# More on waves and electromagnetism

In the last post I mentioned three key aspects connected with the light: waves, photons and (barely touched) electromagnetism. Today I will go more in depth into these concepts.

#### What are waves?

Explaining it in the most simple way it is a transport (movement) of energy without transport of mass. So in that way:

• speeding car is not a wave while
• speeding car that crashed into the tree and making a lot of noise is the source of wave – actually this noise is the wave.

In the latter case crashing made the ambient particles (air) rapidly lean out of their equilibrium position causing neighboring particles to move also. Their movement causes the movement of the others – and so on. This is exactly the wave. Being more precise this is a mechanical (in medium) longitudinal wave (acoustic wave). Each particle vibrates with a certain speed given by the frequency. It tells you how many oscillations it does in a unit of time and is measured in Herz (Hz).

#### So photons which create light are just crazy balls hitting each other?

Not at all. Almost.

First of all, in the classical theory photons, which are portions of energy, do not interact with each other. They can however interact with matter.

Look, I just stated that photons are able to interact with everything that has mass and they are unable to interact with each other at the same time. Your conclusion should be:

Photons are massless particles

This is the level of abstraction which physics uses on daily basis. Still, this is only the first level of it! Even if it does not have mass it has an energy that can be transported even on long distances (look in the sky during the cloudless night).

#### But what is the real cause of the light?

The authors of all that mess are charged particles (electrons, ions, protons etc.). Their movement is the source of electromagnetic field. Any disturbance generated in that field is a wave that moves away from the source and independently of it. Each wave has two vector components – electric $\vec{E}$ and magnetic $\vec{B}$ – bound tightly with each other.

#### The math involved

The depenedencies between electric and magnetic fields are described by the set o 4 equations, called after James Clerk Maxwell Maxwell’s equations. He did not discover them but rather systematized the knowledge about electromagnetism and added improvement to the one of them. Still, this is what governs everything (in vacuum):

1. $\nabla \cdot \vec{E} = 0$
2. ${\displaystyle \nabla \cdot {\vec {B}}=0}$
3. ${\displaystyle \nabla \times {\vec {E}}=-{\frac {\partial {\vec {B}}}{\partial t}}}$
4. ${\displaystyle \nabla \times {\vec {B}}=\mu _{0}\varepsilon _{0}{\frac {\partial {\vec {E}}}{\partial t}}}$

Even if you are not familiar at all with nabla math you clearly understand that these equations exhibit some kind of symmetry. Behaviour (change) of electric field induces reaction of the magnetic field and vice versa.

#### Even more math and interesting properties of electromagnetic waves

First, let’s look at this mathematical identity (where $\vec{R}$ is a whatever vector function):

$\nabla \times \nabla \times \vec{R} = \nabla(\nabla\vec{R}) - \nabla^2\vec{R}$

Now let’s add rotation ($\nabla \times$) on both sides of equations #3 and #4:

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

${\displaystyle \nabla \times \nabla \times {\vec {B}}=\mu _{0}\varepsilon _{0}{\frac {\partial {\nabla \times \vec {E}}}{\partial t}}}$

Quick look at the presented identity and first two Maxwell’s equations allows us to simplify them a lot:

$\nabla^2\vec{E} = \mu _{0}\varepsilon _{0}\frac{\partial^2\vec{E}}{\partial t^2}$

$\nabla^2\vec{B} =\mu _{0}\varepsilon _{0}\frac{\partial^2\vec{B}}{\partial t^2}$

These are just simple wave equations with the propagation constant $c = \frac{1}{\sqrt{\mu _{0}\varepsilon _{0}}}$. Hence, we know now that these two fields behave like waves that are dependent on each other.

Coefficients in these equations are magnetic and electric vacuum permeability. When you calculate the value of $c$ you will get something about $3\cdot 10^8\ \mathrm{\frac{m}{s}}$. This is the Maxwell’s reaction (so taking into account the knowledge of his times) to that result:

This velocity [i.e., his theoretical prediction] is so nearly that of light ($3.153\cdot 10^8\ \mathrm{\frac{m}{s}}$), that it seems we have strong reason to conclude that light itself (including radiant heat, and other radiations if any) is an electromagnetic disturbance in the form of waves propagated through the electromagnetic field according to electromagnetic laws.

#### Electromagnetic waves: longitudinal or transverse?

Until now we know that light is a wave that consists of two components: electrical $\vec{E}$ and magnetical $\vec{B}$. They both meet the wave equation but we still do not know the spatial relationships between them.

Let’s assume that we have an electric field in 3D space (x, y, z) which depends on only one spatial component:

$\vec{E}(x, t) = [E_x(x,t), E_y(x,t), E_z(x, t)]$

Such a wave will be propagating in the x direction of the space. Now we can put it into #1 Maxwell’s equation:

$\nabla\vec{E}(x, t) = \frac{\partial E_x(x, t)}{\partial x} + \frac{\partial E_y(x, t)}{\partial y} + \frac{\partial E_z(x, t)}{\partial z} = \frac{\partial E_x(x, t)}{\partial x} = 0$

Since $E_y$ and $E_z$ components depend only on x they cancel from the sum. The partial derivative that left is equal to 0. Saying that – there are no changes of $\vec{E}(x, t)$ in the direction of propagation (x). Electric field associated with this wave is transverse!

In general, the orientation of $E_y$ and $E_z$ components can be whatever and vary in time. Let’s polarize our field and make it vibrate only in y-direction $E_y\neq 0$ and $E_z=0$. This will simplify calculations a lot preserving the generality of the results.

So, if we take #3 Maxwell’s equation we will eventually come to this solution where $B_x$  and $B_y$ components are constant with respect to the time leaving:

$\frac{\partial E_y(x, t)}{\partial x} = -\frac{\partial B_z(x, y, z, t)}{\partial t}$

Hence magnetic field can only have z-component.

Look, I just showed you that propagation direction (x), electric (y) and magnetic fields (z) are perpendicular to each other (in free space). It can be also easily shown that amount of oscillation (amplitude) at every point in space and time of these two fields is connected with the speed of light:

$E_y=cB_z$

Therefore we say that these fields are oscillating with the same phase.

#### Time to sum up

In that post I wanted to expand the concepts mentioned previously when I was avoiding using full electromagnetic radiation term as much as possible.

From now on you should know that photons, which are massless particles, are convinient level of abstraction for complicated but yet orderly dependencies between electric and magnetic fields. These dependencies are entirely described by the set of Maxwell’s equations.

Having in mind that this blog is devoted to photography topics Maxwell’s equations were presented entirely in the context of light in the simplest form (in vaccum). If you wish, I can think about an article that would give you more mathematical and physical understanding.

As the last thing, I would like to thank everybody for your feedback and I encourage you to subscribe to this blog if you want to be informed about new content first.

#### Useful references

Differential operators

Bible of optics: E. Hecht, “Optics”, Addison Wesely