Lec21 - Scott Hughes 4 May 2004 Massachusetts Institute of...

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Unformatted text preview: Scott Hughes 4 May 2004 Massachusetts Institute of Technology Department of Physics 8.022 Spring 2004 Lecture 21: Polarization & Scattering 21.1 Summary: radiation so far In the last few lectures, we examined solutions of the source free Maxwell equations: ~ ~ E = 0 ~ ~ E =- 1 c ~ B t ~ ~ B = 0 ~ ~ B = 1 c ~ E t . With a little massaging, we discovered that these equations can be rewritten as wave equa- tions for ~ E and ~ B : 2 ~ E t 2- c 2 2 ~ E = 0 2 ~ B t 2- c 2 2 ~ B = 0 . A particularly instructive solution to the wave equations are the plane wave forms: ~ E ( ~ r, t ) = ~ E sin( ~ k ~ r- t ) ~ B ( ~ r, t ) = ~ B sin( ~ k ~ r- t ) . This solution represents an electromagnetic wave propagating in the k = ~ k/k direction (where k = q ~ k ~ k = q k 2 x + k 2 y + k 2 z ). By considering how the wave behaves at some fixed time, we learned that k is simply related to the wavelength : k = 2 / . The requirement that this solution satisfy the wave equation lead us to the condition = ck . From the definition = 2 (angular frequency is 2 radians times regular frequency), we then obtain = c . Finally, requiring that the plane wave solution satisfy all of Maxwells equations leads to some important constraints on the vector amplitudes ~ E and ~ B . These constraints are: The amplitudes are orthogonal to the propagation direction: k ~ E = 0, k ~ B = 0. The amplitudes are orthogonal to each otherL ~ E ~ B = 0. The amplitudes have the same magnitude: | ~ E | = | ~ B | . The propagation direction is parallel to ~ E ~ B . These are important and rather constraining conditions. Nonetheless, they leave us with a great deal of freedom in the amplitudes. This freedom is described in terms of the radiations polarization state . 21.2 Linear polarization Since plane waves propagate in a straight line, we might as well just define their propagation direction as something simple and be done with it. In what follows, we will take k = z , so our wave is of the form ~ E = ~ E sin( kz- t ) ~ B = ~ B sin( kz- t ) . Suppose that the wave is arranged (somehow) so that the electric field is aligned with the x axis: ~ E = E x . Then, our requirement that ~ E ~ B be parallel to k tells us that ~ B = E y . (Were also using | ~ E | = | ~ B | .) This configuration is known as a linearly polarized wave, as the electric (and magnetic) fields at all points align parallel to a line. Such an electromagnetic wave is quite simple to produce: we just need to take a conductor and arrange things so that it has an oscillating charge distribution. Heres an example: + + +--- + + +--- t = 0 t = t = /2 The top part of this figure shows an antenna : a long conductor in which we drive a very rapidly oscillating current, so that the conductor has a charge distribution . As we move to the right, time increases.the right, time increases....
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This note was uploaded on 04/11/2008 for the course PHYSICS 8.022 taught by Professor Hughes during the Spring '08 term at MIT.

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Lec21 - Scott Hughes 4 May 2004 Massachusetts Institute of...

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