Well known problem of finding potential for metal sphere and point charge, solving with "mirror image" trick give you 2 different answers for the force, acting on point charge. One when we use potential of sphere and another, if we use electric field intensity of sphere. This is because mirror charge depends on point charge value and is changing when distance between charge and sphere changes. Any ideas what is a right way to solve this problem?
I can't explain better than it is explained in J. D. Jackson: Classical Electrodynamics, Ch. 2.2 Point Charge in the Presence of a Grounded Conducting Sphere.
This gives the right way (actually two ways, which, however, are shown to give the same result) for the force acting on the point charge.
One point, probably the one which troubles you, is not made as clear by Jackson as it should: When we compute the potential of the conducting sphere in the presence of a point charge q we get a scalar field which only in the limit of vanishing charge gives the force felt by a probing charge q'. In this limit we may neglect the mirror-charge of the probing charge against the mirror-charge of q.
It definitively gives n o t the force on the point charge q for which we included a mirror charge.
I agree with you. Technically we must integrate energy by point charge from zero to real value and we will get correct result.
@Eugene,
unfortunately I see no connection between what I wrote and what you answer.
BTW your text would be more readable if you could afford using articles which, unlike the situation in your language, are expected to be used in English (in my language too, so I cannot really estimate the degree of inconvenience which this would cause you).
Ok, I mean that Jackson don't give arguments, that force is just Coulomb force between point charge and it's image. I shown, that correct way is to calculate energy, needed to bring point charge by infinitesimal parts from infinity to it's position.
As for articles - it's hard for me to understand what can be understood incorrectly in my writing :(
In fact, it's well known problem, that point charge is idealization, in which you through away it's infinite self energy, ~q^2/a, a->0.
That is why I want to work with two spheres with finite energy, but there isn't simple solution for two spheres :(
I did not say that anything in your writing can't be understood. But reading syntactically incomplete texts causes a diffuse feeling of mental discomfort. Trying to avoid this can be seen as a requirement of politeness.
If you want to avoid the infinite self energy of point charges, you also could use small homogeneously charged spheres. Outside their own material extend these generate the same field as a point charge and so Jackson's arguments remain valid if only the charge q is completely outside the metal sphere.
PS. Now I see that what you write is true: Jackson gives no good argument that the force is just the Coulomb force between the point charge and it's image (since shifting the position of the point charge also shifts the image charge.) Thank you for having opened my eyes for this.
I'll try to do my best, writing complete texts.
I don't think, that homogeneously charged sphere will be physical enough for my problem. Main idea is that for metal spheres charge will be only on the surface and it's density will change depending on the distance between spheres.
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