Derivation and Solution of the Electromagnetic Wave Equations

The origin (and solution) of the electromagnetic wave equations

Conservation of charge

Charge conservation requires that the time rate of change of the total charge enclosed within a volume V must equal the net current flowing into the surface S enclosing the volume. Note: For this project, you won’t need to know any physics, but you’ll need to know some physics terminology. So, to make this project doable by those of you who haven’t studied electricity and magnetism yet, I’ve included a chart of terms and symbols you’ll need to know. The information comes from the student’s favorite font of knowledge, Wikipedia.

1. Write the vector equation that expresses this relationship, letting J be the current density (in amperes per square meter) flowing through the surface and ρ the charge density (in coulombs per cubic meter) at each point in the volume. Hint: You’ll have a surface (double) integral (flux) on one side of the equal sign and a volume (triple) integral on the other side. (Because this one is your first question, you can check your response here.)

Assuming the functions are smooth enough, we can bring the differentiation inside the integral to get

骣 r 蝌J�d S - 蝌琪 dV S V 桫 t (M-1 )

The divergence theorem lets us rewrite the left-hand side of this equation in another form.

2. Write down the symbolic statement of the divergence theorem. 3. Apply it to the left-hand side of (M-1) above. 4. Equate the resulting triple integral with the one in (M-1) 5. For sufficiently smooth functions (which we’ll assume these are), if two integrals are equal the integrands are also equal; why? 6. Make use of this property of smooth functions to write an expression for divJ in terms of ρ. 7. State this relationship in words.

Notice that what this process has done is allow us to take information about macroscopically observed quantities and infer something about the properties of the underlying “things”/phenomena that produce them.

Ampere’s Law (as corrected by Maxwell):

The following expression is a mathematical representation of Ampere’s Law as corrected by James Clerk himself:

骣蝌H�d s � 蝌 J d S琪 e E d S C St 桫 S (M-2)

Stokes’ Theorem lets us convert this to differential form:

8. Write Stokes’ Theorem 9. Use it to convert the left-hand side of (M-2) into an expression involving a double integral. 10. Write the equation that results from equating the double integral you just produced with the right-hand side of (M-2). Call this expression (M-3). 11. The two integrals on the right-hand side of (M-3) can be combined under a single integral sign. Why? 12. Make the combination. Here’s another checkpoint for you. 13. Recognizing that  there are no electric charges in free space (and that J is, therefore, 0 under those circumstances); and  that H can be expressed in terms of B; write the expression for curlB that can come from the previous step under these conditions. On to the wave equations…

The difficulty with Maxwell’s equations, if we want to solve for the electric field E and the magnetic field B as functions of position in space and as time after some event, is that they are coupled. That is, while we have an equation for curlE that we could presumably integrate, it is expressed in terms of B rather than E. Hence, we’d have to be able to know how each incremental change in E changed B which in turn would change E again, which in turn would change… well, you get the point.

So, to be able actually to solve these equations, we need to decouple them. In other words, we want an expression for some derivative of E that can be expressed in terms of some other derivatives or functions of E and external constants.

Because I’m a fan of starting with easier things rather than harder things, we’ll look at the way electromagnetic waves behave in free space (i.e., in a vacuum where there aren’t any charges or magnets). Because I am—still—incredibly kind, I’m reproducing Maxwell’s Laws for free space in the table below:

炎 E = 0

* This is different where there are charges! 汛 E= - B t

炎 B = 0

汛 B= m e E 0 0 t * This is different where there are charges!

Maxwell’s Laws (in free space)

14. Show that the following is a solution of each of Maxwell’s equations in free space: E = B = 0.

There might, though, be more interesting solutions than simply the non-existence of magnetic and electric fields. Or, so we can hope. We shall, at least, pursue them hopefully…

The approach we’ll use starts with a key observation: we have expressions for curlB in terms of E (actually, one of its derivatives) and for curlE in terms of B.

Look at the second equation in the table. If we take the curl of both sides, we’ll have an expression for the curl of curlE (on the left) equal to an expression for curlB (or, technically, the curl of its derivative). But we already know (fourth equation in the table) that we can express curlB in terms of E. Hence, if things work out well with these derivatives, we may be able to turn the curl (curlE) into an expression involving some constants and derivatives of E itself. ‘Tis a consummation, to quote Hamlet, devoutly to be wished…

In order to make this work, though, we need to remember (or perhaps learn), what happens when we take the curl of the curl of any general vector function A. Instead of making you figure that out, though, I’ll let you shortcut things a bit and simply demonstrate to me that the result with which I’ll supply you is, in fact, correct.

15. Prove the following vector identity:

2 汛( 汛A) = 蜒( g A) - A

16. Apply the curl operator to both sides of the second of Maxwell’s laws, using his other laws when necessary, to come up with the following way-cool result: 2 2 E � E m0 e 0 2 t (WCR—way cool result)

17. Repeat the process to come up with a similar equation for the magnetic field B. 18. Look up the values of μ0 and ε0 and calculate the constant of proportionality between the two derivatives.

A primer on waves

The basic or general wave equation satisfied by all planar waves is simply

2 2 1 f � f 2 2 v t where f is the function representing the particular wave and v is the velocity of propagation of the wave in the medium under consideration.

From simply the definition of a wave—a disturbance of a continuous medium that propagates with a fixed shape at a constant velocity1—one can come up with a mathematical statement that contains the essence of wave motion:

f(z,t) = f(z-vt,0)=g(z-vt)

Hence, all the following represent waves

A 2 f( z , t ) = f( z , t ) = Ae-b( z - vt ) f( z , t )= A sin[ b ( z - vt )] 3 2 1 2 b( z- vt ) + 1 but the following do not. 2 f( z , t ) = Ae-b( bz + vt ) f( z , t )= A sin( bz )cos( bvt )3 4 5

19. Show that f1 does in fact satisfy the general wave equation 20. Show that f4 does not satisfy the general wave equation.

End of primer…

21. Show that the in the expressions you derived above for E and B, each component of each vector satisfies the general wave equation:

1 2 f �2 f v2 t 2 22. What is the speed of motion of electromagnetic waves in free space according to equation WCR above?

An interlude courtesy of Joseph Fourier and friends…

Of all the possible planar waves (see the Primer above for the properties of functions that can model waves), the most familiar one to most people is the sinusoidal one

f( z , t )= A cos[ k ( z - vt ) +d ]

or (in exponential form)

f( z , t )= Re[ Aei( kt- vt +d ) ] I’ve arbitrarily chosen the z direction for the wave’s motion.

23. To what aspects of a wave do the parameters A, k, v, and δ in the above expressions correspond? Why people generally work with sine/cosine waves (whether they expression the functions in trig form or exponential form) is that they are fairly simple to use and visualize (at least, compared to waves such as f1 and f2 in the Primer above. What Fourier and friends did was to show that any wave can be represented as a linear combination of sinusoidal ones. And hence, you don’t have to worry about whether working with the relatively simply expressed sine/cosine waves will give you results that might not work for other possible electromagnetic waves.

The fancy mathematical way of writing this astounding result is the following, where some of the k’s can be zero if the particular wave corresponding to period k doesn’t contribute to the f(z,t) in which we’re interested.

f( z , t )= A ( k ) ei( kz- vt ) dk -

In other words, any wave can be approximated as closely as we wish by combinations of sine and cosine functions. The graphs in the Appendix give visual support for (though of course they don’t prove) Fourier’s contention.

The solutions to the wave equation (plane waves) traveling in the z direction will be of the following form:

E(z , t ) = E ei( kz-w t ) B(z , t ) = B ei( kz-w t ) 0 and 0

24. Applying Maxwell’s first and third laws to the expressions for E(z,t) and B(z,t) above, implies what about the components of E0 and B0 in the z direction? 25. This result implies that electromagnetic waves are what kind of waves? Hint 26. What does Faraday’s Law (presented in the table in this lab as Maxwell’s second law) imply about the relationship between the y- component of E0 and the x-component of B0 ? Hint 27. What does this result imply about the orientation of E and B to one another as they travel through empty/free space?

Appendix Black curve is the sum of the two simple y=sin x and y= sin 2x curves in blue and red

-6 -4 -2 2 4 6 -0.5

Now, we make it just a tad more complicated by adding y = sin x and y = ½ cos 2x 1

-6 -4 -2 2 4 6

-1.5 A rather “noisy” wave, with its generating components in the graph below it:

-7.5 -5 -2.5 2.5 5 7.5 1.5

-7.5 -5 -2.5 2.5 5 7.5

And finally, a picture that suggests we can combine smooth sine waves and get even saw-tooth waves from them: y

t -3 -2 -1 1 2 3

-1 1 One can actually relax this definition of a wave to fit such things as “wave packets” (that arise in quantum mechanics) and dispersive media, but the overall result is very similar. And this definition of wave fits non-quantum phenomena in a vacuum quite well. And, since that’s what we’re considering, there’s no need to make it more complicated here.