From 2fcf92159cb73bfd7a28e5d2029ef77e270fa1e8 Mon Sep 17 00:00:00 2001 From: luc Date: Wed, 3 Sep 2025 22:03:31 +0200 Subject: [PATCH] docs/physics/electromagnetism/electromagnetic-dynamics.md: add --- .../electromagnetic-dynamics.md | 274 ++++++++++++++++++ mkdocs.yml | 4 +- 2 files changed, 276 insertions(+), 2 deletions(-) create mode 100644 docs/physics/electromagnetism/electromagnetic-dynamics.md diff --git a/docs/physics/electromagnetism/electromagnetic-dynamics.md b/docs/physics/electromagnetism/electromagnetic-dynamics.md new file mode 100644 index 0000000..ac1a0f2 --- /dev/null +++ b/docs/physics/electromagnetism/electromagnetic-dynamics.md @@ -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. diff --git a/mkdocs.yml b/mkdocs.yml index d4080cb..c89ed6d 100755 --- a/mkdocs.yml +++ b/mkdocs.yml @@ -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