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lec20 - Scott Hughes Massachusetts Institute of Technology...

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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 Maxwell’s equations In our last lecture, we finally ended up with Maxwell’s equations, the four equations which encapsulate everything we know about electricity and magnetism. These equations are: Gauss’s law: ~ ∇ · ~ E = 4 πρ Magnetic law: ~ ∇ · ~ B = 0 Faraday’s law: ~ ∇ × ~ E = - 1 c ~ B ∂t Generalized Ampere’s law: ~ ∇ × ~ B = 4 π c ~ J + 1 c ~ E ∂t . In this lecture, we will focus on the source free versions of these equations: we set ρ = 0 and ~ J = 0. We then have ~ ∇ · ~ E = 0 ~ ∇ · ~ B = 0 ~ ∇ × ~ E = - 1 c ~ B ∂t ~ ∇ × ~ B = 1 c ~ E ∂t . The source free Maxwell’s 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. Let’s begin by looking at ~ ∇ × ~ ∇ × ~ E · = ~ ∇ × - 1 c ~ B ∂t . The curl of the left-hand 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 you’d 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 right-hand side: ~ ∇ × - 1 c ~ B ∂t = - 1 c ∂t ~ ∇ × ~ B · = - 1 c 2 2 ~ E ∂t 2 . 186
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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 . This equation is satisfied by ANY function whatsoever 1 provided that the argument of the function is written in the following special way: f = f ( x ± vt ) . This is easy to prove. Let u = x ± vt , so that f = f ( u ). Then, using the chain rule, ∂f ∂x = ∂u ∂x ∂f ∂u = ∂f ∂u ; it follows quite obviously that 2 f ∂x 2 = 2 f ∂u 2 .
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