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7: Electrodynamics - Fields and Waves

Last updated

Mar 27, 2022
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- 6.7: Problems
- 7.1: Maxwell's Equations

Markus Zahn
Massachusetts Institute of Technology via MIT OpenCourseWare

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The electromagnetic field laws, derived thus far from the empirically determined Coulomb-Lorentz forces, are correct on the time scales of our own physical experiences. However, just as Newton's force law must be corrected for material speeds approaching that of light, the field laws must be cor rected when fast time variations are on the order of the time it takes light to travel over the length of a system. Unlike the abstractness of relativistic mechanics, the complete electrodynamic equations describe a familiar phenomenon- propagation of electromagnetic waves. Throughout the rest of this text, we will examine when appropriate the low-frequency limits to justify the past quasi-static assumptions.

7.1: Maxwell's Equations
In the historical development of electromagnetic field theory through the nineteenth century, charge and its electric field were studied separately from currents and their magnetic fields.
7.2: Conservation of Energy
We expand the vector quantity
7.3: Transverse Electromagnetic Waves
Let us try to find solutions to Maxwell's equations that only depend on the $z$ coordinate and time in linear media with permittivity $\varepsilon$ and permeability $\mu$ . In regions where there are no sources so that $\rho _{f}=0$ , $\textbf{J}_{f}=0$ , Maxwell's equations then reduce to
7.4: Sinusoidal Time Variations
If the current sheet of Section 7-3-3 varies sinusoidally with time as $\textrm{Re}\left ( K_{0}e^{j\omega t} \right )$ , the wave solutions require the fields to vary as $e^{j\omega t\left ( t-z/c \right )}$ and $e^{j\omega t\left ( t+z/c \right )}$ .
7.5: Normal Incidence onto a Perfect Conductor
A uniform plane wave with $x$ -directed electric field is normally incident upon a perfectly conducting plane at $z =0$ , as shown in Figure 7-13. The presence of the boundary gives rise to a reflected wave that propagates in the $-z$ direction. There are no fields within the perfect conductor. The known incident fields traveling in the $+z$ direction can be written as
7.6: Normal Incidence onto a Dielectric
We replace the perfect conductor with a lossless dielectric of permittivity $\varepsilon _{2}$ and permeability $\mu _{2}$ , as in Figure 7-14, with a uniform plane wave normally incident from a medium with permittivity $\varepsilon _{1}$ and permeability $\mu _{1}$ . In addition to the incident and reflected fields for $z <0$ , there are transmitted fields which propagate in the $+z$ direction within the medium for $z >0$ :
7.7: Uniform and Nonuniform Plane Waves
Our analysis thus far has been limited to waves propagating in the $z$ direction normally incident upon plane interfaces. Although our examples had the electric field polarized in the $x$ direction., the solution procedure is the same for the $y$ -directed electric field polarization as both polarizations are parallel to the interfaces of discontinuity.
7.8: Oblique Incidence onto a Perfect Conductor
In Figure 7-17a we show a uniform plane wave incident upon a perfect conductor with power flow at an angle $\theta _{i}$ to the normal. The electric field is parallel to the surface with the magnetic field having both $x$ and $z$ components:
7.9: Oblique Incidence Onto a Dielectric
A plane wave incident upon a dielectric interface, as in Figure 7-18a, now has transmitted fields as well as reflected fields. For the electric field polarized parallel to the interface, the fields in each region can be expressed as
7.10: Applications to Optics
Reflection and refraction of electromagnetic waves obliquely incident upon the interface between dissimilar linear lossless media are governed by the two rules illustrated in Figure 7-19:
7.11: Problems

Thumbnail: Plan wave. (Public Domain via Wikipedia)

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