- D.72.1 Existence of a potential
- D.72.2 The Laplace equation
- D.72.3 Egg-shaped dipole field lines
- D.72.4 Ideal charge dipole delta function
- D.72.5 Integrals of the current density
- D.72.6 Lorentz forces on a current distribution
- D.72.7 Field of a current dipole
- D.72.8 Biot-Savart law

D.72 Various electrostatic derivations.

This section gives various derivations for the electromagnetostatic solutions of chapter 13.3.

D.72.1 Existence of a potential

This subsection shows that if the curl of the electric field

That potential can be defined to be

(D.47) |

Now if you evaluate

(D.48) |

Note that if regions are multiply connected, the potential may not be
quite unique. The most important example of that is the magnetic
potential of an infinite straight electric wire. Since the curl of
the magnetic field is nonzero inside the wire, the path of integration
must stay clear of the wire. It then turns out that the value of the
potential depends on how many times the chosen integration path wraps
around the wire. Indeed, the magnetic potential is

D.72.2 The Laplace equation

The homogeneous Poisson equation,

The so-called mean-value property says that the average of

the first equality since

The so called maximum-minimum principle says that either maximum

is not a true
maximum. And then you can start sphere-hopping

to
show that

The only solution of the Laplace equation in all of space that is zero at infinity is zero everywhere. In more general regions, as long as the solution is zero on all boundaries, including infinity where relevant, then the solution is zero everywhere. The reason is the maximum-minimum principle: if there was a point where the solution was positive/negative, then there would have to be an interior maximum/minimum somewhere.

The solution of the Laplace equation for given boundary values is unique. The reason is that the difference between any two solutions must satisfy the Laplace equation with zero boundary values, hence must be zero.

D.72.3 Egg-shaped dipole field lines

The egg shape of the ideal dipole field lines can be found by assuming
that the dipole is directed along the

Change to a new variable

Integrating and replacing

where

D.72.4 Ideal charge dipole delta function

Next is the delta function in the electric field generated by a charge
distribution that is contracted to an ideal dipole. To find the
precise delta function, the electric field can be integrated over a
small sphere, but still large enough that on its surface the ideal
dipole potential is valid. The integral will give the strength of the
delta function. Since the electric field is minus the gradient of the
potential, an arbitrary component

where

For simplicity, take the

D.72.5 Integrals of the current density

In subsequent derivations, various integrals of the current density

Consider an integral like

Now use the fact that the divergence of the current density is zero since the charge density is constant for electromagnetostatic solutions:

where

The first integral in the right hand side can be integrated by parts
in the

with

because only the relative ordering of the indices in the sequence

In quantum applications, it is often necessary to relate the dipole
moment to the angular momentum of the current carriers. Since the
current density is the charge per unit volume times its velocity, you
get the linear momentum per unit volume by multiplying by the ratio

D.72.6 Lorentz forces on a current distribution

Next is the derivation of the Lorentz forces on a given current
distribution

In terms of a current distribution, the moving charge per unit volume
times its velocity is the current density, so the force on a volume
element

The net force on the current distribution is therefore zero, because according to (D.50) with

The moment is not zero, however. It is given by

According to the vectorial triple product rule, that is

The second integral is zero because of (D.50) with

The first of the three integrals is zero because of (D.50) with

and in vector notation that reads

When the (frozen) current distribution is slowly rotated around the
axis aligned with the moment vector, the work done is

where

D.72.7 Field of a current dipole

A current density

where

A magnetic vector potential

that vanishes at infinity. Taking the divergence of this equation shows that the divergence of the vector potential satisfies a homogeneous Poisson equation, because the divergence of the current density is zero, with zero boundary conditions at infinity. Therefore the divergence of the vector potential is zero. It then follows that

because it satisfies the equations for

You might of course wonder whether there might not be more than one magnetic field that has the given divergence and curl and is zero at infinity. The answer is no. The difference between any two such fields must have zero divergence and curl. Therefore the curl of the curl of the difference is zero too, and the vectorial triple product shows that equal to minus the Laplacian of the difference. If the Laplacian of the difference is zero, then the difference is zero, since the difference is zero at infinity (subsection 2). So the solutions must be the same.

Since the integrals of the current density are zero, (D.50)
with

Now the term

with

Note that the expression between brackets is just the

The magnetic field is the curl of

and substituting in for the vector potential from above, differentiating, and cleaning up produces

This is the same asymptotic field as a charge dipole with strength

However, for an ideal current dipole, the delta function at the origin
will be different than that derived for a charge dipole in the first
subsection. Integrate the magnetic field over a sphere large enough
that on its surface, the asymptotic field is accurate:

Using the divergence theorem, the right hand side becomes an integral over the surface of the sphere:

Substituting in the asymptotic expression for

The integrals of

That gives the strength of the delta function for an ideal current dipole.

D.72.8 Biot-Savart law

In the previous section, it was noted that the magnetic field of a
current distribution is the curl of a vector potential

The solution for the vector potential can be written explicitly in terms of the current density using the Green’s function integral (13.29):

The magnetic field is the curl of

or substituting in and differentiating under the integral

In vector notation that gives the Biot-Savart law

Now assume that the current distribution is limited to one or more
thin wires, as it usually is. In that case, a volume element of
nonzero current distribution can be written as

where in the right hand side

where the integration is over all infinitesimal segments