Study Maxwell’s Equations 

Derive the wave equation for a magnetic field from Maxwell's equations.

Enter Maxwell's equations in natural Lorentz–Heaviside units.

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X Inactivate[{gaussE = Div[\[ScriptCapitalE][x, y, z, t], {x, y, z}] == \[Rho][x, y, z, t], gaussB = Div[\[ScriptCapitalB][x, y, z, t], {x, y, z}] == 0, faraday = Curl[\[ScriptCapitalE][x, y, z, t], {x, y, z}] == -D[\[ScriptCapitalB][x, y, z, t], t], ampere = Curl[\[ScriptCapitalB][x, y, z, t], {x, y, z}] == j[x, y, z, t] + D[\[ScriptCapitalE][x, y, z, t], t]}, Div | Curl | D];

Create a nicely formatted table of the equations.

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X Inactivate[{gaussE = Div[\[ScriptCapitalE][x, y, z, t], {x, y, z}] == \[Rho][x, y, z, t], gaussB = Div[\[ScriptCapitalB][x, y, z, t], {x, y, z}] == 0, faraday = Curl[\[ScriptCapitalE][x, y, z, t], {x, y, z}] == -D[\[ScriptCapitalB][x, y, z, t], t], ampere = Curl[\[ScriptCapitalB][x, y, z, t], {x, y, z}] == j[x, y, z, t] + D[\[ScriptCapitalE][x, y, z, t], t]}, Div | Curl | D];; Grid[Transpose[{{"Gauss's Law for Electricity", "Gauss's Law for Magnetism", "Faraday's Law", "Ampere's Law"}, %}], Alignment -> Left, Dividers -> All, Spacings -> {4, 2}] // TraditionalForm

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Take the curl of Ampere's law in vacuum ( and ).

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X Inactive[Curl][#, {x, y, z}] & /@ ampere /. j -> (0 &)

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Interchange order of differentiation.

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X Inactive[Curl][#, {x, y, z}] & /@ ampere /. j -> (0 &); % /. Inactivate[Curl[D[v_, t_], x_] :> D[Curl[v, x], t]]

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Substitute Faraday's law.

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X Inactivate[{gaussE = Div[\[ScriptCapitalE][x, y, z, t], {x, y, z}] == \[Rho][x, y, z, t], gaussB = Div[\[ScriptCapitalB][x, y, z, t], {x, y, z}] == 0, faraday = Curl[\[ScriptCapitalE][x, y, z, t], {x, y, z}] == -D[\[ScriptCapitalB][x, y, z, t], t], ampere = Curl[\[ScriptCapitalB][x, y, z, t], {x, y, z}] == j[x, y, z, t] + D[\[ScriptCapitalE][x, y, z, t], t]}, Div | Curl | D];; % /. Inactivate[Curl[D[v_, t_], x_] :> D[Curl[v, x], t]]; % /. Solve[faraday, Inactive[Curl][\[ScriptCapitalE][x, y, z, t], {x, y, z}]][[1]]

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Activating the equation results in the wave equation for the magnetic field.

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X Inactivate[{gaussE = Div[\[ScriptCapitalE][x, y, z, t], {x, y, z}] == \[Rho][x, y, z, t], gaussB = Div[\[ScriptCapitalB][x, y, z, t], {x, y, z}] == 0, faraday = Curl[\[ScriptCapitalE][x, y, z, t], {x, y, z}] == -D[\[ScriptCapitalB][x, y, z, t], t], ampere = Curl[\[ScriptCapitalB][x, y, z, t], {x, y, z}] == j[x, y, z, t] + D[\[ScriptCapitalE][x, y, z, t], t]}, Div | Curl | D];; % /. Solve[faraday, Inactive[Curl][\[ScriptCapitalE][x, y, z, t], {x, y, z}]][[1]]; wave = Activate[%]

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This is the equation expressed using traditional notation.

A plane-wave solution of the equation can be verified by a simple substitution.

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X wave /. \[ScriptCapitalB] -> (A Exp[{kx, ky, kz}.{#1, #2, #3} - I Sqrt[kx^2 + ky^2 + kz^2] #4] &) // Simplify

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