### Physics NO. 1 : The momentum of light

29Jun08

Well, everyone of us probably heard in high school (or earlier) that the photon has it’s own momentum.
And I think that more then a half of us kept wondering then about it. Namely, without deep knowledge of Physics, such statements like “light have no rest mass”, or “light is an electromagnetic wave, but it has a momentum” can be confusing, and as Richard Feynman once said – it’s an awful feeling 🙂

Here I want to explain shortly the mechanism of passing the momentum by a photon through classic physics.

Firstly, light is an electromagnetic wave which as the names says is composed from magnetic and electric field. Those two fields are perpendicular to each other and to the direction in which the wave propagates. What makes them propagate? Here my answer can be given only by those two equations:

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

$\nabla \times \bf{B} = - \mu_0 \epsilon_0 \frac{\partial \bf{E}}{ \partial t}$

Those equations mean that whenever there exists a circulation of electric field there should be created a changeable magnetic field and vice versa.

Using Maxwell equations and some mathematics we can derive the following equation $\bf{B} = - \bf{e_{r'}} \times \bf{E} /c$ wchich gives us the direct relationship between the electric and the magnetic field!

Now lets assume that an electric field is vertically positioned in space and the magnetic field horizontally. What happens when an electromagnetic wave passes an electron? The electric field of the electromagnetic wave makes it swing back and forth in the vertical direction what leads the electron to create a new electromagnetic wave in every perpendicular direction to it’s movement. That’s *exactly how the reflection of the light works!

Hold on, but what does the magnetic field of the electromagnetic wave to the electron? Well, I suppose that everyone remember the famous Lorentz Force $\bf{F}= q\bf{v} \times \bf{F}$.The Lorentz force keeps affecting the electron only when it’s in movement, and it is when the electric field affects it. Both fields work then together, but in what direction affects the Lorentz force? In the direction of light propagation. Though when light go over an charged object, which starts to move in the reaction of the affecting field, then a pushing force happens in the direction in which light propagates!

That forced is called the pressure of radiation now, we’ll derive the amount of that pressure. The force equals of course $F=qvB$, but because everything moves, the $\overline{F}$ is an average of force $F$. Moreover $\overline{B}=\overline{E}/c$ and the speed $\overline{v}$ is also taken an avarage, so we have $\overline{F}=q\overline{vE}/c$… but wait a minute! $\overline{qE} = \overline{F'}$ and $\overline{F'v}= \frac{dW}{dt}$ and now we can write $p=\overline{F}dt= \frac{dW}{c}.$ What’s more, from school we know that light keeps with itself energy $W=hv=\hbar \omega$ This leads us to the ultimate result $p=W/c=\hbar \omega /c = \hbar k$ where $k$ is known as wave wavenumber.

Even though we finished, there’s a tousend more things to explore, for example we can find out that $W=\hbar \omega$ and $\bf{p} = \hbar \bf{k}$ create a four-vector like the energy and the momentum in special relativity, but it’s a whole new topic… 😉

I hope you enjoied  my article, feel free to ask if anything’s unclear. I will try to help you as much as I can.

#### 4 Responses to “Physics NO. 1 : The momentum of light”

1. 1 Kuba

Czesc Michał! Oczywiście zepsuje Ci cały blog bo napsize kilka słow po polsku. Blog masz bardzo ladny.. 😛 Ale ja w troche innej sprawie… Mianowicie jak sam widzisz, skuszony Twoimi argumentami o przydatności takiego bloga postanowiłem sam sobie takiego założyć… Oczywiście w celu przechowywania co cenniejszych tożsamości i twierdzeń. Ale gdy dla spróbowania chciałem napisac pierwszego posta zadałem sobie pytanie “Czy oni mają tu zainstalowanego LaTeXa? ” I z tym pytaniem zwracam sie o Ciebie. Minowicie w jaki sposob umieszczasz tu takie ładne wzory? jesli tu jest LaTeX to pewnie przeniosę sie tutaj.. No, dzięki w każdym razie za wszelką informację i pzodrawiam serdecznie! Do zobaczenia!

P.S Wybacz wszystkie błedy ale z przyzwyczajenia nie pisze polskich liter no i przecierz mam dysortografie.. 😛

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