Schwartz's Derivation of Electric Field in Conducting Medium

[sloneranger]

Hello,
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!
sloneranger

 

[sloneranger]

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.
sloneranger

 

[astrokreng]

Dear sloneranger,

Thank you for your question regarding Schwartz's derivation of the electric field in a conducting medium. I would be happy to help clarify the term that you are having trouble understanding.

First, it is important to note that in this derivation, Schwartz is considering a conducting medium that is placed in a uniform electric field, with the electric field vector pointing in the z-direction. The material is assumed to be infinite in extent in the x and y directions.

In order to calculate the electric field inside the material, Schwartz starts by considering the electric field at a point z inside the material, which is a result of the incident field from the left and the contributions from all the slices of material to the left and right of z. The electric field at z can be written as a sum of these contributions, which is essentially an integral over all the slices of material.

To calculate the first derivative of Ez, Schwartz uses the chain rule and the fact that the electric field is continuous across the interfaces between the slices of material. However, when calculating the second derivative, he also needs to take into account the discontinuity in the electric field at the interfaces. This is where the additional term, 4πσikEx(z), comes in.

This term represents the contribution from the surface charge density σ at the interface between two slices of material. This surface charge density is a result of the polarization of the material due to the incident electric field. When calculating the second derivative, this term arises because the electric field at z is discontinuous at the interface, and thus has an effect on the second derivative.

In short, this term is necessary in order to account for the discontinuity in the electric field at the interfaces between the slices of material. It may seem to come out of the blue, but it is a crucial part of the derivation. If you were to use the same math to calculate the second derivative without taking into account the discontinuity at the interfaces, you would not get the correct result.

I hope this helps clarify the term you were having trouble with. If you have any further questions, please let me know. Additionally, if you would like to discuss this further, I would suggest seeking out a textbook or other resources that cover this topic in more detail. Thank you for your question and happy studying!