"Showing E&B Obey Wave Equation w/ Maxwell's Curl

[knowlewj01]

Homework Statement

This question is closely related to physics but it's in a maths assignment paper i have so here it is:

By taking curls of the following equations:

$\nabla \times \bf{E} = -\frac{1}{c}\frac{\partial\bf{B}}{\partial t}$

$\nabla \times \bf{B} = \frac{1}{c}\frac{\partial\bf{E}}{\partial t}$

show that both E and B obey the wave equation with speed c, that is:

$\nabla^2\bf{E} = \frac{1}{c^2}\frac{\partial^2\bf{E}}{\partial t^2}$

and

$\nabla^2\bf{B} = \frac{1}{c^2}\frac{\partial^2\bf{B}}{\partial t^2}$

Homework Equations

$\nabla\cdot\bf{E} = 0$
$\nabla\cdot\bf{B} = 0$
$\nabla \times \bf{E} = -\frac{1}{c}\frac{\partial\bf{B}}{\partial t}$
$\nabla \times \bf{B} = \frac{1}{c}\frac{\partial\bf{E}}{\partial t}$

The Attempt at a Solution

I don't really know how to start this, so i took a wild guess. I started by looking at the first equation with $\nabla \times \bf{E}$ in it. and did the cross product, needless to say i got some horrible expression.

given that the divergence of both vector fields are 0, we should be able to write these vector fields as the curl of another vector field, such that:

$\bf{E} = \nabla \times \bf{A}$

iff

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

i think this is the starting point for the question. But i hit a brick wall at this point, anyone know what is next? thanks

 

[knowlewj01]

i have got a bit further with this, not sure how to finish it off though.

take curls of the equations:

$\nabla \times \left(\nabla\times\bf{E}\right) = -\frac{1}{c}\frac{\partial}{\partial t}\left(\nabla\times\bf{B}\right)$ [1]

$\nabla \times \left(\nabla\times\bf{B}\right) = \frac{1}{c}\frac{\partial}{\partial t}\left(\nabla\times\bf{E}\right)$ [2]

Use the identity: $\nabla\times\left(\nabla\times A\right) = \nabla\left(\nabla\cdot A\right) - \nabla^2 A$

and since $\nabla\cdot \bf{E} = \nabla\cdot \bf{B} = 0$

eqn's [1] and [2] become:

$\frac{1}{c}\frac{\partial}{\partial t}\left(\nabla\times\bf{B}\right) = \nabla^2\bf{E}$ [3]
$\frac{1}{c}\frac{\partial}{\partial t}\left(\nabla\times\bf{E}\right) = -\nabla^2\bf{B}$ [4]

is this correct? i don't know wether i should curl something that is already operated by a d/dt also, where should the second time derivative come into it?