Energy and Momentum in Electromagnetic Waves, Papers of Physics

Download Energy and Momentum in Electromagnetic Waves and more Papers Physics in PDF only on Docsity! Prelecture 22: Slide 2 Today we will look in more detail at some properties of the electromagnetic waves we introduced in the last prelecture. We’ll begin by discussing the energy that is carried by electromagnetic waves. We’ll introduce the Poynting vector that contains all of the energy information. We will define the intensity of the wave as the average value over space and time of the magnitude of the Poynting vector. We will then move on to discuss the momentum that is carried by electromagnetic waves and its use in calculating the radiation pressure that a wave can exert on objects. We will close by discussing frequencies in electromagnetic waves. We will explore the vast range of the electromagnetic spectrum and calculate the Doppler shift, the change in frequency due to relative motion between the wave source and the observer. Prelecture 22: Slide 3 We begin today with a brief discussion of the relationship between the electric and magnetic fields in an electromagnetic wave. Our primary example will be a plane harmonic wave since the mathematics is simple and many waves, far from the source, can be approximated as plane waves. We will assume the electric field has only an x-component given by: Ex = E0sin(kz-ωt). Here the oscillations in space are specified by the wave number k and the oscillations in time are specified by the angular frequency ω. These quantities are related by the velocity of the wave c = ω/k. The velocity of the wave is given by 1/sqrt(µ0ε0) and is numerically equal to the measured speed of light. If the wave travels in the positive z-direction, then the magnetic field will have only a y-component and it will oscillate in phase with Ex. i.e., By = B0sin(kz-ωt). The amplitudes E0 and B0 are also related by the velocity of the wave, c. Namely, c = E0/B0. Such a plane wave is often represented on the printed page by a diagram such as the one you see here. Care needs to be taken in interpreting this picture since lots of different kinds of information are being presented here and it is very easy to get confused. The picture represents a snapshot at a given time, with one spatial axis (the direction of propagation, z, in this case) and two field axes (Ex and By in this case). It’s possible to construct this picture for a plane wave since neither Ex nor By have any x or y dependence. The wave you see in Ex or By at a given z-value exists at all x and at all y values. We’ll close by noting that if we define s-hat as the unit vector in the direction of propagation and e-hat and b-hat as the unit vectors for the E and B fields, we see that s-hat = e-hat X b-hat at any time t. This expression is perfectly general and we will use it to develop a formalism for the energy transport in electromagnetic waves. Note that s-hat = e-hat X b-hat implies that had we assumed a wave propagating in the positive z- direction with Ey and Bx components, we would have Ey and Bx oscillating 180 o out of phase with each other. i.e., if Ey = E0sin(kz-ωt), then Bx = -B0sin(kz-ωt). Prelecture 22: Slide 6 Electromagnetic waves carry energy described by the Poynting vector that specifies the power per unit area in the wave. It can be shown that electromagnetic waves carry momentum also described by the Poynting vector. In particular, the momentum density (the momentum per unit volume) is given by dp/dV = S/c2. We can convert this momentum density into a flow rate by considering the volume dV swept out by a cross-sectional area A of the wave over a time dt, where dV = Acdt. We then obtain 1/A dp/dt = S/c. Once again we will average over time to obtain a more physically relevant quantity. Now, dp/dt is the time rate of change of momentum which is a force. Therefore, if we obtain the average value of dp/dt and divide by A, we get the force per unit area, or pressure that an electromagnetic wave can exert on any object it may encounter. The result is that this radiation pressure is just given by the average value of the magnitude of S divided by c which is just the intensity I divided by c. It should be noted that while this discussion has been totally classical in the sense that the wave is considered to be a continuous form; in fact, the energy and momentum in an electromagnetic form are carried by photons, the quanta of electromagnetic radiation. Photons are quantum objects that exhibit, at different times, properties of both waves and particles. Individual photons have an energy that is proportional to their frequency; namely E = hf, where h is Planck’s constant = 1.24 X 10-6 eV-m. Photons also carry momentum = E/c = h/λ. Prelecture 22: Slide 7 To this point, our discussion has been abstract and general. We will now consider a specific ordinary example to get some idea of the magnitude of these quantities. Let’s look at the energy and momentum present in the sunlight that arrives on Earth. A typical value for the intensity of sunlight here is I = 100 mW/cm2. Note that this solar energy is non-negligible. I you were able to produce a 100% efficient solar cell with area 1 square meter, it would produce power at the rate of 1 kilowatt! Though the energy is sizable, the pressure exerted by sunlight is pretty small. If the sunlight is totally absorbed, the pressure it would exert is equal to I/c = 3.3 X10-6 N/m2. If the radiation is totally reflected, the pressure will increase by a factor of two since the momentum transfer to the object must be twice as big since the wave has reversed its direction. Finally, let’s look at the size of the amplitude of the electric field oscillations in sunlight. We can obtain this value directly from the intensity. Namely, since I = ½ E0 2/377Ω, we obtain E0 = 868 V/m. Note that this field is equivalent to that produced by connecting your power outlet (120 V) across conductors that are separated by 14 cm. Prelecture 22: Slide 8 All electromagnetic waves travel at the speed of light in empty space. We usually associate “light” with visible light, having wavelengths in the 380 – 750 nm range which correspond to frequencies in the range of 8 – 4 X 1014 Hz. The energy of individual photons that comprise visible light range from 3 – 1.6 eV. Visible light, though, makes up only a tiny fraction of the electromagnetic spectrum. To illustrate this statement, let’s consider examples of electromagnetic radiation with energies much lower and much higher than visible light. On the low energy end, for example, we have radio waves. Your i-clicker operates at a frequency of 900 MHz which corresponds to a wavelength of about 1/3 of a meter. The energy of an individual i-clicker photon is just 4 X 10-7 eV. On the other end of the spectrum we have gamma rays. At Fermilab, for example, photons are produced with energies approaching 1 TeV (that’s 1012 eV !). The corresponding frequency is quite high (2 X 1026 Hz) and the wavelength is tiny (10-18 m). These examples span almost 20 orders of magnitude. Electromagnetic radiation is indeed pervasive, playing an important role in almost everything we experience.