Energy and Momentum Analysis in Radiation

Electromagnetic Radiations in the Classical Model

Electromagnetic radiation can essentially be described as stream of photons. These photons are defined as chargeless and massless, however they have both energy and surprisingly, given their lack of mass, momentum, which can be calculated from their wave properties.

Waves were poorly understood until the 1900s, when Max Planck and Albert Einstein developed modern corrections to classical theory.

Planck theorized that "black bodies" or thermal radiators and other forms of electromagnetic radiation existed not as a continuous spectrum but rather in discrete or "quantized" form. This implied that there were only certain energy values that an electromagnetic wave could have.

In the classical model that we study now it is mentioned that electromagnetic radiation carries both momentum and energy, and can impart both energy and momentum to matter. We shall discuss these ideas in more detail.

Energy in Electromagnetic Radiation

Let's start by understanding how a pulse of electromagnetic radiation would interact with ordinary matter. To see this effect, we shall look at a single charged particle interacting with the electromagnetic radiation which is composed of perpendicular electric and magnetic fields and see what happens when the pulse of radiation goes by.

Until the pulse reaches the charged particle nothing happens. Now let's assume that the electromagnetic radiation has a width of "w" so the pulse would last for a short time (w/c) where c is the speed of light. Now upon the arrival of the pulse, the charged particle (assuming it is positive) experiences a force (F = qE, where E is the magnitude of the electric field in the electromagnetic radiation) in the direction of the electric field that comprises the radiation and this brief impulse i.e. the product of force and time duration gives the charged particle a momentum. According to the Newton's Second Law of Motion:

[math]\displaystyle{ \Delta p = p - 0 = F \Delta t = (qE)*(w/c) }[/math]

Now this interaction is over so quickly that the charged particle hardly has the time to move a significant distance in the direction of the field. However, if attach the charged particle to a spring then it will oscillate to and fro in the direction of applied force. Since the kinetic energy of the charged particle has to come from somewhere, we conclude that it was imparted to the charged particle by the radiation which means that the there must be some energy carried by the electromagnetic radiation which was transferred.

Since the momentum is proportional to the magnitude of the electric field E in the pulse, we have the following:

[math]\displaystyle{ \Delta K = K - 0 \approx \frac{p^2}{2m} = (qE \frac{w}{c})^2(\frac{1}{2m}) }[/math]

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• Source: Boundless. “Energy and Momentum.” Boundless Physics. Boundless, 21 Jul. 2015. Retrieved 25 Nov. 2015 from https://www.boundless.com/physics/textbooks/boundless-physics-textbook/electromagnetic-waves-23/electromagnetic-waves-and-their-properties-166/energy-and-momentum-601-11184/