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Unformatted text preview: Scott Hughes 27 April 2004 Massachusetts Institute of Technology Department of Physics 8.022 Spring 2004 Lecture 19: Displacement current. Maxwells equations. 19.1 Inconsistent equations Over the course of this semester, we have derived 4 relationships between the electric and magnetic fields on the one hand, and charge and current density on the other. They are: Gausss law: ~ ~ E = 4 Magnetic law: ~ ~ B = 0 Faradays law: ~ ~ E =- 1 c ~ B t Amperes law: ~ ~ B = 4 ~ J c As written, these equations are slightly inconsistent. We can see this inconsistency very easily we just take the divergence of both sides of Amperes law. Look at the left hand side first: Left hand side: ~ ~ ~ B = 0 . This follows from the rule that the divergence of the curl is always zero (as you proved on pset 3). Now look at the right hand side: Right hand side: ~ 4 ~ J c =- 4 c t . Here, Ive used the continuity equation, ~ ~ J =- /t . Amperes law is inconsistent with the continuity equation except when /t = 0 !!! This is actually a fairly common circumstance in many applications, so its not too surprising that we can go pretty far with this incomplete version of Amperes law. But, as a matter of principle and, as we shall soon see, of practice as well its just not right. We need to fix it somehow. 19.2 Fixing the inconsistency We can fix up this annoying little inconsistency by inspired guesswork. When we take the divergence of Amperes left hand side, we get zero no uncertainty about this whatsoever, its just ZERO. But, we should be able to add a function to the right hand side such that the divergence of the right side is forced to be zero as well. Lets suppose that our generalized Amperes law takes the form ~ ~ B = 4 ~ J c + ~ F . Our goal now is to figure out what ~ F must be. Taking the divergence of both sides, we find = 4 ~ ~ J c + ~ ~ F- ~ c ~ F = 4 t . Our mystery function has the property that when we take its divergence (and multiply by c ), we get the rate of change of charge density. This looks a lot like Gausss law! If we take the time derivative of Gausss law, we have t ~ ~ E = 4 t- 4 t = ~ ~ E t On the last line, Ive use the fact that it is OK to exchange the order of partial derivatives: t ~ = x t x + y t y + z t z = x x t + y y t + z z t = ~ t ....
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