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- Jan 26, 2012

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**Background Info**: Maxwell's equations relating the electric field $\mathbf{E}$ and magnetic field $\mathbf{H}$ as they vary with time in a region containing no charge and no current can be stated as follows:

\[\begin{array}{ccc} \begin{aligned}\mathrm{div}\,\mathbf{E} &= 0 \\ \mathrm{curl}\,\mathbf{E} &= -\frac{1}{c}\frac{\partial \mathbf{H}}{\partial t} \end{aligned} & & \begin{aligned}\mathrm{div}\,\mathbf{H} &= 0 \\ \mathrm{curl}\,\mathbf{H} &= \frac{1}{c}\frac{\partial\mathbf{E}}{\partial t}\end{aligned}\end{array}\]

where $c$ is the speed of light.

**Problem**: Use the above equations to prove the following:

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

(b) $\displaystyle \nabla\times (\nabla\times \mathbf{H}) = -\frac{1}{c^2}\frac{\partial^2 \mathbf{H}}{\partial t^2}$

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

(d) $\displaystyle\nabla^2\mathbf{H} = \frac{1}{c^2}\frac{\partial^2 \mathbf{H}}{\partial t^2}$

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**Note**: $\nabla \times\mathbf{F}$ and $\nabla\cdot\mathbf{F}$ also denotes the curl and divergence of a vector field $\mathbf{F}$ respectively.

**Hint**:

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