docs/physics/electromagnetism/electromagnetic-dynamics.md: add
This commit is contained in:
parent
508c5c2449
commit
2fcf92159c
2 changed files with 276 additions and 2 deletions
274
docs/physics/electromagnetism/electromagnetic-dynamics.md
Normal file
274
docs/physics/electromagnetism/electromagnetic-dynamics.md
Normal file
|
|
@ -0,0 +1,274 @@
|
|||
# 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.
|
||||
|
|
@ -196,8 +196,8 @@ nav:
|
|||
# - 'Statistical mechanics':
|
||||
- 'Electromagnetism':
|
||||
- 'Electrostatics': physics/electromagnetism/electrostatics.md
|
||||
# - 'Magnetostatics':
|
||||
# - 'Electrodynamics':
|
||||
- 'Magnetostatics': physics/electromagnetism/magnetostatics.md
|
||||
- 'Electromagnetic dynamics': physics/electromagnetism/electromagnetic-dynamics.md
|
||||
- 'Optics':
|
||||
- 'Waves': physics/electromagnetism/optics/waves.md
|
||||
- 'Electromagnetic waves': physics/electromagnetism/optics/electromagnetic-waves.md
|
||||
|
|
|
|||
Loading…
Add table
Add a link
Reference in a new issue