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

$$\mathbf{P}={\chi }_e{ϵ}_0\mathbf{E}$$

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

$$\int {\rho }_{p}dV+\int \mathrm{\nabla }\cdot \mathbf{P}dV=0$$

$${\rho }_p=-\mathrm{\nabla }\cdot \mathbf{P}$$

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

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

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

Now consider Gauss’s law, $\mathrm{\nabla }\cdot \mathbf{E}=\rho /{ϵ}_0$. 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

$${ϵ}_0\mathrm{\nabla }\cdot \mathbf{E}={\rho }_f-\mathrm{\nabla }\cdot \mathbf{P}$$

$$\mathrm{\nabla }\cdot ({ϵ}_0\mathbf{E}+\mathbf{P})={\rho }_f$$

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

$$\mathbf{D}={ϵ}_0\mathbf{E}+\mathbf{P}$$

Using $\mathbf{P}={\chi }_e{ϵ}_0\mathbf{E}$ we can also write $\mathbf{D}=ϵ\mathbf{E}$, where $ϵ={ϵ}_0{ϵ}_r$ and ${ϵ}_r=1+{\chi }_e$ = relative permittivity.

Thus we can write Gauss’s law as

$$\mathrm{\nabla }\cdot \mathbf{D}={\rho }_f$$

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

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

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

$$\mathrm{\nabla }\times \mathbf{B}={\mu }_0\mathbf{J}+{\mu }_0{ϵ}_0\frac{d\mathbf{E}}{dt}$$

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

$$\mathrm{\nabla }\times \left(\frac{\mathbf{B}}{{\mu }_0}\right)={\mathbf{J}}_c+\mathrm{\nabla }\times \mathbf{M}+\frac{\partial \mathbf{P}}{\partial t}+{ϵ}_0\frac{d\mathbf{E}}{dt}$$

$$\mathrm{\nabla }\times (\frac{\mathbf{B}}{{\mu }_0}-\mathbf{M})={\mathbf{J}}_c+\frac{\partial }{\partial t}({ϵ}_0\mathbf{E}+\mathbf{P})$$

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

$$\mathbf{H}=\frac{\mathbf{B}}{{\mu }_0}-\mathbf{M}$$

and recall that $\mathbf{D}={ϵ}_0\mathbf{E}+\mathbf{P}$. Thus, Maxwell’s fourth equation becomes

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

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.