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Just a wave

In this post I continue exploring Maxwell’s insight regarding the nature of light as an electromagnetic wave.

Consider the solution to the wave equation:

\mathbf{E}=\mathbf{E_0}e^{i(\mathbf{k}\cdot\mathbf{r}-\omega t)}

\mathbf{k} is the direction of propagation of the wave and length equal to the wavenumber, \omega is the angular frequency of the light.

\nabla\cdot\mathbf{E}= \frac{\partial}{\partial x_j}E_{0j}e^{i(k_mx_m-\omega t)}= i k_jE_{0j}e^{i(k_m x_m-\omega t)}= i\mathbf{k}\cdot\mathbf{E}

Similarly,

\nabla\cdot\mathbf{B}=i\mathbf{k}\cdot\mathbf{B}

Further,

(\nabla\times\mathbf{E})_m=\epsilon_{mnp}\frac{\partial}{\partial x_n}E_{0p}e^{i(k_sx_s-\omega t)}=\epsilon_{mnp}i k_nE_{0p}e^{i(k_sx_s-\omega t)}

So

\nabla\times\mathbf{E}=i\mathbf{k}\times\mathbf{E}

and

\nabla\times\mathbf{B}=i\mathbf{k}\times\mathbf{B}

Now looking at Maxwell’s equations in vacuo:

\nabla\cdot\mathbf{E}=\nabla\cdot\mathbf{B} =i\mathbf{k}\cdot\mathbf{E}=i\mathbf{k}\cdot\mathbf{B}=0

Hence the direction of propagation is orthogonal to both the direction of the electric field and magnetic field.

Now,

\nabla\times\mathbf{E} =-\frac{\partial\mathbf{B}}{\partial t} = i\omega\mathbf{B}

Hence,

\mathbf{B} =\frac{\mathbf{k}\times\mathbf{E}}{\omega}

Note \frac{\omega}{|k|} =c,

|\mathbf{B}|= |\mathbf{E}|/c

Finally,

\mathbf{E}\cdot\mathbf{B} = \mathbf{E}\cdot(\mathbf{k}\times\mathbf{E}) =0

Hence, the electric and magnetic field are orthogonal.

Insights from some symbolic manipulations…

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Categories: Mathematics
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