notes/docs/physics/electromagnetism/electromagnetic-dynamics.md

Electromagnetic dynamics

We let loose the electrostatic and magnetostatic regime and consider the interplay of electric and magnetic fields as electromagnetic dynamics.

Electromagnetic dynamics in vacuum

Axiom 1: The electric \mathbf{E}: (\mathbf{r},t) \mapsto \mathbf{E}(\mathbf{r},t) and magnetic field \mathbf{B}: (\mathbf{r},t) \mapsto \mathbf{B}(\mathbf{r},t) are described by

\begin{align*} \nabla \cdot \mathbf{E} &= \frac{\rho}{\epsilon_0},\ \nabla \cdot \mathbf{B} &= 0,\ \nabla \times \mathbf{E} + \partial_t \mathbf{B} &= \mathbf{0},\ \nabla \times \mathbf{B} - \mu_0 \epsilon_0 \partial_t \mathbf{E} &= \mu_0 \mathbf{J},
\end{align*}

with \rho the volume charge density, \mathbf{J} the volume current density and \varepsilon_0 and \mu_0 the permittivity and permeability of vacuum.

The equations in Axiom 1 are called the Maxwell equations and their integral form can be obtained from the curl and divergence theorem, respectively.

It follows that from the definition of the electric force and magnetic force on a point charge q at a field point \mathbf{r} moving with a velocity \mathbf{v} that the electromagnetic force \mathbf{F} is given by

\mathbf{F} = q(\mathbf{E} + \mathbf{v} \times \mathbf{B}),

for all \mathbf{E}: (\mathbf{r},t) \mapsto \mathbf{E}(\mathbf{r},t) and \mathbf{B}: (\mathbf{r},t) \mapsto \mathbf{B}(\mathbf{r},t) the electric and magnetic field (Lorentz law).

In the linear assumption we may express the volume current density \mathbf{J} in terms of the electromagnetic force per unit charge

\mathbf{J} = \sigma (\mathbf{E} + \mathbf{v} \times \mathbf{B}),

with \sigma the conductivity of the medium. In the particular case that \mathbf{v} = \mathbf{0} we obtain Ohm's law.

Definition 1: Let the emf \epsilon be defined as

\varepsilon = \oint_L \mathbf{E} \cdot d\mathbf{l},

for all \mathbf{E}: (\mathbf{r},t) \mapsto \mathbf{E}(\mathbf{r},t).

Definition 1 imposes that

\varepsilon = - d_t \int_S \mathbf{B} \cdot d\mathbf{a},

which is called the Flux rule. According to Lenz and Axiom 1 nature abhors a change in flux.

Inductance

Consider two loops of wire, by proportionality we may define \Phi_2 = M_{21} I_1 as the flux of the magnetic field created by loop 1 through loop 2 and \Phi_1 = M_{12} I_2 the opposite with M_{ij} the mutual inductance.

It then follows (without stating the proof) that

M_{21} = M_{12} = \frac{\mu_0}{4\pi} \oiint \frac{1}{\mathfrak{r}} d\mathbf{l}_1 \cdot d\mathbf{l}_2,

is the Neumann formula.

By proportionality we may as well define \Phi = L I with L the self-inductance.

Energy in magnetic fields

Consider

d_t W = - \varepsilon I = L I d_t I,

then W = \frac{1}{2} L I^2 is the work required to build up the line current density from zero to I.

It follows then that for a volume current density \mathbf{J} the work W needed to construct the system is given by

W = \frac{1}{2} \int_\mathscr{V} \mathbf{A} \cdot \mathbf{J} d\tau.

From this we may state:

Theorem 1: The work W required to construct the system can be expressed in terms of the magnetic field \mathbf{B} as

W = \frac{1}{2\mu_0} \int_{\mathbb{R}^3} |\mathbf{B}|^2 d\tau.

??? note "Proof:"

Rewrite the work in terms of the magnetic field $\mathbf{B}$:

$$
    \begin{align*}
        W &= \frac{1}{2\mu_0} \int_\mathscr{V} \mathbf{A} \cdot (\nabla \times \mathbf{B})d\tau,\\
          &= \frac{1}{2\mu_0} \Bigg(\int_\mathscr{V} \|\mathbf{B}\|^2 d\tau - \int_\mathscr{V} \nabla \cdot (\mathbf{A} \times \mathbf{B}) d\tau \Bigg),\\
          &= \frac{1}{2\mu_)} \Bigg(\int_\mathscr{V} \|\mathbf{B}\|^2 d\tau - \int_{\partial \mathscr{V}} (\mathbf{A} \times \mathbf{B}) \cdot d\mathbf{a} \Bigg).
    \end{align*}
$$

If we now set $\mathscr{V} = \mathbb{R}^3$ then the integral over $\partial \mathscr{V}$ goes to zero and we are left with:

$$
    W = \frac{1}{2\mu_0} \int_{\mathbb{R}^3} \|\mathbf{B}\|^2 d\tau.
$$

That is performing the integral over all space.

Conservation of charge

Formally, the charge in a domain \mathscr{V} is

Q(t) = \int_\mathscr{V} \rho(\mathbf{r},t) d\tau,

and the current out of the boundary of the domain \partial \mathscr{V} is

Q'(t) = - \oint_{\partial \mathscr{V}} \mathbf{J}(\mathbf{r},t) \cdot d\mathbf{a},

such that

\int_\mathscr{V} \partial_t \rho(\mathbf{r},t) d\tau = - \int_\mathscr{V} \nabla \cdot \mathbf{J}(\mathbf{r},t) d\tau,

and since this is true for any domain \mathscr{V}, it follows that

\partial_t \rho + \nabla \cdot \mathbf{J} = 0,

conservation of charge.

Theorem 2: Axiom 1 imposes that the volume charge density \rho: (\mathbf{r},t) \mapsto \rho(\mathbf{r},t) and volume current density \mathbf{J}: (\mathbf{r},t) \mapsto \mathbf{J}(\mathbf{r},t) adhere to

\partial_t \rho + \nabla \cdot \mathbf{J} = 0.

??? note "Proof:"

$$
    \begin{align*}
        \nabla \cdot \mathbf{J} &= \frac{1}{\mu_0} \nabla \cdot \Big(\nabla \times \mathbf{B} - \mu_0 \epsilon_0 \partial_t \mathbf{E}\Big),\\ 
        &= - \epsilon_0 \partial_t (\nabla \cdot \mathbf{E}),\\
        &= - \partial_t \rho.
    \end{align*}
$$

Conservation of energy

The work W done by the electromagnetic force may be expressed as

\begin{align*} W &= \mathbf{F} \cdot d\mathbf{l},\ &= q(\mathbf{E} + \mathbf{v} \times \mathbf{B}) \cdot \mathbf{v} dt,\ &= q \mathbf{E} \cdot \mathbf{v} dt, \end{align*}

such that the rate at which work is done on a domain \mathscr{V} is

d_t W = \int_\mathscr{V} \mathbf{E} \cdot \mathbf{J} d\tau.

Theorem 3: Axiom 1 imposes that the electric field \mathbf{E}: (\mathbf{r},t) \mapsto \mathbf{E}(\mathbf{r},t), magnetic field \mathbf{B}: (\mathbf{r},t) \mapsto \mathbf{B}(\mathbf{r},t) and volume current density \mathbf{J}: (\mathbf{r},t) \mapsto \mathbf{J}(\mathbf{r},t) adhere to

\mathbf{E} \cdot \mathbf{J} + \frac{1}{2} \partial_t \bigg(\epsilon_0 |\mathbf{E}|^2 + \frac{1}{\mu_0} |\mathbf{B}|^2\bigg) + \frac{1}{\mu_0} \nabla \cdot \Big(\mathbf{E} \times \mathbf{B}\Big) = 0.

??? note "Proof:"

$$
    \begin{align*}
        \mathbf{E} \cdot \mathbf{J} &= \frac{1}{\mu_0} \mathbf{E} \cdot (\nabla \times \mathbf{B}) - \epsilon_0 \mathbf{E} \cdot \partial_t \mathbf{E},\\
        &= \frac{1}{\mu_0} \Big(-\mathbf{B} \cdot \partial_t \mathbf{B} - \nabla \cdot (\mathbf{E} \times \mathbf{B})\Big) -\epsilon_0 \mathbf{E} \cdot \partial_t \mathbf{E},\\
        &= -\frac{1}{2} \partial_t \bigg(\epsilon_0 \|\mathbf{E}\|^2 + \frac{1}{\mu_0} \|\mathbf{B}\|^2\bigg) - \frac{1}{\mu_0} \nabla \cdot (\mathbf{E} \times \mathbf{B}).
    \end{align*}
$$

We may thus write this result as

d_t W + d_t \int_\mathscr{V} u d\tau + \oint_{\partial \mathscr{V}} \mathbf{S} \cdot d\mathbf{a} = 0,

with u = \frac{1}{2}\big(\epsilon_0 \|\mathbf{E}\|^2 + \frac{1}{\mu_0} \|\mathbf{B}\|^2\big) the electromagnetic energy density and \mathbf{S} = \frac{1}{\mu_0} \big(\mathbf{E} \times \mathbf{B}\big) the electromagnetic flux density, called Poynting's theorem.

Conservation of momentum

We may write the divergence of the energy-momentum tensor of the electromagnetic field \nabla \cdot \mathbf{T} in terms of the electric and magnetic field:

\nabla \cdot \mathbf{T} = \epsilon_0 \bigg(\big(\nabla \cdot \mathbf{E}\big) \mathbf{E} + \big(\mathbf{E} \cdot \nabla\big) \mathbf{E} \bigg) + \frac{1}{\mu_0} \bigg(\big(\nabla \cdot \mathbf{B}\big) + \big(\mathbf{B} \cdot \nabla\big) \mathbf{B}\bigg) - \frac{1}{2} \nabla \bigg(\epsilon_0 |\mathbf{E}|^2 + \frac{1}{\mu_0}|\mathbf{B}|^2\bigg),

and define a momentum density \mathbf{g} in terms of \mathbf{S}:

\mathbf{g} = \mu_0 \epsilon_0 \mathbf{S},

then it follows from Newton's second law that:

Theorem 4: Axiom 1 imposes that the divergence of the energy-momentum tensor of the electromagnetic field \nabla \cdot \mathbf{T} and the momentum density \mathbf{g} adhere to

\nabla \cdot \mathbf{T} - \partial_t \mathbf{g} = \mathbf{0}.

??? note "Proof:"

Should be rewritten/reconsidered in Lagrangian/Hamiltonian formalism.

Electromagnetic dynamics in matter

Let \mathbf{J}_p = \partial_t \mathbf{P} denote the polarisation volume current density such that

\begin{align*} \nabla \cdot \mathbf{J} &= \nabla \cdot (\mathbf{J}_f + \mathbf{J}_b + \mathbf{J}_p),\ &= \nabla \cdot (\mathbf{J}_f + \nabla \times \mathbf{M} + \partial_t \mathbf{P}),\ &= -(\partial_t \rho_f + \partial_t \rho_p),\ &= - \partial_t \rho, \end{align*}

complying to the conservation of charge.

Theorem 5: The electric field \mathbf{E}: (\mathbf{r},t) \mapsto \mathbf{E}(\mathbf{r},t), electric displacement \mathbf{D}: (\mathbf{r},t) \mapsto \mathbf{D}(\mathbf{r},t), magnetic field \mathbf{B}: (\mathbf{r},t) \mapsto \mathbf{B}(\mathbf{r},t) and auxiliary field \mathbf{H}: (\mathbf{r},t) \mapsto \mathbf{H}(\mathbf{r},t) in a medium are described by

\begin{align*} \nabla \cdot \mathbf{D} &= \rho_f,\ \nabla \cdot \mathbf{B} &= 0,\ \nabla \times \mathbf{E} + \partial_t \mathbf{B} &= \mathbf{0},\ \nabla \times \mathbf{H} - \partial_t \mathbf{D} &= \mathbf{J}_f, \end{align*}

with \rho_f the free volume charge density and \mathbf{J}_f the free volume current density.

??? note "Proof:"

In a medium we have

$$
    \begin{align*}
        \nabla \cdot \mathbf{E} &= \frac{1}{\epsilon_0} (\rho_f + \rho_b),\\
        &= \frac{1}{\epsilon_0}(\rho_f - \nabla \cdot \mathbf{P}),
    \end{align*}
$$

or $\nabla \cdot \mathbf{D} = \rho_f$, and

$$
    \begin{align*}
        \nabla \times - \mu_0 \epsilon_0 \partial_t \mathbf{E} &= \mu_0(\mathbf{J}_f + \mathbf{J}_b + \mathbf{J}_p),\\
        &= \mu_0(\mathbf{J}_f + \nabla \times \mathbf{M} + \partial_t \mathbf{P}),
    \end{align*}
$$

or $\nabla \times \mathbf{H} = \mathbf{J}_f + \partial_t \mathbf{D}$.

Which require constitutive relations for closure.