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Schwartz's Derivation of Electric Field in Conducting Medium

  1. Jul 18, 2009 #1
    I have a question regarding page 241 of “Principles of Electrodynamics” by Melvin Schwartz. He is deriving the electric field inside a conducting medium as a function of position z; by summing the incident field, contributions from slices of material to the left of z, and from slices to the right of z. I follow what he is doing (I think) up through calculating the first derivative of Ez. When taking the derivative of dEx/dz, he adds a term at the end of his expression for the second derivative that I don’t understand. The term is 4πσikEx(z) and it seems to come out of the blue (to me). Using the same math to get the second derivative as was used to calculate the first derivative, does not result in this term appearing. Please help me find what I am missing here; be it math and/or physics.
    I have tried finding a similar derivation in other books without success. Later on the same page, Mr. Schwartz states that his derivation could be done much more rapidly by starting with Maxwell’s Equations and he then shows how this is done. His reason for the preceding derivation is to not, “lose the beautiful insight into the origin of the fields in terms of flowing currents.” Based on this and similar derivations from Maxwell’s Equations in other books, it seems that the term I don’t understand is real and necessary. I just don’t understand how he went from his expression for the first derivative to his results for the second derivative.
    I would post the 2 equations, but I don’t know a good way to include the integral and exponential terms in this posting. If someone knows of a text that uses a similar approach to Schwartz’s, that may be all I need.
    Thanks for your help!
  2. jcsd
  3. Jul 20, 2009 #2
    Good news! I found that my error was in ignoring the derivative of the integrals appearing in Ex(z), and therefore not using the product rule when taking the first derivative with respect to z. Note that z is one of the limits of integration. The “new” terms arising from the product rule cancel each other in the first derivative, so it appears that they are not needed. Oops. Due to a sign change that appears in the 1st derivative, the similar terms do not cancel when taking the second derivative.
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