5D (Electromagnetism)

\nabla \times (\nabla \times B) = μ_{0} ε_{0} \frac{\partial}{\partial t} (\nabla \times E) = - μ_{0} ε_{0} B_{tt} .

Since

\nabla \cdot B = 0

, the left side is

- \nabla^{2} B

. Hence

B_{tt} = c^{2} \nabla^{2} B, c = (μ_{0} ε_{0})^{- 1/2} .

b

B_{3} = 0

, and the wave equation requires

ω^{2} = c^{2} k^{2} .

For propagation in the positive

z

direction, Faraday's law gives

E = \frac{ω}{k} (B_{2}, - B_{1}, 0) cos (k z - ω t) .

Thus $E$ , $B$ , and the propagation direction are mutually perpendicular.

S = \frac{1}{μ _{0}} E \times B = \frac{ω}{μ _{0} k} (B_{1}^{2} + B_{2}^{2}) cos^{2} (k z - ω t) e_{z} .

It is the electromagnetic energy flux in the propagation direction; its time average is

c (B_{1}^{2} + B_{2}^{2}) / (2 μ_{0})

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