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Unformatted text preview: 8 Electrodynamics Read: Boas Ch. 6, particularly sec. 10 and 11. 8.1 Maxwell equations Some of you may have seen Maxwell’s equations on tshirts or encountered them briefly in electromagnetism courses. These equations were written down for the first time by Scottish physicist James Clerk Maxwell in his “Treatise on Electricity and Magnetism” (1873), and they caused a stir because his new equations proved that light was an electromagnetic phenomenon. Imagine that you had no clue that light and all the phenomena of electricity and magnetism you knew from the laboratory were related, and someone showed you that you could calculate the speed of some weird electromagnetic wave solutions to these differential equations, and show that the speed was exactly that of light (which had been measured astronomically). Spectacular! With the new tools you possess you can understand the equations at a deeper level. Suppose charge is increasing at some rate within a given volume τ . Assume that we have no “sources” or “sinks” of charge in the system. This means that it has to come from outside the region. The amount created inside per time has to show up as a flux of the charge through the boundary of the volume coming from outside (make sure you understand the sign): ∂ ∂t Z vol τ ρ ( ~ r, t ) dτ = Z surf . ∂τ ~ j · d~a = Z τ ~ ∇ · ~ j dτ (1) where the last equality follows from the divergence theorem. Now since we did this for an arbitrary function ρ ( ~ r, t ) and current density j ( ~ r, t ), it must hold locally: ∂ρ ∂t + ~ ∇ · ~ j = 0 , (2) the socalled “ equation of continuity ”. This is not thought of as one of Maxwell’s equations, because it doesn’t contain the electromagnetic fields ~ E and ~ B , but merely expresses the conservation of charge. Here are the standard 4: 1. Gauss’s law . For pt. charge, ~ E = q 4 π² r 2 ˆ r ⇒ Z closed ~ E · d~a = q 4 π² Z ˆ r · d~a r 2  {z } = q ² d Ω 1 Divergence theorem then says Z τ ( ~ ∇ · ~ E ) dτ = 1 ² Z τ ρ ( ~ r ) dτ  {z } ⇒ ~ ∇ · ~ E = ρ ²  {z } (3) q Maxwell I There are two mathematical subtleties I swept under the rug in this “proof”. First I did it for a point charge, but expressed things in terms of a general charge density at the end. You can go back and convince yourself that if you say the ~ Efield is a sum of many small charge elements dq each producing a field falling off like 1/ r 2 from itself, you get the same answer. Secondly, in thefrom itself, you get the same answer....
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 Fall '07
 HIRSHFELD
 Charge, James Clerk Maxwell

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