### Trial lecture: Maxwell's equations

David Weir

helsinki.fi/~weir/slides/trial-lecture/

2nd May 2018

### Scope

• 25-minute trial lecture
• Aimed at the audience for Electrodynamics course
• Assumed knowledge:
• Integral form of Maxwell's equations
• Vector calculus theorems

Image: CC-BY-SA 2.0 Зеленый кабинет

### Aims

• Motivate, derive differential form of Maxwell's equations
• Discuss implications of Maxwell's equations, including:
• No monopoles
• Wave equation
• To place Maxwell's equations in a broader context
(e.g. as an example of unification of forces)

### Recap: Gauss's law - integral form

"The total electric flux flowing through a closed surface is proportional to the electric charge inside"

$${\large\unicode{x222F}}_S \mathbf{E} \cdot \mathrm{d}\mathbf{A} = \frac{Q}{\varepsilon_0} \vphantom{\iiint_V}$$

• $\unicode{x222F}_S$: integral over a closed surface (called $S$)
• $\mathbf{E}$: electric field
• $\mathrm{d}\mathbf{A}$: infinitesimal element of surface $S$
• $\varepsilon_0$: permittivity of free space
• $Q$: total electric charge inside $S$

$$\require{color} {\large\unicode{x222F}}_S \mathbf{E} \cdot \mathrm{d}\mathbf{A} = {\color{red}\frac{1}{\varepsilon_0}\iiint_V \rho \, \mathrm{d}V}$$

• $\unicode{x222F}_S$: integral over a closed surface (called $S$)
• $\mathbf{E}$: electric field
• $\mathrm{d}\mathbf{A}$: infinitesimal element of surface $S$
• $\varepsilon_0$: permittivity of free space
• $\rho$: charge density
• $\iiint_V$: integral over volume $V$ enclosed by $S$

### Gauss's law

$$\underbrace{{\large\unicode{x222F}}_S \mathbf{E} \cdot \mathrm{d}\mathbf{A}}_\text{Total electric flux through surface S} = \frac{1}{\varepsilon_0} \underbrace{\iiint_V \rho \, \mathrm{d}V}_\text{Charge in volume V bounded by surface S}$$

### Divergence theorem

"The flux of a vector field through a surface depends on how the vector field behaves inside the surface"

$$\iiint_V (\nabla \cdot \mathbf{F}) \, \mathrm{d}V = {\large \unicode{x222F}}_S \mathbf{F} \cdot \mathrm{d}\mathbf{A}$$

• True for any $V$ and $S$
• $\mathbf{F}$ must be smooth

Other sources: Khan Academy, Feynman lectures

### Applying the divergence theorem

• We have: $$\iiint_V (\nabla \cdot \mathbf{F}) \, \mathrm{d}V = {\large \unicode{x222F}}_S \mathbf{F} \cdot \mathrm{d}\mathbf{A}$$
• And we want to apply it to Gauss's law for electric fields: $${\large\unicode{x222F}}_S \mathbf{E} \cdot \mathrm{d}\mathbf{A} = \frac{1}{\varepsilon_0}\iiint_V \rho \, \mathrm{d}V$$
• We can immediately use it to rewrite the left hand side: $$\iiint_V (\nabla \cdot \mathbf{E}) \, \mathrm{d}V = \frac{1}{\varepsilon_0}\iiint_V \rho$$
• As it is true for any $V$, we can just write: $$\fbox{\nabla \cdot \mathbf{E} = \frac{\rho}{\varepsilon_0}}.$$

### Gauss's law - differential form

"Electric field lines start and end on electric charges"

$$\nabla \cdot \mathbf{E} = \frac{\rho}{\epsilon_0}$$

Images: CC-BY-SA 3.0 Nicoguaro [1], [2] after Gonfer.

Other sources: Hyperphysics, Wikipedia, Bleaney & Bleaney, Feynman lectures.

### Gauss's law for magnetic fields

"The total magnetic flux flowing through a closed surface is always zero"

$${\large\unicode{x222F}}_S \mathbf{B} \cdot \mathrm{d}\mathbf{A} = 0$$

• $\unicode{x222F}_S$: integral over a closed surface (called $S$)
• $\mathbf{B}$: magnetic field
• $\mathrm{d}\mathbf{A}$: infinitesimal element of surface $S$

### Applying the divergence theorem again

• We have: $$\iiint_V (\nabla \cdot \mathbf{F}) \, \mathrm{d}V = {\large \unicode{x222F}}_S \mathbf{F} \cdot \mathrm{d}\mathbf{A}$$
• And we want to apply it to Gauss's law for magnetic fields: $${\large\unicode{x222F}}_S \mathbf{B} \cdot \mathrm{d}\mathbf{A} = 0$$
• Again, use it to rewrite the left hand side: $$\iiint_V (\nabla \cdot \mathbf{B}) \, \mathrm{d}V = 0$$
• As it is true for any $V$, we can drop the integral: $$\nabla \cdot \mathbf{B} = 0.$$

### Gauss's law for magnetic fields

"Magnetic field lines never end"

$$\nabla \cdot \mathbf{B} = 0$$

• No magnetic monopoles!
• All magnets are (at their simplest) dipoles

Other sources: Hyperphysics, Wikipedia, Bleaney & Bleaney, Feynman lectures.

### Recap: Faraday's law of induction

"A time-varying magnetic flux induces an electric field"

$${\large\unicode{x222E}}_{\partial \Sigma} \mathbf{E} \cdot \mathrm{d} \mathbf{l} = - \frac{\partial \Phi_B}{\partial t} \vphantom{\frac{\mathrm{d}}{\mathrm{d}t} \iint_\Sigma}$$

• $\unicode{x222E}_{\partial \Sigma}$: integral over a closed loop ($\partial \Sigma$, boundary of surface $\Sigma$)
• $\mathbf{E}$: electric field
• $\mathrm{d}\mathbf{l}$: infinitesimal line element along loop $\partial \Sigma$
• $\Phi_B$: total magnetic flux through $\Sigma$

$$\require{color}{\large\unicode{x222E}}_{\partial \Sigma} \mathbf{E} \cdot \mathrm{d} \mathbf{l} = - {\color{red} \frac{\mathrm{d}}{\mathrm{d}t} \iint_\Sigma \mathbf{B} \cdot \mathrm{d} \mathbf{A}}$$

• $\unicode{x222E}_{\partial \Sigma}$: integral over a closed loop ($\partial \Sigma$, boundary of surface $\Sigma$)
• $\mathbf{E}$: electric field
• $\mathrm{d}\mathbf{l}$: infinitesimal line element along loop $\partial \Sigma$
• $\iint_\Sigma$: integral over surface $\Sigma$ bordered by $\partial \Sigma$
• $\mathbf{B}$: magnetic field
• $d\mathbf{A}$: infinitesimal element of surface $\Sigma$

### Faraday's law of induction

$$\underbrace{{\large\unicode{x222E}}_{\partial \Sigma} \mathbf{E} \cdot \mathrm{d} \mathbf{l}}_\text{Voltage around closed loop \partial \Sigma} = \underbrace{ - \frac{\mathrm{d}}{\mathrm{d} t} \iint_\Sigma \mathbf{B} \cdot \mathrm{d} \mathbf{A}}_\text{Rate of change of magnetic flux through surface \Sigma}$$

### Stokes' Theorem

"The flux of the curl of a vector field through a surface is equal to the integral of the vector field along the boundary of the surface"

$$\iint_\Sigma (\nabla \times \mathbf{F}) \cdot \mathrm{d}\mathbf{A} = {\large\unicode{x222E}}_{\partial \Sigma} \mathbf{F} \cdot \mathrm{d}\mathbf{l}$$

• True for any $\Sigma$ bounded by $\partial \Sigma$.
• Surface $\Sigma$ is not (in general) closed

### Applying Stokes' Theorem

• We have: $$\iint_\Sigma (\nabla \times \mathbf{F}) \cdot \mathrm{d}\mathbf{A} = {\large\unicode{x222E}}_{\partial \Sigma} \mathbf{F} \cdot \mathrm{d}\mathbf{l}$$
• And we want to apply it to Faraday's law: $${\large\unicode{x222E}}_{\partial \Sigma} \mathbf{E} \cdot \mathrm{d} \mathbf{l} = - \frac{\mathrm{d}}{\mathrm{d} t} \iint_\Sigma \mathbf{B} \cdot \mathrm{d} \mathbf{A}$$
• Use it to rewrite the left hand side: $$\iint_\Sigma (\nabla \times \mathbf{E}) \cdot \mathrm{d}\mathbf{A} = - \frac{\mathrm{d}}{\mathrm{d} t} \iint_\Sigma \mathbf{B} \cdot \mathrm{d} \mathbf{A}$$
• Take the surfaces $\Sigma$ to be the same: $$\fbox{\nabla \times \mathbf{E} = -\frac{\partial \mathbf{B}}{\partial t}}$$

### Faraday's law of induction

"A time-varying magnetic field induces an electric field"

$$\nabla \times \mathbf{E} = -\frac{\partial \mathbf{B}}{\partial t}$$

Direction of the magnetic field? Lenz'z law.

"Nature works to oppose a change in flux"

Other sources: Hyperphysics, Wikipedia, Bleaney & Bleaney, Feynman lectures.

### Lenz's law - video

Video: CC-BY-SA 4.0 Andrejdam.

### Recap: Ampère's law

(actually due to Maxwell!)

$${\large\unicode{x222E}}_{\partial \Sigma} \mathbf{B} \cdot \mathrm{d} \mathbf{l} = \mu_0 \iint_\Sigma \mathbf{J} \cdot \mathrm{d}\mathbf{A}$$

• $\unicode{x222E}_{\partial \Sigma}$: integral over a closed loop ($\partial \Sigma$, boundary of surface $\Sigma$)
• $\mathbf{B}$: magnetic field
• $\mathrm{d}\mathbf{l}$: infinitesimal line element along loop $\partial \Sigma$
• $\mu_0$: permeability of free space
• $\iint_\Sigma$: integral over surface $\Sigma$
• $\mathbf{J}$: electric current density
• $\mathrm{d}\mathbf{A}$: infinitesimal surface element of $\Sigma$

### Applying Stokes' Theorem

• We have: $$\iint_\Sigma (\nabla \times \mathbf{F}) \cdot \mathrm{d}\mathbf{A} = {\large\unicode{x222E}}_{\partial \Sigma} \mathbf{F} \cdot \mathrm{d}\mathbf{l}$$
• And we want to apply it to Ampère's law: $${\large\unicode{x222E}}_{\partial \Sigma} \mathbf{B} \cdot \mathrm{d} \mathbf{l} = \mu_0 \iint_\Sigma \mathbf{J} \cdot \mathrm{d}\mathbf{A}$$
• Use it to rewrite the left hand side: $$\iint_\Sigma (\nabla \times \mathbf{B}) \cdot \mathrm{d}\mathbf{A} =\mu_0 \iint_\Sigma \mathbf{J} \cdot \mathrm{d}\mathbf{A}$$
• Take the surfaces $\Sigma$ to be the same: $$\fbox{\nabla \times \mathbf{B} = \mu_0\mathbf{J}}$$

### Ampère's law - video

Video: CC-BY-SA 4.0 Andrejdam.

### Something is missing

• For any vector field $\mathbf{F}$, the following is true: $$\nabla \cdot (\nabla \times \mathbf{F}) = 0$$
• This works for Faraday's law: $$\nabla \cdot (\nabla \times \mathbf{E}) = \nabla\cdot\left( - \frac{\partial \mathbf{B}}{\partial t} \right) = - \frac{\partial}{\partial t} (\nabla \cdot \mathbf{B}) = 0.$$
• But for Ampère's law: $$\nabla \cdot (\nabla \times \mathbf{B}) = \mu_0 (\nabla\cdot\mathbf{J}).$$ 🤔 $\nabla \cdot \mathbf{J} \neq 0$ in general (capacitors?)
• In fact, there is the continuity equation $$\nabla \cdot \mathbf{J} = -\frac{\partial \rho}{\partial t}$$ and we must take this into account.

### The displacement current

• Starting with the continuity equation $$\nabla \cdot \mathbf{J} = -\frac{\partial \rho}{\partial t}$$
• Now apply Gauss's law $$\nabla \cdot \mathbf{J} = -\frac{\partial \rho}{\partial t} = -\frac{\partial}{\partial t}(\varepsilon_0 \nabla \cdot \mathbf{E}) = -\nabla \cdot \left( \varepsilon_0 \frac{\partial \mathbf{E}}{\partial t} \right)$$
• The bit $\epsilon_0 \frac{\partial \mathbf{E}}{\partial t}$ will fix the problem with Ampère's law
• Maxwell called this the displacement current...
...but it is not a real current

### Ampère's law

"A time-varying electric flux induces a magnetic field"

With the displacement current piece added: $$\fbox{ \nabla \times \mathbf{B} = \mu_0 \mathbf{J} + \mu_0 \varepsilon_0 \frac{\partial \mathbf{E}}{\partial t} }$$

(quick check...) $$\nabla \cdot (\nabla \times\mathbf{B}) = \mu_0 \big(\underbrace{\nabla \cdot \mathbf{J} + \frac{\partial}{\partial t} \underbrace{\varepsilon_0 \nabla \cdot \mathbf{E}}_{= \rho}}_{ = 0} \big)$$

Other sources: Hyperphysics, Wikipedia, Bleaney & Bleaney, Feynman lectures.

### Summary: Maxwell's equations

\begin{align} \nabla \cdot \mathbf{E} & = \frac{\rho}{\varepsilon_0} & \text{Gauss's law}\\ \nabla \cdot \mathbf{B} & = 0 \vphantom{\rho_m} \\ \nabla \times \mathbf{E} & = - \frac{\partial \mathbf{B}}{\partial t} & \text{Faraday's law} \\ \nabla \times \mathbf{B} & = \mu_0 \mathbf{J} + \mu_0 \varepsilon_0 \frac{\partial \mathbf{E}}{\partial t} & \text{Ampère's law} \end{align}

Other sources: Hyperphysics, Wikipedia, Bleaney & Bleaney, Feynman lectures.

### Maxwell's equations in context

• Unification of forces
• Electricity and magnetism are facets of a single theory
• Later, we incorporated weak nuclear interactions into an expanded theory - electroweak theory
• Light as travelling electromagnetic waves
• Maxwell came up with the "displacement current" in 1861 as a purely theoretical construct
• It had the consequence of predicting that travelling electromagnetic waves existed
• The existence of electromagnetic travelling waves was then shown by Hertz in 1888

Other sources: Hyperphysics on unification, EM waves; Feynman lectures on EM waves and unification.

### No free monopoles

Picture: Corinne Mucha [source and further reading]

### No free monopoles

$$\nabla \cdot \mathbf{B} = 0$$
• In 1931 Dirac showed that if at least one magnetic monopole existed somewhere, electric charge would be quantised
• Searches for magnetic monopoles have been going on for decades but we have never seen one
• The MoEDAL experiment at the LHC is looking for evidence of magnetic monopoles

### How do Maxwell's equations change?

\begin{align} \nabla \cdot \mathbf{E} & = \frac{\rho}{\varepsilon_0} \\ \nabla \cdot \mathbf{B} & = 0 \vphantom{\rho_m} \\ \nabla \times \mathbf{E} & = - \frac{\partial \mathbf{B}}{\partial t} \\ \nabla \times \mathbf{B} & = \mu_0 \mathbf{J} + \mu_0 \varepsilon_0 \frac{\partial \mathbf{E}}{\partial t} \end{align}

\require{color} \begin{align} \nabla \cdot \mathbf{E} & = \frac{\rho_{\color{red}e}}{\varepsilon_0} \\ \nabla \cdot \mathbf{B} & = {\color{red}\rho_m} \\ \nabla \times \mathbf{E} & = - \frac{\partial \mathbf{B}}{\partial t} {\color{red} - \mathbf{J}_m}\\ \nabla \times \mathbf{B} & = \mu_0 \mathbf{J}_{\color{red} e} + \mu_0 \varepsilon_0 \frac{\partial \mathbf{E}}{\partial t} \end{align}

More symmetrical!

### Maxwell's equations - summary

• Describe time evolution of $\mathbf{E}$ and $\mathbf{B}$ fields
• Their differential form implies conservation of charge - through the "displacement current"
• Led to the prediction of electromagnetic waves
• Unification of electricity and magnetism
• Further developments led to special relativity

### Next time

• Travelling electromagnetic waves
(check out quick review question!)
• Maxwell's equations in media
• Boundary conditions
• Snell's law