7: Time Dependent Electromagnetic Fields.
- Page ID
- 22838
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)Chapters 2-5 treated the problem of how to calculate electric and magnetic field distributions given time independent charge and current distributions. This chapter discusses the more general problem of how to calculate electric and magnetic fields given time varying charge and current distributions. It turns out that the solution to this general problem is most easily developed using the scalar and vector potentials discussed in chapters 2 and 4. By way of example, the formalism is applied to the generation of radio waves by currents flowing in an antenna, and to the generation of light waves by oscillating atomic dipole moments.
- 7.2: Time Dependent Maxwell’s Equations
- This page covers the formulation of Maxwell's equations through vector and scalar potentials, highlighting the role of charge and current densities. It establishes the Lorentz gauge condition to simplify the equations, deriving wave equations linked to these potentials and illustrating the interrelation between electric and magnetic fields.
- 7.3: A Simple Radio Antenna
- This page covers the analysis of a center-fed linear antenna, focusing on the current distribution and its sinusoidal approximation, forming a standing wave with zero current at the ends. It details vector potential calculations and assumptions for electromagnetic field analysis. Additionally, it discusses far-field electromagnetic fields produced by the antenna, describing how the magnetic and electric fields behave with distance.
- 7.4: An Electric Dipole Radiator
- This page explains how excited atoms or molecules create oscillating electric dipoles that emit electromagnetic fields upon returning to their ground state. It details the derivation of equations for resulting electric and magnetic fields using the point dipole approximation and Maxwell's equations.
- 7.5: A Point Magnetic Dipole
- This page covers the vector potential of an oscillating magnetic dipole oriented along the z-axis, addressing both static and time-retarded scenarios. It derives magnetic field components from this vector potential, illustrating their dependence on distance and time. The summary emphasizes the proportional and orthogonal relationship between electric and magnetic fields in the far-field approximation, highlighting their correlation with the observer's position relative to the dipole.
- 7.6: A Moving Point Charge in Vacuum
- This page covers key concepts in electromagnetic theory, focusing on the charge density of a point charge and its potential through the Dirac delta function, emphasizing retarded time's effect on calculations. It discusses Lienard-Wiechert potentials for moving charges and the integration of current density to derive electric and magnetic fields, noting complexity that simplifies with slow charge motion.
General Reference: The Feynman Lectures in Physics, Vol.(II), by R.P.Feynman, R.B.Leighton, and M.Sands, Addison Wesley, Reading, Mass., 1964.