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# Unit 14 - Physics 225 Relativity and Math Applications...

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Physics 225 Relativity and Math Applications Spring 2010 Unit 14 Solving Maxwell’s Equations N.C.R. Makins University of Illinois at Urbana-Champaign © 2010

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Physics 225 14.2 14.2
Physics 225 14.3 14.3 Unit 14: Solving Maxwell’s Equations You saw them initially last week, and here they are again: Maxwell’s Equations , the four equations that define all of electromagnetic theory. Drum roll please … here they are! ! ! " ! E = # \$ 0 ! ! " ! B = 0 ! ! " ! E = # \$ ! B \$ t ! ! " ! B = μ 0 ! J + μ 0 # 0 \$ ! E \$ t This electromagnetic theory, with its quantum-mechanical extension QED = Quantum ElectroDynamics, is arguably the most precisely-tested theory in the world. Experiment and theory have been pushed to more than 7 significant digits, and they still agree. The reason that the four Maxwell equations are “all you need” comes from a mathematical principle known as the Helmoltz Theorem : The behavior of any vector field is defined entirely by its divergence and curl. In other words: if you know the curl of a vector field ! E and you know its divergence as well, then you know everything about it. 1 So how does one solve Maxwell’s Equations? How does one use these master equations to actually obtain the electric and magnetic fields produced by any charge or current distribution? In general, there are three techniques: 1. Brute-force integration over the charge and current distributions. FYI: In your later courses, you will find this technique under the heading Green’s functions . Green’s functions are solutions of inhomogeneous differential equations for point sources . Once you know the field created by a point source, you just add together — integrate — all the contributions to the field at any given field point from every source point in the universe. Easy. We know the Green’s function solutions for Maxwell’s equations in the static case when nothing is changing with time: they are Coulomb’s Law and the Biot-Savart Law . We’ve been practicing this technique and we’ll continue to do so today. 2. Apply Gauss-Green-Stokes for cases with high symmetry . This is the solution technique you have been using in Physics 212 to obtain (a) the electric field of charged spheres, infinite cylinders, and infinite sheets or slabs from Gauss’ Law, and (b) the magnetic field of infinite solenoids, infinitely long wires, toroids, and current sheets from Ampere’s Law. This is a very elegant technique, but it only applies in those seven high-symmetry cases. The material at the end of this unit goes into more detail about this solution technique. 3. Find classes of solutions for the differential equations in different circumstances, and patch them together using boundary conditions . You will study this technique extensively in Physics 435. 1 To be perfectly precise, there is one additional piece of information needed, which is boundary conditions on the field.

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Unit 14 - Physics 225 Relativity and Math Applications...

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