# Lecture 29 - EM Waves and Introduction to Optics

## Maxwell's equations in Vacuum

$\oint\vec{E}\cdot d\vec{A}=\frac{Q}{\varepsilon_{0}}$     →     $\nabla\cdot\vec{E}=\frac{\rho}{\varepsilon_{0}}$

$\oint \vec{B}\cdot d\vec{A}=0$     →     $\nabla\cdot\vec{B}=0$

$\oint\vec{E}\cdot d\vec{l}=-\frac{d\Phi_{B}}{dt}$     →     $\nabla\times\vec{E}=-\frac{\partial\vec{B}}{\partial t}$

$\oint\vec{B}\cdot d\vec{l}=\mu_{0}I+\mu_{0}\varepsilon_{0}\frac{d\Phi_{E}}{dt}$     →      $\nabla\times\vec{B}=\mu_{0}\vec{J}+\mu_{0}\varepsilon_{0}\frac{\partial \vec{E}}{\partial t}$

In the absence of charges and currents:

$\oint\vec{E}\cdot d\vec{A}=0$     →     $\nabla\cdot\vec{E}=0$

$\oint \vec{B}\cdot d\vec{A}=0$     →     $\nabla\cdot\vec{B}=0$

$\oint\vec{E}\cdot d\vec{l}=-\frac{d\Phi_{B}}{dt}$     →     $\nabla\times\vec{E}=-\frac{\partial\vec{B}}{\partial t}$

$\oint\vec{B}\cdot d\vec{l}=\mu_{0}\varepsilon_{0}\frac{d\Phi_{E}}{dt}$     →      $\nabla\times\vec{B}=\mu_{0}\varepsilon_{0}\frac{\partial \vec{E}}{\partial t}$

## Production of Electromagnetic Waves

Historically, transmission of electromagnetic waves was put to use first for the development of radar during World War II.

It involved the development of the klystron, which is at the heart of particle accelerators. See here for example.

## Electromagnetic Wave Equations

We now have two equations

$\frac{\partial E_{y}}{\partial x}=-\frac{\partial B_{z}}{\partial t}$ and $-\frac{\partial B_{z}}{\partial x}=\mu_{0}\varepsilon_{0}\frac{\partial E_{y}}{\partial t}$

which can be combined to give

$$\frac{\partial^{2} E_{y}}{\partial t^{2}}=\frac{1}{\mu_{0}\varepsilon_{0}}\frac{\partial^{2} E_{y}}{\partial x^{2}}$$

$$\frac{\partial^{2} B_{z}}{\partial t^{2}}=\frac{1}{\mu_{0}\varepsilon_{0}}\frac{\partial^{2} B_{z}}{\partial x^{2}}$$

Compare that to:

$$v^2\frac{\partial^{2} D}{\partial x^2}=\frac{\partial^{2} D}{\partial t^2}$$

with a solution

$D(x,t)=A\sin\frac{2\pi}{\lambda}(x-vt)$

where $D$ is some kind of displacement.

Our equations

$\frac{\partial^{2} E_{y}}{\partial t^{2}}=\frac{1}{\mu_{0}\varepsilon_{0}}\frac{\partial^{2} E_{y}}{\partial x^{2}}$ and $\frac{\partial^{2} B_{z}}{\partial t^{2}}=\frac{1}{\mu_{0}\varepsilon_{0}}\frac{\partial^{2} B_{z}}{\partial x^{2}}$

thus give us two waves in phase with one another, but at right angles to one another

$E_{y}(x,t)=E_{0}\sin\frac{2\pi}{\lambda}(x-\sqrt{\frac{1}{\mu_{0}\varepsilon_{0}}}t)$

$B_{z}(x,t)=B_{0}\sin\frac{2\pi}{\lambda}(x-\sqrt{\frac{1}{\mu_{0}\varepsilon_{0}}}t)$

This leads to the prediction for the speed of light (all electromagnetic waves in vacuum):

$v=\sqrt{\frac{1}{\mu_{0}\varepsilon_{0}}}=3.00\times10^{8}\,\mathrm{m/s}=c$

## The electromagnetic spectrum

The equation we used for waves in general $v=f\lambda$ can be applied to electromagnetic waves, $c=f\lambda$, to find the frequency from the wavelength and vice versa. The full electromagnetic spectrum is much broader than the relatively narrow range we can see.

A frequency of 30 MHz corresponds to a wavelength of 10 m.

A frequency of 300 MHz corresponds to a wavelength of 1 m.

A frequency of 3 GHz corresponds to a wavelength of 10 cm.

## Energy in an EM wave

The energy stored in an electric field per unit volume is

$u_{E}=\frac{1}{2}\varepsilon_{0}E^{2}$

This is true in general, but think about a parallel plate capacitor:

$U=\frac{1}{2}CV^2=\frac{1}{2}(\frac{\epsilon_0A}{d})(Ed)^2$,

and since the volume is $Ad$, one gets the formula.

Similarly, one can show that for a magnetic field per unit volume (for example by calculating $1/2LI^2$ for an ideal solenoid)

$u_{B}=\frac{1}{2}\frac{B^{2}}{\mu_{0}}$

The total energy of an EM wave per unit volume is

$u=u_{E}+u_{B}=\frac{1}{2}\varepsilon_{0}E^{2}+\frac{1}{2}\frac{B^{2}}{\mu_{0}}$

and using

$\frac{E}{B}=c=\sqrt{\frac{1}{\mu_{0}\varepsilon_{0}}}$

$u=\varepsilon_{0}E^{2}=\frac{B^{2}}{\mu_{0}}=\sqrt{\frac{\varepsilon_{0}}{\mu_{0}}}EB$

In an electromagnetic wave the fields are moving with velocity $c$ the amount of energy passing through a unit area at any given time is

$S=\varepsilon_{0}cE^{2}=\frac{cB^{2}}{\mu_{0}}=\frac{1}{\mu_{0}}EB$

More generally, the Poynting vector is a vector $\vec{S}$ which represents the flux of energy in an electromagnetic field

$\vec{S}=\frac{1}{\mu_{0}}\vec{E}\times\vec{B}$

As an EM wave carries energy it should be able to exert a force also. The force per unit area exerted by an EM wave is called radiation pressure and was predicted by Maxwell. If light is absorbed by a material the change in momentum which is transferred to the material is

$\Delta p = \Delta F\Delta t = \frac{\Delta U}{\Delta x/\Delta t}=\frac{\Delta U}{c}$

whereas if it is fully reflected

$\Delta p =\frac{2\Delta U}{c}$

Energy is transferred to the object at a rate

$\frac{dU}{dt}=\bar{S}A$

The force is

$F=\frac{dp}{dt}$

so the pressure is

$P=\frac{1}{A}\frac{dp}{dt}$

When the light is absorbed

$P=\frac{1}{Ac}\frac{dU}{dt}=\frac{\bar{S}}{c}$

When it is reflected

$P=\frac{2}{Ac}\frac{dU}{dt}=\frac{2\bar{S}}{c}$

So, this analysis motivates the use of a solar sail.

Check out this description of IKAROS, which successfully deployed a solar sail. Radiometers are devices that react to electromagnetic radiation.

## Ray model of light

We have seen that light is an electromagnetic wave.

We can also reason that if a wave is moving in the $x$ direction there is no reason for it to change it's amplitude or direction if it is traveling in free space.

We can thus consider light to travel in straight lines, which we can also see, especially with a collimated light source such as a laser.

This property means that the behavior of mirrors and lenses can be worked out with geometrical methods and a couple of fundamental rules about reflection and refraction.

This method is described as geometrical optics and will be our focus for the next 4 lectures.

## Reflection

The first rule we have is for reflection. We define an angle of incidence $\theta_{i}$ relative to the surface normal and find that the angle of reflection $\theta_{r}$, also defined relative to the surface normal is given by

$\theta_{r}=\theta_{i}$

## Specular and diffuse reflection

The reflection we have just considered is what we can expect from a smooth surface and is referred to as specular reflection, in addition to this there is reflection from the surface and sub-surface region which occurs in all directions and is called diffuse reflection. When a surface is rough the reflections from the surface occur in many different orientations and the specular reflection is not observed.

## Parabolic mirror

A parabolic reflector has the special property that any ray coming in to the mirror parallel to the principal axis of the parabola will be reflected through the focus of the parabola.

For a parabola $y=ax^{2}$ the focus is at $y=\frac{1}{4a}$