Fundamental Equations of Electrostatics

(The following are my notes on Chapter 1 of "Classical Electrodynamics" by Jackson)

In this post I am going to develop two very important and fundamental equations of electrostatics.  The mathematics involved will be advanced vector calculus, so this isn't exactly and introductory post.  These equations are two of Maxwell's Equations. First I will introduce and develop Gauss' Law in both integral and differential form.  From there I will develop the curl equation for the electric field, and finally I will go over Poisson's and Laplace's equation and delve into them using Greens Functions, and use these results to prove the uniqueness of their respective solutions.

Gauss' Law

Gauss' Law states that the total electric flux from a dielectric field from a charged body is equal to the total charge of the object.  Or in other mathematical terms, the total flux from the electric field is equal to the charge of the body divided by the permittivity of free space.

Consider any arbitrary Gaussian surface. With a point charge within it.

We will project the electric field on the surface of the Gaussian surface.  The above relationship is described by..

We notice the following relationship..

Where the differential Omega is equal to the solid angle subtended by the projection of the electric field vector on the area.  We can now integrate both sides over the closed surface.

We have therefore proved Gauss' Law for an arbitrary Gaussian Surface.  If the charge is located on the outside of the surface, the integral on the right hand side is equal to zero.

But what if we had a multiple charges inside the surface? Or a charged body?  Well it is easy to see that we would only have to make one modification to the above formula.  For N amount of charges...

If we have a charged body, the sum becomes a volume integral.

Where we have taken the total charge Q to be the volume integral of the volume charge density. (d^3x is simply another way of writing dV)

Using the above expression and the Divergence theorem, we can transform this integral equation into a differential equation.  It is done in the following way.

Where we have used the fact that we can combine both integral since they are being integrated over the same volume.  We then noted that the volume integral was equal to zero and therefore the integrand must be equal to zero.

Curl Equation for E and the Scalar Potential:

To derive the curl equation for the electric field, we start with another one of Maxwell's Equations.  Specifically, the one that states that the closed path integral over the electric field is equal to zero. (Meaning the static electric field is a conservative field.)

We now develop a very useful concept, the scalar potential.  The scalar potential is useful in the sense that it is much easier to describe something with a scalar equation than a vector equation.  So we can develop what exactly the scalar potential should be equal to in order to make the above equation true.

Recall that the electric field of any body is..

We will now prove a very important and necessary formula by doing the following..

We can now use this in the above expression for the Electric Field.

The term inside the brackets will be defined to be the Scalar Potential of the electric field.  This is evident because...

Poisson and Laplace Equations

So far we have developed the following three relations. (The last two are really saying the same thing..)

Plugging the third equation into the first we get..

This is known as Poisson's Equation.  When dealing with a space in the absence of a charge density, the equation reduces to Laplace's Equation.

We now want to show that the expression we obtained earlier is indeed a solution to Poisson's Equation.

So it does indeed satisfy the Poisson Equation.  We can use a property of the above derivation to state another equation.

Where we have used the Dirac Delta Function.  This follows from the properties of the Dirac Delta Function and the fact that the Laplacian of the above function is zero for all points where x-x' is not zero.  We will use this soon.

Green's Theorem

When dealing with Poisson's and Laplace's Equation, we often times, need to satisfy some sort of boundary condition dealing with a finite space.  In order to later prove that the solutions to these equations are in fact, unique, we need to develop some new mathematical tools to use on the equations.  These "tools"I am referring to are Green's Identities, or more generally Greens Theorem.

We start out with the Divergence Theorem for any vector field A.

We now set A equal to the product of a scalar field and the gradient of another scalar field and use a vector identity.

The last line is what is known as the "normal derivative" It is simply the directional derivative along the normal component.  The notation with the partial derivative is simply a notation that means take the derivative of the scalar along the direction of the normal vector.  We can now plug these relations into the Divergence Theorem.

This is known as Greens First Identity.

We now use the above identity and swap psi and phi and then subtract the two equations.

This is known as Greens Second Identity.

Our next goal is to convert the Poisson differential equation into an integral equation using Greens Theorem.  We will do this by making the following substitutions.

If the point x lies outside the volume, the left hand side of this equation is equal to zero.

In the words of Jackson directly from the book.

"If the surface S goes to infinity and the electric field on S falls off faster than 1/R, then the surface intrgral vanishes and we are left with the familiar expression for the potential.  Secondly, for a charge free volume, the potential anywhere inside the volume (a solution of the Laplace equation) is expressed in the above formula in terms of the potential and its normal derivative only on the surface of the volume.  This rather surprising result is not a solution to a boundary-value problem, but only an integral statement, since the arbitrary specification of both the potential and its normal derivative (Cauchy boundary conditions) is an over-specification of the problem."

Uniqueness of the Solution with Dirichlet or Neumann Boundary Conditions

Say you and I have a boundary value problem coming from Poisson's (or for that matter Laplace's equation.) You, being the quick witted physicist solve it analytically and get a nice elegant solution, while I a little slower than you, break down in trying to solve the problem analytically and solve it using a series solution. (AKA numerically.)  Our solutions will look very much different, how can we be sure that they are indeed the same solution?

That is the point of proving uniqueness of a differential equation.  We want to know that any solution we get, not matter how we get it is THE solution.  Meaning there is only one.

For the Poisson equation we will be dealing with two boundary conditions. Dirichlet conditions, whereby we specify the potential on a closed surface, and Neumann conditions whereby we specify the electric field (normal derivative of the potential) everywhere on the surface.

For the Poisson equations we will suppose that there exists two solutions.