Capacitance between point charge and finite plate

[Alex555]

I need help trying to set up this problem. I want to find the capacitance between a point charge and a finite plate (or disk) as the point moves from above the center of the plate to some distance off the plate. I have been able to simulate this problem using FEM, however, there should be a way to solve this by using Gauss's law. In the simulated model I used a potential difference of 1 V between the plate and point.
I don't know if this forms an actual capacitor or if it is a self-capacitance of the plate with the point charge inducing a surface charge distribution on the plate. Any insight into this problem would be helpful.

 

[JoePhysics]

Well, a capacitor requires two conductors. I don't think a point charge can take the place of an actual conductor. However, you can speak of the (self-)capacitance of a conductor, such as your plate. Now while your point charge will induce a surface charge distribution in the plate such that the E-field inside the plate is 0 and the boundary condition for the electric field is satisfied at the surface, the total charge is going to be zero assuming you started with an electrostatically neutral plate, so I am not sure how you would go about calculating its self-capacitance if the net charge is vanishing.

The reason I'm not sure the point charge and the plate form a capacitor is as follows: assume the point charge is very close to the plate so that it looks like an infinite plane, assume it has a uniform surface charge density such that the total charge on the plate is that of the point charge but the other sign, and then calculate the potential difference between the two. But the point charge has an electric field that goes as $1/r^2$, so the integral would blow up at the origin. Now, if your point charge is actually an electrode or something whose radius is not vanishingly small, that's another issue, but your wording ("point charge") implied a mathematical point charge.

Do you have more details about what exactly you're trying to do?

 

[Alex555]

Well, a capacitor requires two conductors. I don't think a point charge can take the place of an actual conductor. However, you can speak of the (self-)capacitance of a conductor, such as your plate. Now while your point charge will induce a surface charge distribution in the plate such that the E-field inside the plate is 0 and the boundary condition for the electric field is satisfied at the surface, the total charge is going to be zero assuming you started with an electrostatically neutral plate, so I am not sure how you would go about calculating its self-capacitance if the net charge is vanishing.

The reason I'm not sure the point charge and the plate form a capacitor is as follows: assume the point charge is very close to the plate so that it looks like an infinite plane, assume it has a uniform surface charge density such that the total charge on the plate is that of the point charge but the other sign, and then calculate the potential difference between the two. But the point charge has an electric field that goes as $1/r^2$, so the integral would blow up at the origin. Now, if your point charge is actually an electrode or something whose radius is not vanishingly small, that's another issue, but your wording ("point charge") implied a mathematical point charge.

Do you have more details about what exactly you're trying to do?

In my simulation I treated the plate as a terminal set at 1 V relative to a point above set as "ground".

 

[tech99]

I need help trying to set up this problem. I want to find the capacitance between a point charge and a finite plate (or disk) as the point moves from above the center of the plate to some distance off the plate. I have been able to simulate this problem using FEM, however, there should be a way to solve this by using Gauss's law. In the simulated model I used a potential difference of 1 V between the plate and point.
I don't know if this forms an actual capacitor or if it is a self-capacitance of the plate with the point charge inducing a surface charge distribution on the plate. Any insight into this problem would be helpful.

To have capacitance we require a capacitor, which is a device able to store charge, like a tank of water.
The capacitor might be any object, like a sphere, plate disc etc. But in your case you have some water but no tank! So I don't think you can define capacitance for the situation you describe.

 

[mic*]

I am really not sure what the issue is here beyond the basics of electrostatics.

Two parallel plates create the effect of parallel field lines between the plates, orthogonal to the plane of the plates.

Charged stored is given by the relation

C = epsilon*A/d

Where C is capacitance in Farads or Coulombs.V^-1, epsilon is the dielectric permittivity of the separating medium, A is area, and d is separation distance.

A point charge will NOT create a similar field, except in the limiting scenario where d approaches infinity. In this case the electric field goes to zero.