# The four-field formalism#

Gravity, Magnetism, and Orbital Dynamics

The standard theoretical formalism for treating electromagnetic fields in matter introduces two new fields, $$\mathbf{D}$$ and $$\mathbf{H}$$. This notebook will introduce these fields and other basic properties associated with them, as well as their role in Maxwell’s equations.

## Polarisation and the D field#

When an electric field is applied on a dielectric, the atoms become aligned electric dipoles. To study this property, we define the polarisation, $$\mathbf{P}$$, as the atomic dipole moment per volume

$\mathbf{P} = N\mathbf{p}$

where $$N$$ is the number of atoms per volume, and $$\mathbf{p}$$ is the atomic dipole moment. Using this definition we can write

$\mathbf{P} = \chi_e \epsilon_0 \mathbf{E}$

where $$\chi_e$$ is the electric susceptibility of the material. Now consider a small volume element inside the dielectric. Since the dielectric is neutral in charge we can write $$Q_{inside} + Q_{surf} = 0$$. This can be written as

$\int \rho_p \,dV + \oint \mathbf{P} \cdot d\mathbf{S} = 0$

where $$\rho_p$$ is the polarisation charge density. Using Stoke’s theorem we can write

$\int \rho_p \,dV + \int \nabla \cdot \mathbf{P} \,dV = 0$
$\rho_p = -\nabla \cdot \mathbf{P}$

If $$\mathbf{P}$$ is time-varying then

$\frac{∂\mathbf{P}}{∂t} = N\frac{∂\mathbf{p}}{∂t} = Nq\mathbf{v}$

But this is just the current density due to the movement of the bound charges, thus $$\mathbf{J}_p$$ = polarization current density = $$∂\mathbf{P}/{∂t}$$.

Now consider Gauss’s law, $$\nabla \cdot \mathbf{E} = \rho/{\epsilon_0}$$. We can write the charge density as $$\rho = \rho_f + \rho_p$$ where $$\rho_f$$ is the charge density of free charges. Thus, Gauss’s law becomes

$\epsilon_0 \nabla \cdot \mathbf{E} = \rho_f - \nabla \cdot \mathbf{P}$
$\nabla \cdot (\epsilon_0 \mathbf{E} + \mathbf{P}) = \rho_f$

Now we define the $$\mathbf{D}$$ field (sometiems called electric displacement) as

$\mathbf{D} = \epsilon_0 \mathbf{E} + \mathbf{P}$

Using $$\mathbf{P} = \chi_e \epsilon_0 \mathbf{E}$$ we can also write $$\mathbf{D} = \epsilon \mathbf{E}$$, where $$\epsilon = \epsilon_0 \epsilon_r$$ and $$\epsilon_r = 1 + \chi_e$$ = relative permittivity.

Thus we can write Gauss’s law as

$\nabla \cdot \mathbf{D} = \rho_f$

## Magnetisation and the H field#

To introduce the $$\mathbf{H}$$ field we must first talk about atomic magnetic dipole moments. These are in reality a quantum effect, but it can be helpful to visualise the atom as a tiny current loop. The atomic magnetic dipole moment is defined as

$\mathbf{m} = I\mathbf{A}$

where $$I$$ is the current and $$\mathbf{A}$$ is the vector area enclosed by the loop (right-hand-rule). Define the magnetisation, $$\mathbf{M}$$ as the atomic magnetic dipole moment per volume.

$\mathbf{M} = N\mathbf{m}$

Consider two adjacent current loops. It can be shown that a net current can be produced from the two loops. Thus, atomic magnetic dipole moments result to the formation of current. This is related to the magnetisation of the material by

$\mathbf{J}_m = \nabla \times \mathbf{M}$

where $$\mathbf{J}_m$$ is the magnetisation current density. Now lets consider Maxwell’s fourth equation

$\nabla \times \mathbf{B} = \mu_0\mathbf{J} + \mu_0 \epsilon_0 \frac {d\mathbf{E}}{dt}$

We can write the current density as $$\mathbf{J} = \mathbf{J}_c + \mathbf{J}_m + \mathbf{J}_p$$ where $$\mathbf{J}_c$$ is the conduction current density. Therefore we can write

$\nabla \times \left(\frac{\mathbf{B}}{\mu_0}\right) = \mathbf{J}_c + \nabla \times \mathbf{M} + \frac{∂\mathbf{P}}{∂t} + \epsilon_0 \frac {d\mathbf{E}}{dt}$
$\nabla \times \left(\frac{\mathbf{B}}{\mu_0} - \mathbf{M}\right) = \mathbf{J}_c + \frac{∂}{∂t}\left(\epsilon_0 \mathbf{E} + \mathbf{P}\right)$

Now we define the $$\mathbf{H}$$ field as

$\mathbf{H} = \frac{\mathbf{B}}{\mu_0} - \mathbf{M}$

and recall that $$\mathbf{D} = \epsilon_0 \mathbf{E} + \mathbf{P}$$. Thus, Maxwell’s fourth equation becomes

$\nabla \times \mathbf{H} = \mathbf{J}_c +\frac{∂\mathbf{D}}{∂t}$

We can also define $$\mathbf{M} = \chi_m \mathbf{H}$$ where $$\chi_m$$ is the magnetic susceptibility of the material. Thus, from the definition of $$\mathbf{H}$$, we can relate $$\mathbf{H}$$ strictly to $$\mathbf{B}$$:

$\mathbf{H} = \frac{\mathbf{B}}{\mu_0} - \chi_m \mathbf{H}$
$\mathbf{H} = \frac{\mathbf{B}}{\mu_0 (1 + \chi_m)} = \frac{\mathbf{B}}{\mu}$

where $$\mu = \mu_0 \mu_r$$ and $$\mu_r = 1 + \chi_m$$ = relative permeability.