Talk:PlanetPhysics/Derivation of Wave Equation From Maxwell's Equations

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%%% This file is part of PlanetPhysics snapshot of 2011-09-01 %%% Primary Title: derivation of wave equation from Maxwell's equations %%% Primary Category Code: 40. %%% Filename: DerivationOfWaveEquationFromMaxwellsEquations.tex %%% Version: 1 %%% Owner: invisiblerhino %%% Author(s): invisiblerhino %%% PlanetPhysics is released under the GNU Free Documentation License. %%% You should have received a file called fdl.txt along with this file. %%% If not, please write to gnu@gnu.org. \documentclass[12pt]{article} \pagestyle{empty} \setlength{\paperwidth}{8.5in} \setlength{\paperheight}{11in}

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Maxwell was the first to note that Amp\`ere's Law does not satisfy conservation of \htmladdnormallink{charge}{http://planetphysics.us/encyclopedia/Charge.html} (his corrected form is given in \htmladdnormallink{Maxwell's equation}{http://planetphysics.us/encyclopedia/MaxwellsEquations.html}). This can be shown using the equation of conservation of electric charge:

\[ \nabla \cdot \mathbf{J} + \frac{\partial \rho}{\partial t} = 0 \]

Now consider Faraday's Law in differential form: \[ \nabla \times \mathbf{E} = -\frac{ \partial \mathbf{B}}{\partial t} \] Taking the \htmladdnormallink{curl}{http://planetphysics.us/encyclopedia/Curl.html} of both sides: \[ \nabla \times (\nabla \times \mathbf{E}) = \nabla \times (- \frac{ \partial \mathbf{B}}{\partial t}) \]

The right-hand side may be simplified by noting that \[ \nabla \times (\frac{ \partial \mathbf{B}}{\partial t}) = - \frac{ \partial}{\partial t} (\nabla \times \mathbf{B}) \] Recalling Amp\`ere's Law, \[ - \frac{ \partial}{\partial t} (\nabla \times \mathbf{B}) = -\mu_0 \epsilon_0 \frac{ \partial^2 \mathbf{E}}{\partial t^2} \] Therefore \[ \nabla \times (\nabla \times \mathbf{E}) = -\mu_0 \epsilon_0 \frac{ \partial^2 \mathbf{E}}{\partial t^2} \] The left hand side may be simplified by the following \htmladdnormallink{Vector Identity}{http://planetphysics.us/encyclopedia/VectorRelationships.html}: \[ \nabla \times (\nabla \times \mathbf{E}) = -\nabla^2 \mathbf{E} \] Hence \[ \nabla^2 \mathbf{E} = \mu_0 \epsilon_0 \frac{ \partial^2 \mathbf{E}}{\partial t^2} \] Applying the same analysis to Amp\'ere's Law then substituting in Faraday's Law leads to the result \[ \nabla^2 \mathbf{B} = \mu_0 \epsilon_0 \frac{ \partial^2 \mathbf{E}}{\partial t^2} \] Making the substitution $\mu_0 \epsilon_0 = 1/c^2$ we note that these equations take the form of a transverse \htmladdnormallink{wave}{http://planetphysics.us/encyclopedia/CosmologicalConstant2.html} travelling at constant \htmladdnormallink{speed}{http://planetphysics.us/encyclopedia/Velocity.html} $c$. Maxwell evaluated the constants $\mu_0$ and $\epsilon_0$ according to their known values at the time and concluded that $c$ was approximately equal to 310,740,000 $\mbox{ms}^{-1}$, a value within ~3\% of today's results!

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