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 mag dt d dt B dA 4 Ampere Maxwell B dL 0Ienclosed 0 0 d Elec dt 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 I I gap above that midpoint so you d think from Ampere s law that no B current enclosed means no B field B B 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 0 d Elec dt 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 B E B But this new B field is itself shaking B E B E E so Faraday s law says this in turn wiggle charge 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 34 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 and B dA 0 B dL 0 0 d dt d E dL dt B dA E 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 t j B B0 sin k x t 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 E0 here s what E looks like just along a single line x If you could watch a time lapse you d see it traveling E 0 in the x direction It s a plane wave because this wave is exactly the same everywhere in the yz 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 …
View Full Document
Unlocking...