Electrostatic uniqueness theorem

[facenian]

There is one thing I don't understand about this and is that besides the Dirichlet and Neumann conditions there seems to be a third one which is important when the method of images is used and is never mentioned. The problem is that Newmann condition requires especification of $\frac{\partial\phi }{\partial \eta}$ at each point on the surface while in electrostaics problems one many times knows two conditions,
a) the potencial is constant on the limit surfaces, and b) you don't know the normal derivative of phi at each point on the surface but you know the total charge of the conductor,ie, $\oint_S\frac{\partial\phi }{\partial \eta}ds$ in which case the uniqueness theorem also seems to follow easily.
Can someone please tell me where I'm wrong or if this is not the case why this
importan observation is never even mentioned in texts?

 

[Meir Achuz]

That case is covered in Sec. 3.1.1 of Franklin "Classical Electromagnetism", which states
"the electric field is uniquely specified ... if either the potential or charge is specified on each conductor."

 

[facenian]

Thank you, I will look for that