## The Energy of Beamed Photons

The Doppler Effect and Beaming combined.

### introduction

In this section we shall develop the concepts necessary to understand Einstein's first derivation of

### E = mc2

In the following section we shall go through the logic of his derivation.

### animation

The accompanying animation is a combination of the Doppler Effect and Relativistic Beaming. In the one illustration there is a photon depiction of light and a continuous wave depiction of light.

In the animation there are two photons. Both travel perpendicular to the direction of travel of the spacecraft. They also travel in opposite directions. Ultimately only consider the two photons. The wave depiction is used to help illustrate the magnitude of the Doppler Effect on the photons.

In the animation • there are two light wave crests moving out from the light source
• when the spacecraft is stationary the light is of one colour, or in other words, one wavelength.
• it is irrelevant the actual value of the wavelength.
• the wavelength of the light waves near the photons illustrates the wavelength of the photons.
• the wavelength of the photons are plotted for the different selected spacecraft velocities.

### animation controls

Select different velocities to develop a detailed plot of photon wavelength versus spacecraft velocity.

• wavelength: When the wavelength button is selected the photon wavelength is plotted for the selected velocities. This option is the default and will also be selected each time a new velocity is selected.
• energy: When the energy button is selected the photon energy is plotted for all the selected velocities.

### the graphs

The graphics illustrate how the wavelength, and energy, of the photons change for different velocities of the spacecraft. The default plot is the wavelength versus spacecraft velocity plot. As mentioned before the wavelength of the light emitted when the spacecraft is stationary is irrelevant. The plot is the same for all wavelengths. To highlight this the value that is plotted is actually

( wavelength for a specific spacecraft velocity ) / ( wavelength at zero spacecraft velocity )

This always results in a value of 1 for zero spacecraft velocity irrespective of wavelength.

The energy plot is also scaled in a similar way to the wavelength so the energy plotted for zero spacecraft velocity is alway 1. So the value plotted for the energy is

( energy of the photon for a specific spacecraft velocity ) / ( energy at zero spacecraft velocity )

### description

The main observations to make from the animation are

• because of beaming and Doppler Effect combined the wavelength of the photons decreases with increasing velocity of the spacecraft
• the energy of the photons increases with increasing velocity of the spacecraft

If you are patient enough to plot all the wavelengths for all the spacecraft velocities you will notice it follows the same path as that for the time dilation plot. Now this only applies for the two photons depicted in this animation. If the photons were to follow any other path the plot would be different.

It is more useful to consider what the energies of the photons are telling us. As the spacecraft velocity increases both the kinetic energy of the spacecraft and the energy of the photons increases. This will be an important concept for the next section.

### why two photons 180o apart?

A more general statement can be made about the total energy carried away by the photons. As long as the photons are emitted 180o apart the total energy emitted by the two photons will follow the same plot as illustrated in the animation. For example in the animation the photons are emitted at 90o(above) and 270o (below) to the direction of travel. However if we chose 45o and 125o, once again 180o difference, the total energy of the photons for all the different velocities of the spacecraft still follow the same energy plot. The energy plot is not dependent on the angle of emission as long as there is 180o difference and the combined energy of the two photons is plotted. In that way we have got rid of any dependence on angle of emission. We are free to concentrate just on the dependence of the photon energy on the velocity of the spacecraft.