derivation of wave equation from Maxwell’s equations

Maxwell was the first to note that Ampère’s Law does not satisfy conservation of charge (his corrected form is given in Maxwell’s equation). This can be shown using the equation of conservation of electric charge:


Now consider Faraday’s Law in differential formMathworldPlanetmath:


Taking the curl of both sides:


The right-hand side may be simplified by noting that


Recalling Ampère’s Law,




The left hand side may be simplified by the following vector identityMathworldPlanetmath:




Applying the same analysis to Ampére’s Law then substituting in Faraday’s Law leads to the result


Making the substitution μ0ϵ0=1/c2 we note that these equations take the form of a transverse wave travelling at constant speed c. Maxwell evaluated the constants μ0 and ϵ0 according to their known values at the time and concluded that c was approximately equal to 310,740,000 ms-1, a value within  3% of today’s results!

Title derivation of wave equation from Maxwell’s equations
Canonical name DerivationOfWaveEquationFromMaxwellsEquations
Date of creation 2013-03-22 17:52:09
Last modified on 2013-03-22 17:52:09
Owner invisiblerhino (19637)
Last modified by invisiblerhino (19637)
Numerical id 10
Author invisiblerhino (19637)
Entry type Derivation
Classification msc 35Q60
Classification msc 78A25