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Unformatted text preview: Scott Hughes 28 April 2005 Massachusetts Institute of Technology Department of Physics 8.022 Spring 2005 Lecture 20: Wave equation & electromagnetic radiation 20.1 Revisiting Maxwells equations In our last lecture, we finally ended up with Maxwells equations, the four equations which encapsulate everything we know about electricity and magnetism. These equations are: Gausss law: ~ ~ E = 4 Magnetic law: ~ ~ B = 0 Faradays law: ~ ~ E = 1 c ~ B t Generalized Amperes law: ~ ~ B = 4 c ~ J + 1 c ~ E t . In this lecture, we will focus on the source free versions of these equations: we set = 0 and ~ J = 0. We then have ~ ~ E = ~ ~ B = ~ ~ E = 1 c ~ B t ~ ~ B = 1 c ~ E t . The source free Maxwells equations show us that ~ E and ~ B are coupled : variations in ~ E act as a source for ~ B , which in turn acts as a source for ~ E , which in turn acts as a source for ~ B , which ... The goal of this lecture is to fully understand this coupled behavior. To do so, we will find it easiest to first uncouple these equations. We do this by taking the curl of each equation. Lets begin by looking at ~ ~ ~ E = ~  1 c ~ B t . The curl of the lefthand side of this equation is ~ ~ ~ E = ~ ~ E  2 ~ E = 2 ~ E . (We used this curl identity back in Lecture 13; it is simple to prove, albeit not exactly something youd want to do at a party.) The simplification follows because we have restricted ourselves to the source free equations we have ~ ~ E = 0. Now, look at the curl of the righthand side: ~  1 c ~ B t = 1 c t ~ ~ B = 1 c 2 2 ~ E t 2 . 186 Putting the left and right sides together, we end up with 2 ~ E t 2 c 2 2 ~ E = 0 . Repeating this procedure for the other equation, we end up with something that is essentially identical, but for the magnetic field: 2 ~ B t 2 c 2 2 ~ B = 0 . As we shall now discuss, these equations are particularly special and important. 20.2 The wave equation Let us focus on the equation for ~ E ; everything we do will obviously pertain to the ~ B equation as well. Furthermore, we will simplify things initially by imagining that ~ E only depends on x and t . The equation we derived for ~ E then reduces to 2 ~ E t 2 c 2 2 ~ E x 2 = 0 . 20.2.1 General considerations At this point, it is worth taking a brief detour to talk about equations of this form more generally. This equation for the electric field is a special case of 2 f t 2 v 2 2 f x 2 = 0 ....
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This note was uploaded on 07/19/2010 for the course 8 8.022 taught by Professor Scotthughes during the Spring '10 term at MIT.
 Spring '10
 ScottHughes

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