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.