem based on it therefore I am not posting it in homework section.

 

[Dale]

Your best approach will be Jefimenko's equations.

 

[cnh1995]

How to find electrostatic interaction energy between two uniformly charged conducting spheres /uniformly charged non conducting spheres or between a charge and uniformly charged spherical shell I mean what is general method of finding electrostatic energy in a given system.I don't have any specific problem based on it therefore I am not posting it in homework section.

Force between the charges=kq₁q₂/r².
Interaction energy=force between charges*distance between them.
E=kq1q2/r.

 

[jbriggs444]

How to find electrostatic interaction energy between two uniformly charged conducting spheres /uniformly charged non conducting spheres or between a charge and uniformly charged spherical shell I mean what is general method of finding electrostatic energy in a given system.I don't have any specific problem based on it therefore I am not posting it in homework section.

It makes little sense to say that a sphere is both uniformly charged and conducting. If it is conducting, it will not remain uniformly charged. So if it is uniformly charged, it must not be conducting.

By "interaction energy", I assume that you want to know the electrostatic potential energy that the assembly of charged objects has when assembled as compared to the electrostatic potential energy that they would have if the objects were all infinitely far apart. In particular, you want to ignore the self-energy of each charged sphere or shell.

You appear to be interested in the simplest case -- two objects only and are probably interested in the case where the objects do not overlap/contain each other.

Why would you not use the formula for the potential energy of two point charges in that case?

 

[gracy]

It makes little sense to say that a sphere is both uniformly charged and conducting. If it is conducting, it will not remain uniformly charged. So if it is uniformly charged, it must not be conducting.

I meant surface charge distribution is uniform.Surface of a conducting sphere is uniformly charged.

 

[jbriggs444]

I meant surface charge distribution is uniform.Surface of a conducting sphere is uniformly charged.

So we will pretend for the sake of this exercise that the surface charge distribution remains approximately uniform even in the face of an externally imposed field. That's fine. Though it might have been simpler to skip the "conducting" adjective and go with "uniformly charged".

 

[cnh1995]

This might help..
http://www.feynmanlectures.caltech.edu/II_08.html

 

[gneill]

Gracy, if you allow for charge movement due to interaction of the fields of the spheres (i.e. if you assume conducting spheres) then the problem is not at all trivial. Relative sphere sizes and separations can have interesting effects on the behavior (where "interesting" can mean non-intuitive or complicated).

Remember how a charged object can be attracted to an uncharged (neutral) conductive object due to induced charges? Well induced charges can also affect the interaction of charged objects like conducting spheres and produce some somewhat counter-intuitive results (like similarly charged spheres actually attracting each other under certain circumstances).

For example, have a look here: Electrostatics of Two Charged Conducting Spheres

 

[gracy]

Force between the charges=kq1q2/r2.
Interaction energy=force between charges*distance between them.
E=kq1q2/r.

How can I apply it for two spheres and for one sphere and charge q?By treating two spheres as if whole charge of these spheres is concentrated in centre and then will multiply it by distance between the centers of the two spheres.This was for two spheres case for one sphere and one charge system we will assume the same for sphere whole charge of sphere is kept on centre and then for distance we will take distance between that charge and centre of the sphere.
Right?

 

[cnh1995]

we will assume the same for sphere whole charge of sphere is kept on centre and then for distance we will take distance between that charge and centre of the sphere.
Right?

I don't follow.. If there is only one charge present, the sphere will have self energy only. For interaction energy, there must be more then one charges present.

 

[gracy]

for one sphere and one charge system

one sphere along with charge q will form a system , charge q isn't alone!

 

[cnh1995]

one sphere along with charge q will form a system , charge q isn't alone!

Yes. I believe there will only be self energy since there is only one charge q present on the sphere.

 

[gracy]

no sphere is with it's charge say Q which is uniformly distributed on it's surface and there is also charge q

 

[cnh1995]

there is also charge q

At the center?

 

[gracy]

it can be anywhere

 

[cnh1995]

it can be anywhere

Ok.

for one sphere and one charge system we will assume the same for sphere whole charge of sphere is kept on centre and then for distance we will take distance between that charge and centre of the sphere.
Right?

Right.

 

[gracy]

And what about this

How can I apply it for two spheres and for one sphere and charge q?By treating two spheres as if whole charge of these spheres is concentrated in centre and then will multiply it by distance between the centers of the two spheres.

 

[jbriggs444]

And what about this

When you treat a charged sphere as if the charge were concentrated in its center, treat the charged sphere as if the charge were concentrated at its center. Do not complicate things. Yes, the potential is still given by $k\frac{q_1 q_2}{r}$ where r is the distance between the centers.

The caveats is that this only applies as long as the charge on the sphere(s) is uniform (or at least spherically symmetric) and that the two objects do not overlap. If your charge $q_1$ is in the interior of a sphere with charge $q_2$ then the above formula will not apply.

[All of which was already said in post #4 above]

 

[cnh1995]

And what about this

I believe that's right too.

 

[nasu]

As long as the charge distribution is uniform on the spheres (or just have spherical symmetry, you can treat them as point charges in the center, when dealing with fields outside the spheres.
The condition in bold letters will not apply in most cases with conductive spheres in external field of other spheres or point charges.

 

[gracy]

So how am I going to apply formula mentioned in post #3 in system of two spheres or in system of one charged sphere and charge q?

 

[jbriggs444]

By treating the spheres as if they were point charges with all the charge at their center.

Edit: And realizing that if they are conducting spheres that the formula in post #3 does not apply and that, accordingly, it cannot be used.

 

[gracy]

By treating the spheres as if they were point charges with all the charge at their center.

But We cannot make that assumption

 

[jbriggs444]

But We cannot make that assumption

Under what circumstances may we not treat the spheres that way?

 

[gracy]

Under what circumstances may we not treat the spheres that way?

I think we can only treat the sphere that way in case of isolated sphere and non-conducting sphere with its charges fixed in place.

 

[jbriggs444]

I think we can only treat the sphere that way in case of isolated sphere and non-conducting sphere with its charges fixed in place.

Right. So what is your question?

 

[gracy]

So how am I going to apply formula mentioned in post #3 in system of two spheres or in system of one charged sphere and charge q?

 

[jbriggs444]

So how am I going to apply formula mentioned in post #3 in system of two spheres or in system of one charged sphere and charge q?

That is the wrong question.

Before you get that far you have to ask "Can I apply the formula mentioned in post #3 to easily determine the force between two charged, conducting spheres".

 

[gracy]

Can I apply the formula mentioned in post #3 to easily determine the electrostatic interaction energy between two charged, conducting spheres?

 

[jbriggs444]

Can I apply the formula mentioned in post #3 to easily determine the electrostatic interaction energy between two charged, conducting spheres?

No.

 

[gracy]

Did you notice the bold letters?

 

[jbriggs444]

I noticed them but discounted them because they were meaningless and substituted "electrostatic potential energy" in their place.

 

[jbriggs444]

If you re-read this thread, you may notice that in post #8, gneill said (paraphrasing), "with conducting spheres, it's complicated and not intuitive". That is an extremely strong hint that you cannot blindly apply the formula $PE = k\frac{q_1 q_2}{r}$ to the case of two charged conducting spheres.

What you could do would be to use the formula $PE = k\frac{q_1 q_2}{r}$ and integrate it over a distribution of charges on the two charged spheres, searching for the distribution that gives the lowest possible result. But that is more easily said than done.

Edit: There is a tricky piece in that process. We had assumed that you are concerned only with the potential energy of the two charged objects in relation to each other and not with the potential energy of each objects's charge distribution with itself. But the minimization process described above must take into account each object's own self potential energy. That means that the result depends on exactly what question you want answered.

Edit: Unless I am mistaken, it gets even worse in the case of more than two bodies because one might be faced with the possibility of multiple local minima. A naive optimization approach based on incremental relaxation may not lead to the global minimum.

 

[gracy]

how he took interaction energy between charge q and a charged sphere

 

[jbriggs444]

how he took interaction energy between charge q and a charged sphere

Without looking at that video -- did he assume a conducting sphere? Or, instead, a sphere with a spherically symmetric charge distribution?