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Let There Be Light

In this post I explore Maxwell’s equations to enrich my understanding of the amazing insight that light is an electromagnetic wave and using plane wave solutions obtain simple insights into the characteristics of these waves.

The following is a typical differential form representation of Maxwell’s equations for non-magnetic and non-polarisable media.

\begin{array}{lll}\nabla\cdot\vec{E}&=&\frac{\rho}{\epsilon_0}\\\nabla\cdot \vec{B} &=&0\\\nabla\times \vec{E}&=&-\frac{\partial\vec{B}}{\partial t}\\\nabla\times\vec{B}&=&\mu_0 \vec{j} +\mu_0\epsilon_0\frac{\partial\vec{E}}{\partial t}\end{array}

Now consider whether these equations have non-trivial solutions in vacuo (no charges or currents). The equations simplify:

\begin{array}{lll} \nabla\cdot\vec{E}&=&0\\\nabla\cdot\vec{B}&=&0\\\nabla\times\vec{E}&=&-\frac{\partial\vec{B}}{\partial t}\\\nabla\times\vec{B}&=&\mu_0\epsilon_0\frac{\partial\vec{E}}{\partial t}\end{array}

Apply the curl to the last two equations yields:

\begin{array}{lll}\nabla\times(\nabla\times\vec{E})&=&-\frac{\partial\nabla\times\vec{B}}{\partial t}\\\nabla\times(\nabla\times\vec{B})&=&\mu_0\epsilon_0\frac{\partial\nabla\times\vec{E}}{\partial t}\end{array}

Using \nabla\times(\nabla\times\vec{A})=\nabla(\nabla\cdot\vec{A})-\nabla^2\vec{A} and substituting the last two equations:

\begin{array}{lll}\nabla^2\vec{E}&=&\mu_0\epsilon_0\frac{\partial^2\vec{E}}{\partial t^2}\\\nabla^2\vec{B}&=&\mu_0\epsilon_0\frac{\partial^2\vec{B}}{\partial t^2}\end{array}

or equivalently:

\begin{array}{lll}\frac{\partial^2\vec{E}}{\partial t^2}&=&\frac{1}{\mu_0\epsilon_0}\nabla^2\vec{E}\\\frac{\partial^2\vec{B}}{\partial t^2}&=&\frac{1}{\mu_0\epsilon_0}\nabla^2\vec{B}\end{array}

These are recognisable as wave equations. The speed of propagation, c, is:

\begin{array}{lll}c& =& \frac{1}{\sqrt{\mu_0\epsilon_0}}\\&=& \frac{1}{\sqrt{8.854187817620 \times 10^{-12}\times 4\pi\times 10^{-7}}}\\&\approx& 3 \times 10^8 \text{m}\cdot\text{s}^{-1}\end{array}


In a future post, I aim to look at simple solutions to explore properties of  light…

Categories: Mathematics
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  1. July 23, 2011 at 6:46 pm

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