Characterizing Total Charge of Conductor A in an External Electrical Field

[coquelicot]

Assume that a certain charge distribution $\rho$ generates an electrical field $E_{ext}$ in the surrounding space. We also note the corresponding generated potential $V_{ext}$.

Assume furthermore that a conductor A, with a definite shape and volume, is placed in field $E_{ext}$, and is connected to an infinitely far ground $G$ with a very thin electrical wire. The potential of G is supposed to be $0.$

By electrostatic induction, free charges in A move to G, and conductor A, after equilibrium, experiences a distribution of charge $\sigma$ on its surface.

Assume furthermore that the charge $\sigma$ does not influence $\rho$, so that $E_{ext}$ and $V_{ext}$ are given and fixed in this problem.

Question: how to characterize the total charge $Q = \int_{S(A)} \sigma dS$ of A, as a function of $E_{ext}$ or $V_{ext}$, using, e.g. surface integrals on A etc. ?

 

[vanhees71]

This is a pretty tough boundary-value problem, which usually you have to solve numerically. For some simple, i.e., very symmetric, problems you can solve it with the method image charges, e.g., for a infinite plane or a sphere. See, e.g., Jackson, Classical Electrodynamics for details.

 

[coquelicot]

@vanhees71. Yes, probably. The question is how to formulate this boundary value problem with equations, integrals etc. in order it become solvable (in other words, I'm not interested in "solving" the problem, but in formulating it).

 

[vanhees71]

You have the usual boundary conditions for the Poisson equation,
$$\Delta \Phi=-\frac{1}{\epsilon_0} \rho,$$
i.e., along the surfaces of the conductors the potential should be continuous and the tangential components of the electric field must vanish (not to have surface currents), i.e., the surfaces of the conductors are equipotential surfaces.

 

[coquelicot]

@vanhees71. This does not help much, because the distribution of charge $\sigma$ (that you call $\rho$) is unknown, and is what has to be determined.
The potential $\Phi$ is also unknown, except inside and at the surface of the conductor where it is equal to 0. But this is not sufficient to provide its differential orthogonal component at the surface.

 

[vanhees71]

I know that this doesn't help much, but you asked for the formulation of the problem. As I said before, you can only find a solution in special simple cases (infinite plane, sphere) or for 2D problems using conformal mappings. See, e.g., Jackson, Classical Electrodynamics.

 

[coquelicot]

@vanhees71. That's not the point. I'm still seeking a formulation of the problem. In my last message, I simply meant that there is not sufficiently many equations to solve the problem (e.g. where is the exterior field used?), but maybe you know how to complete them.

 

[vanhees71]

I don't know, what you still look for. You have the boundary conditions which make the solution of the Poisson equation unique. For uniqueness of the solutions of the Maxwell equations, see, e.g., Sommerfeld, Lectures on Theoretical physics vol 3.

Let's to the most simple example, the plane. Take the $xy$ plane of a Cartesian coordinate system as the conductor. We look for the Green's function
$$-\Delta G(\vec{x},\vec{x}')=\delta^{(3)}(\vec{x}-\vec{x}').$$
The boundary condition for the grounded conductor is
$$G(\vec{x},\vec{x}')|_{x_3=0}=0.$$
Then it's clear that you can solve the problem of a point charge (modulo factors) at $\vec{x}'$ fulfilling the boundary conditions by putting a mirror charge at $\vec{x}''=(x_1',x_2',-x_3')$. So the solution is
$$G(\vec{x},\vec{x}')=\begin{cases} \frac{1}{4 \pi |\vec{x}-\vec{x}'|}-\frac{1}{4 \pi |\vec{x}-\vec{x}''|} & \text{for} \quad x_3 x_3'>0 \\ 0 & \text{for} \quad x_3 x_3'<0. \end{cases}$$
For a general charge distribution the solution for the potential thus is
$$\Phi(\vec{x})=\frac{1}{\epsilon_0} \int_{\mathbb{R}^3} \mathrm{d}^3 x' G(\vec{x},\vec{x}') \rho(\vec{x}').$$
From this you get the induced surface-charge density by calculating the jump of the normal component of the electric field, i.e.,
$$\sigma(\vec{x})=-\epsilon_0 \left [\partial_3 \Phi(\vec{x})|_{x_3=+0^+}-\partial_3 \Phi(\vec{x})|_{x_3=-0^+} \right].$$

 

[coquelicot]

Thanks. That looks close to the answer I expected. I have to work out the theoretical solution of the Poisson equation with boundary conditions by mean of the Green function thought. I will do that, and if necessary ask another question here.