Ring to infinite plate capacitance calculation I don't know how to calculate the capacitance between a conductive ring and a conductive infinite plate.
The ring has internal radius R1 and external radius R2 and is placed on a parallel plane with distance D from the infinite plate.
I would appreciate if anyone can help me in this matter. 


 A: What you are asking is far from simple. There is no obvious Gaussian Surface which will make this problem easy, because the electric field does not follow a simple pattern as it does in the case of parallel plane or coaxial cylindrical or concentric spherical conducting plates. The ring (actually an annulus) is conducting and has a finite width, so the distribution of charge will not generally be uniform, though it will have circular symmetry. You could get a numerical answer by solving Laplace's Equation with boundary conditions, but you seem to be asking for an algebraic formula.
A simpler scenario is to assume that the annulus is infinitesimally thin - ie $R_1\approx R_2$. The obvious solution uses the Method of Image Charges. The ring induces a mirror image with opposite charge in the plate; the scenario is equivalent to 2 rings carrying charges $-Q, +Q$ a distance $2D$ apart. The solution is still far from easy.
The steps involved are :


*

*Calculate the electric field at any off-axis point due to a charged ring. 

*Hence find the force between two identical coaxial charged rings. 

*Calculate the work done to move a ring of charge $+\delta q$ from ring 2 carrying charge $-q$ to ring 1 carrying charge $+q$, which are distance $2D$ apart. 

*Integrate again, from $0$ to $Q$, to find the total work done $U$ to create the charged rings $-Q,+Q$.

*Use $U=\frac{Q^2}{2C}$ to find capacitance.
