34-1 (SJP, Phys 1120) Ch. 34: Electromagnetic waves. We’ve seen some of the ideas/discoveries of Ampere, Faraday, and others. So far, we've treated E and B as distinct (if related). But, in what is perhaps one of a small handful of truly triumphant intellectual breakthroughs in the history of physics, James Clerk Maxwell (a Scot, in the mid 1800’s) put it all together and came up with the four equations which described all electromagnetic phenomena! We've seen most of them already: 1) Gauss’ Law : Charges create E, in specific “patterns”. E fields superpose. (Coulomb’s Law is a “special case”) 2) The analogue of 1 for B fields (but, there are no magnetic monopoles ) 3) Faraday’s Law: Changing B makes E. 4) Ampere’s Law: Currents make B “New and Improved” (Maxwell) : Changing E will also make B. This last piece was Maxwell’s insight. It was not based on experiments (like all the rest). Maxwell argued as a “theorist”, arguing from mathematical symmetry. (It was only later demonstrated in the lab.) We'll discuss it soon. The math of those 4 equations is a little tough (vector calculus is required). Here's how we've been writing them: 1) (Gauss) ! E " dA##= Qenc/$0 2) (Gauss for magnetism)! B " dA##= 0 3) Faraday ! E " dL#= $d%magdt = $ddtB##" dA 4) Ampere-Maxwell:! B " dL#=µ0Ienclosed+µ0$0d%Elecdt 1) Tells you that E field lines "emanate" from charges 2) Tells you that B field lines aren't created (there are no magnetic monopoles), but they form loops, with no start or stop. 3) Tells you that changing a magnetic field will create an electric field (which is where our AC electric power comes from, and Eddy currents, and inductors...) 4) Tells you that currents create magnetic fields (that's the first term, "Ampere's law") and also, much like #3, that changing Electric fields create magnetic fields too. (This last one is new - we haven't discussed it yet) This is everything we know about Electricity and Magnetism. There is nothing else, really. It's a spectacular synthesis! These 4 equations tell you the result of any classical experiment or phenomenon involving (macroscopic) electricity or magnetism.34-2 (SJP, Phys 1120) We still need to talk about that "last" of Maxwell's equations, the part he came up with. It's the extra term in the last equation, that says changing Electric flux generates magnetic fields. Consider a wire that has a gap in it. (Looks like a tiny capacitor, the ends of the wire on either side of the gap are charging up): There's no physical current in the "gap" above that midpoint, so you'd think (from Ampere's law) that no current enclosed means no B field. But that just doesn't seem right - in fact, if you pick an Amperian loop running through the "B?" point and "bend" the flat surface like a soap bubble either way, you DO get an "I_enclosed" so there SHOULD be a B field there.... So Maxwell had this insight - even though no charges flow through the gap, we can still imagine a sort of current flowing there (just as we talked about the current "through" a capacitor in the RC chapter) He called it the "displacement current", and argued that it should have the form: ! "0d#Elecdt (arguing from analogy and looking at the other equations) Knight 34.4 talks about displacement current. Using it is mathematically very much analogous to what we did with Faraday's law to find B fields. Adding this term in has made the equations very much symmetrical between E and B. (The only lack of symmetry arising from the fact that there are no magnetic monopoles, and hence e.g. no "magnetic currents" term in Faraday's law) There are many consequences, but one in particular is quite remarkable: Imagine shaking a charge “q” up & down. The E-field is thus “shaking” too. Maxwell’s big insight was that a changing E induces (creates) a B-field. But this new B-field is itself “shaking”, so Faraday’s law says this in turn creates a new E-field, which creates a new B, which… It's a little bit like wiggling a water molecule, which makes a neighbor wiggle, which makes its neighbor wiggle…= a traveling wave. But here, what exactly is waving? It’s nothing physical, exactly, it’s the E and B fields themselves turning on and off. You need a charge to start it, but the wave can them propagate through empty space (vacuum). You would call this an “Electromagnetic Wave” or “EM Wave”. People also call this “EM Radiation.” B wiggle charge B B B E E E E ++++---IBBB?I34-3 (SJP, Phys 1120) Maxwell derived this mathematically. Perhaps he wondered, are there any examples of these EM waves in nature? Could we produce/observe such a wave in the lab? If we did, what would it “look” like? How fast would it go? Maxwell derived the speed of these "EM waves" himself (See Knight Section 34-6). Let's sketch out some of the ideas here, though: If we're in free (empty) space, then there are no charges or currents. So Q_enc and I_enc =0, and the set of Maxwell's equations simplify a lot, to: ! E " dA##= 0, ! B " dA##= 0, ! E " dL#= $ddtB##" dA, and ! B " dL#=µ0$0ddtE " dA## We're looking for (vector!) functions E(x,t) and B(x,t) that satisfy these equations. One solution is E = B = 0 everywhere. Not very interesting. But there are other solutions. I won't derive it here, but will just state that the following fields DO satisfy the equations everywhere and for all time: E = E0 sin(k x - ωt) j B = B0 sin(k x - ωt) k (No, it's not obvious! Take a look at knight if you want help convincing yourself that these DO satisfy Maxwell's equations. It's also not unique, there are lots of other solutions.) This one is called a "plane wave". Why? Let's try to picture it. First of all, E always points in the j direction (the y direction). But it does not depend on y or z. It wiggles as you move along the x direction, getting bigger and smaller as you move along x (but always pointing in the y direction). It's hard to sketch, because it's a vector in 3-D, but here's what E looks like just along a single line... (If you could watch a time lapse you'd see it traveling in the +x direction. It's a "plane wave" because this wave is exactly the same everywhere in the y-z plane, off to infinity. Meanwhile, there's also a B field, which looks similar, except it points in the z direction. Knight 34.26 sketches it (except you have to realize that that figure only shows E and B along the x axis, it's
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