Application of the Divergence Theorem: Laplace's Equation

The combination div grad , () or� is called the "Laplacian" differential operator,
The equation () f = 0 is called Laplace's equation. Static electric and steady state magnetic fields obey this equation where there are no charges or current.
Any solution to this equation in R has the property that its value at the center of a sphere within R is the average of its value on the sphere's surface. If g(r) obeys Laplace's equation inside a spherical surface, S, of radius a, centered at r', we have

Laplace's Equation

Thus solutions to Laplace's equation� are very smooth: they� have no bumps maxima or minima in R and essentially "interpolate" smoothly between their values on the boundaries of R. We prove this important fact as an application of the divergence theorem.
This result also implies that if we know the divergence of a vector v and its curl everywhere, these are differentiable everywhere, and v vanishes� at infinity, then v is uniquely determined. proof box( if there were two solutions v and v' with the same divergence and curl, then on applying the double cross product identity we find that every component of their difference obeys Laplace's equation everywhere. Its value anywhere is then its average value on a huge circle at infinity, which is 0 by assumption. The same conclusion holds if v and v' are required to behave at infinity in the same way, so that v - v' must approach 0 for large arguments.)
Proof
Suppose our function, f(r), obeys Laplace's equation within some� sphere S centered at r':

div grad f = () f = 0 inside S

We apply the divergence theorem to the vector fg - gf in the sphere with surface S excluding a tiny sphere of radius b with surface S'� having the same center. We obtain

the latter being obtained by substituting for g. The second integral on the second line vanishes here as can be seen by applying the divergence theorem again within S and noticing that the� Laplacian applied to f is 0.
The right hand side here is the average value of f on S. The similar integral over S' is evaluated in exactly the same way and is the average value of f on S'. We conclude that the average value of v on any sphere with center at r' is the same. Obviously as the sphere approaches radius 0 that average value becomes the value of f at r'.
The method used in this argument is a very important and general one that is used in dealing with many differential equations. In fact the use of the divergence theorem in the form used above is often called "Green's Theorem." And the function g defined above is called a "Green's function" for Laplaces's equation.
We can use this function g to find a vector field v that vanishes at infinity obeying div v = , �curl v = 0. (we assume that r is sufficently well behaved, integrable, vanishes at infinity etc...)� Suppose we write v as grad f.
We get

(This equation is called Poisson's equation and is obeyed by the potential produced by a charge distribution with charge density .)
A solution that vanishes at infinity is then given by

Proof
The same approach can be used to obtain a vector potential A and determine a vector v obeying

v = 0, v= j

�With v =A we obtain

((A)) = j

and with the double cross identity

((A)) = (A) - () A

and the "gauge" condition (which we are free to assume) ) A = 0, we find that each component of A obeys Poisson's equation with source - the corresponding component of j.
We may therefore find a formula for each coordinate of A exactly like the corresponding formula for the scalar potential V. Again v may be recovered from A by differentiating.
These results are of some use in the study of electromagnetic fields but they� don't solve all problems. Often the known charges and currents induce unknown charges and currents in conducting surfaces, and one wants to determine the fields, in circumstances in which there are some unknown charges and /or currents and you know conditions on the field at the conductor surfaces instead of the charges and currents in them.
Exercises

The Gradient in General Orthogonal Coordinate System

Given a scalar field f, its gradient is a vector normal to the surface on which the field is constant, of length given by f's directional derivative in that direction.
We can always write

df

or , after trivial manipulation,

We may identify ds_j� = u_jdw_j , and obtain

Suppose we set two of the ds_i's to zero; then the directional derivative of f in the third direction must be the component of� the gradient in that direction. We therefore obtain

The Curl in General Orthogonal Coordinate System

We may use Stokes' Theorem to deduce this by applying it to a tiny rectangle normal to the w₁ direction with sides obeying one of dw_2� = 0 and dw_3� = 0, and then cyclically permuting the indices.
The surface integral in Stokes' theorem then becomes

u₂u₃dw₂dw₃(v)₁

while the path integrals become

which gives us:

 

Maxwell's Equations

Maxwell raised the question: what are the magnetic fields produced by non-steady currents? He noticed that Ampere's Law contradicts the conservation of charge and required modification when currents are not steady. These two laws, as we have noted, correspond to the equations

and

cB = j

Taking the divergence of both sides of the latter and using the former gives:

0 =B =j = -/ t

Thus, Ampere's Law implies that the current in it is steady.
Maxwell noted that the electrostatic equation,E =, provides a way to modify Ampere's Law so that it is consistent with non-steady currents. If we add a term E / t to its left hand side, we can eliminate the condition that be constant, and obtain a consistent set of equations: These equations are called Maxwell's Equations.
They are:
B +E / ct = j / c (Maxwell's modified Ampere's Law)
B = 0 (No magnetic sources or sinks)
E -B / ct = 0 (Faraday's Law)
E =                       (Gauss's Theorem)