The discussion revolves around the method of images in electrostatics, particularly focusing on the concept of the region of interest in relation to conducting shells and the placement of image charges. Participants explore how the placement of real charges and image charges affects the application of Poisson's equations.
The conversation is ongoing, with participants providing insights into the uniqueness theorem and its relation to the method of images. Some participants express confusion regarding the implications of image charge placement on Poisson's equation, while others clarify the conditions under which the uniqueness theorem applies.
There is a focus on the assumptions regarding the distribution of charge and the implications for the potential in different regions. Participants are considering the effects of real charges on the electrostatic field and the conditions necessary for applying the uniqueness theorem.
Ohh alright this makes sense. We were doing examples of having a point charge at x=a outside a sphere of radius R with constant potential and that the image charge should be placed at a position of x=R^2/a. This suggests that it’s possible for the image charge to be outside which is what made me confused because then wouldn’t rho in Poisson’s equation be different. But when the image charge is outside this means the real charge is inside the conductor so the region of interest is only r<R and Poisson’s equation isn’t changed.kuruman said:Say you have a spherical conducting shell of finite radius ##R## which is an equipotential. Put the origin at the center of the shell. Don't forget that you have another equipotential surface at infinity where the potential is usually defined to be zero. Now consider point P at ##r<R##. Is it inside or outside the equipotential surfaces at ##R## and at infinity?
The region of interest is the region where the real charge is placed at point Q. Any other point P that can be reached from Q without crossing a conducting boundary is in the region of interest. So if Q is at ##r<R##, the region of interest is ##r<R##; if Q is at ##r>R##, the region of interest is ##r>R.##
No. The uniqueness theorem for the region of interest only involves the real charges. In order to invoke the uniqueness theorem for the images you would have to assume things about the solution outside the region in which the problem is to be solved.lys04 said:Oh and by the Uniqueness theorem, the placement of image charges are unique too right?
You could reason this out yourself.lys04 said:Oh and by the Uniqueness theorem, the placement of image charges are unique too right?