Electromagnetic Wave Equation - The Origin of The Electromagnetic Wave Equation

The Origin of The Electromagnetic Wave Equation

In his 1864 paper titled A Dynamical Theory of the Electromagnetic Field, Maxwell utilized the correction to Ampère's circuital law that he had made in part III of his 1861 paper On Physical Lines of Force. In Part VI of his 1864 paper titled Electromagnetic Theory of Light, Maxwell combined displacement current with some of the other equations of electromagnetism and he obtained a wave equation with a speed equal to the speed of light. He commented:

The agreement of the results seems to show that light and magnetism are affections of the same substance, and that light is an electromagnetic disturbance propagated through the field according to electromagnetic laws.

Maxwell's derivation of the electromagnetic wave equation has been replaced in modern physics by a much less cumbersome method involving combining the corrected version of Ampère's circuital law with Faraday's law of induction.

To obtain the electromagnetic wave equation in a vacuum using the modern method, we begin with the modern 'Heaviside' form of Maxwell's equations. In a vacuum- and charge-free space, these equations are:

\begin{align} \nabla \cdot \mathbf{E} \;&=\; 0\\ \nabla \times \mathbf{E} \;&=\; -\frac{\partial \mathbf{B}} {\partial t}\\ \nabla \cdot \mathbf{B} \;&=\; 0\\ \nabla \times \mathbf{B} \;&=\; \mu_0 \varepsilon_0 \frac{ \partial \mathbf{E}} {\partial t}\\ \end{align}

where ρ = 0 because there's no charge density in free space.

Taking the curl of the curl equations gives:

\begin{align} \nabla \times \left( \nabla \times \mathbf{E} \right) \;&=\; -\frac{\partial } {\partial t} \nabla \times \mathbf{B} = -\mu_0 \varepsilon_0 \frac{\partial^2 \mathbf{E} } {\partial t^2}\\ \nabla \times \left( \nabla \times \mathbf{B} \right) \;&=\; \mu_0 \varepsilon_0 \frac{\partial } {\partial t} \nabla \times \mathbf{E} = -\mu_o \varepsilon_o \frac{\partial^2 \mathbf{B}}{\partial t^2} \end{align}

We can use the vector identity

where V is any vector function of space. Since

\begin{align} \nabla \cdot \mathbf{E} \;&=\; 0\\ \nabla \cdot \mathbf{B} \;&=\; 0\\ \end{align}

then the first term on the right in the identity vanishes and we obtain the wave equations:

\begin{align} {\partial^2 \mathbf{E} \over \partial t^2} - {c_0}^2 \cdot \nabla^2 \mathbf{E} \;&=\; 0\\ {\partial^2 \mathbf{B} \over \partial t^2} - {c_0}^2 \cdot \nabla^2 \mathbf{B} \;&=\; 0 \end{align}

where

is the speed of light in free space.

Read more about this topic:  Electromagnetic Wave Equation

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