# Curl of the curl?

1. Aug 27, 2010

### magnifik

given curl E = -1/c*($$\partial$$H/$$\partial$$t)
div E = 0
div H = 0
curl H = 1/c*($$\partial$$E/$$\partial$$t), find

$$\nabla$$ x ($$\nabla$$ x E)

how do i take $$\nabla$$ x curl E? i tried to do it by determinants, but i'm not sure which values correspond to the i, j, and k. so my next assumption is that there is some property that i can take advantage of to solve the problem. please help. thanks.

2. Aug 27, 2010

### lanedance

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

you could do it directly
$$\nabla \times (\nabla \times E) =\nabla \times ( -\frac{1}{c} \frac{\partial H}{\partial t}) =-\frac{1}{c}(\nabla \times \frac{\partial H}{\partial t})$$

and i think it should be ok to take the time derivative outside the curl, though you may want to confirm that...
$$-\frac{1}{c}(\nabla \times \frac{\partial H}{\partial t})=-\frac{1}{c}\frac{\partial}{\partial t}(\nabla \times H)$$

and it should follow, otherwise, if you know the original field you could make use of the equality
$$\nabla \times (\nabla \times E) =\nabla (\nabla \cdot E) - \nabla^2 E$$

which should simplify with some of the other info

Last edited: Aug 27, 2010