Question

Starting from Maxwell’s field equations in free space, derive wave equation

Answer 1

Answer on Question 39086, Physics, Optics Maxwells field equations in free space are

$\nabla\cdot\mathbf{E}=0$

$\nabla\times\mathbf{E}=\frac{\partial\mathbf{B}}{\partial t}$

$\nabla\cdot\mathbf{B}=0$

$\nabla\times\mathbf{B}=\frac{1}{c^{2}}\frac{\partial\mathbf{E}}{\partial t}$

To derive wave equation we will take curl of the curl equations and use curl of the curl identity:

$\nabla\times(\nabla\times\mathbf{E})=\nabla\cdot(\nabla\cdot\mathbf{E})-\nabla^{2}\mathbf{E}$

We know that $\nabla\cdot\mathbf{E}$. From the other hand, from equations we see that

$\nabla\times(\nabla\times\mathbf{E})=-\nabla\times(\frac{\partial\mathbf{B}}{\partial t})=-\frac{\partial(\nabla\times\mathbf{B})}{\partial t}=-\frac{1}{c^{2}}\frac{\partial^{2}E}{\partial t^{2}}$

All together give us

$\frac{1}{c^{2}}\frac{\partial^{2}\mathbf{E}}{\partial t^{2}}-\nabla^{2}\mathbf{E}=0$

In the very same way we get

$\frac{1}{c^{2}}\frac{\partial^{2}\mathbf{B}}{\partial t^{2}}-\nabla^{2}\mathbf{B}=0$