EMLecture14 - Lecture 14 Notes, 95.657 Electromagnetic...

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Lecture 14 Notes, 95.657 Electromagnetic Theory I, Fall 2011 Dr. Christopher S. Baird, UMass Lowell 1. The Maxwell Equations - In the previous lecture, we showed that in order for Ampere's law to hold even for non-static fields and for the continuity equation to hold, an extra term had to be added. The result was: ∇× B = 0 J total  0 0 E t Ampere's Law in Complete Form - Maxwell added this term for reasons of mathematical consistency, but it has profound physical implications. - The additional term means that a changing electric field can create a curling magnetic field, even in the absence of any currents. - Because Faraday's law says magnetic fields can create electric fields and Ampere's law says that electric fields can create magnetic fields, there is a feedback process where they can create each other cyclically, independent of any charges, currents, or materials. This is the basis behind electromagnetic radiation. - The final set of equations are known as the Maxwell equations: ∇⋅ E = ρ total ϵ 0 , ∇⋅ B = 0 Maxwell Equations for Total Fields ∇× E =− B t , ∇× B 0 J total 0 ϵ 0 E t - The Maxwell equations can be cast in a more compact, but less intuitive form, in terms of partial fields instead of total fields, and in terms of free currents/charges instead of total current/charges: ∇⋅ D = , ∇⋅ B = 0 Maxwell Equations for Partial Fields ∇× E =− B t , ∇× H = J D t - Now it becomes obvious why the partial fields were defined in different units than the total fields: it makes the constants disappear in the final form of the Maxwell Equations. - In electrodynamics, the partial fields still have the same definition as long as the electric dipole and magnetic dipole are the dominant ones in the material's response. E = 1 0 D 1 0 P B = 0 H  0 M - All of these fields are now dependent on time and space. - Let us look at the boundary conditions at the interface between two regions. - The divergence equations are no different than in electrostatics and magnetostatics, so we can apply the Gaussian pillbox method in the usual way and end up with the boundary conditions:
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D 2 D 1 ⋅ n = B 2 B 1 ⋅ n = 0 - Let us now draw an Amperian loop straddling the surface and integrate the curl equations over the loop in the usual way. E d l =− B t n da H d l = J D t n da - As we shrink the area of the loops to zero, the finite fields go to zero as well. The volume current density J becomes a surface current density K . n × E 2 E 2 = 0 n × H 2 H 2 = K - So it turns out that the boundary conditions for electrodynamics is the same as those we used for electrostatics and magnetostatics. - It should be noted that the Maxwell equations give a full description of the electromagnetic
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This note was uploaded on 02/13/2012 for the course PHYSICS 95.657 taught by Professor Staff during the Fall '11 term at UMass Lowell.

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EMLecture14 - Lecture 14 Notes, 95.657 Electromagnetic...

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