Introduction to Electrodynamics/Chapter 8/2/2
Contents
8.2.2 Energy and Momentum of Electromagnetic Waves
We can calculate the energy and momentum of waves. I also introduce a new quantity, intensity.
Hi, this is Jonathan Gardner. We're in section 8.2.2 of Griffith's Introduction to Electrodynamics, 2nd Edition. We just figured out how monochromatic plane waves electromagnetic waves behave in a vacuum, and how to calculate the E and B fields at any given point in time for a given wave of specific frequency. Now we're going to look at the energy and momentum, and introduce intensity.
Keep in mind, what follows applies only to monochromatic plane waves; there are other types of waves, and we're not talking about them.
Total Energy
The Energy is simple to calculate:
<math> U = {1 \over 2} \left( \epsilon_0 E^2 + {1 \over \mu_0} B^2\right) </math>
Substituting in the fact that for EM waves <math>B = E/c = \sqrt{\mu_0 \epsilon_0} E\;</math>:
<math> \begin{align} U &= {1 \over 2} \left(\epsilon_0 E^2 + \epsilon_0 E^2\right)\\
&= \epsilon_0 E^2 \\ &= \epsilon_0 E_0^2 \cos^2(\kappa x - \omega t + \delta) \\
\end{align} </math>
Note that the E field fluctuates between 0 and E0, so the energy at a point fluctuates between 0 and <math>\epsilon_0 E_0^2</math>. Where does the energy go? It follows the maximum which propagates in the direction of motion -- the x axis in this form.
(draw a graph)
Energy Flux Density
The Poynting Vector gives us the energy flux density, the rate of energy passing through a surface.
<math> \vec{S} = {1 \over \mu_0} (\vec{E} \times \vec{B}) </math>
The E and B fields are always perpendicular, so the cross product is easy. It points in the direction of travel -- the x axis, if you write things out like this:
<math> \begin{align} \vec{S} &= {1 \over \mu_0} E_0 \cos (\kappa x - \omega t + \delta) {E_0 \over c} \cos (\kappa x - \omega t + \delta) \hat{i}\\
&= c \epsilon_0 E_0^2 \cos^2(\kappa x - \omega t + \delta) \hat{i} \\
&= c U \hat{i} \\
\end{align} </math>
Evidently, the point at which E and B are maximum represent when the greatest flow of energy is occurring, and it points along the direction of motion. The energy moves in pulses. This matches what we saw with the energy being greatest when the fields are strongest. The factor of c is obvious -- that's the speed the wave is moving at, and so it is the rate of energy flow.
(Add to the graph)
Momentum
Recall that EM fields can carry momentum.
<math> \vec{\mathit{p}} = {1 \over c^2}\vec{S} = {1 \over c} U \hat{i} </math>
The magnitude of U is simply pc. Where have we seen this before? Nudge nudge, wink wink.
Intensity
This is all well and good, but in real life, you often don't care about the phase constant or the energy at a specific point. What you really want to know is how much energy is transferred on average over many cycles. What you can do is calculate how much energy is transferred on average for one complete cycle, or just grab an arbitrarily large amount of data over time and that will give you the average, even if it isn't an exact number of cycles. The partial cycle won't make a meaningful difference.
The average of cos^2 over one cycle is 1/2. So calculating the averages are simple:
<math> \begin{align} &= {1 \over 2} \epsilon_0 E^2 \\ <\vec{S}> &= c <U> \hat {i} \\ <\vec{p}> &= {1 \over c} <U> \hat{i} \\ \end{align} </math>
The average pf the Poynting vector represents the average rate of energy flux, or the average power flux, or the average power per unit area, depending on how you want to say it. This is the intensity I.
<math>I = \;</math>
In the next section, we'll cover how monochromatic plane waves behave in linear media. It's really short and easy. After that, we'll talk about what happens when a wave meets a boundary between two different linear media.
Thanks for watching, and leave your questions in the comments or a video response.
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