PlanetPhysics/Derivation of Wave Equation From Maxwell's Equations

Maxwell was the first to note that Amp\`ere's Law does not satisfy conservation of [[../Charge/|charge]] (his corrected form is given in [[../MaxwellsEquations/|Maxwell's equation]]). This can be shown using the equation of conservation of electric charge:

$\mathrm{\nabla }\cdot 𝐉+\frac{\partial \rho }{\partial t}=0$

Now consider Faraday's Law in differential form:

$\mathrm{\nabla }\times 𝐄=-\frac{\partial 𝐁}{\partial t}$

Taking the [[../Curl/|curl]] of both sides:

$\mathrm{\nabla }\times (\mathrm{\nabla }\times 𝐄)=\mathrm{\nabla }\times (-\frac{\partial 𝐁}{\partial t})$

The right-hand side may be simplified by noting that

$\mathrm{\nabla }\times (\frac{\partial 𝐁}{\partial t})=-\frac{\partial }{\partial t}(\mathrm{\nabla }\times 𝐁)$

Recalling Amp\`ere's Law,

$-\frac{\partial }{\partial t}(\mathrm{\nabla }\times 𝐁)=-{\mu }_0{ϵ}_0\frac{{\partial }^2𝐄}{\partial t^2}$

Therefore

$\mathrm{\nabla }\times (\mathrm{\nabla }\times 𝐄)=-{\mu }_0{ϵ}_0\frac{{\partial }^2𝐄}{\partial t^2}$

The left hand side may be simplified by the following [[../VectorRelationships/|Vector Identity]]:

$\mathrm{\nabla }\times (\mathrm{\nabla }\times 𝐄)=-{\mathrm{\nabla }}^2𝐄$

Hence

${\mathrm{\nabla }}^2𝐄={\mu }_0{ϵ}_0\frac{{\partial }^2𝐄}{\partial t^2}$

Applying the same analysis to Amp\'ere's Law then substituting in Faraday's Law leads to the result

${\mathrm{\nabla }}^2𝐁={\mu }_0{ϵ}_0\frac{{\partial }^2𝐄}{\partial t^2}$

Making the substitution ${\mu }_0{ϵ}_0=1/c^2$ we note that these equations take the form of a transverse [[../CosmologicalConstant2/|wave]] travelling at constant [[../Velocity/|speed]] $c$. Maxwell evaluated the constants ${\mu }_0$ and ${ϵ}_0$ according to their known values at the time and concluded that $c$ was approximately equal to 310,740,000 ${\text{ms}}^{-1}$, a value within ~3\