- #1
CAF123
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Homework Statement
Consider the following representation of Maxwell's eqns: $$\nabla \cdot \underline{E} =0,\,\,\, \nabla \cdot \underline{B} = 0,\,\,\, \nabla \times \underline{E} = -\frac{\partial \underline{B}}{\partial t}, \,\,\,\frac{1}{\mu_o}\nabla \times \underline{B} = \epsilon_o \frac{\partial \underline{E}}{\partial t}.$$
By considering ##\nabla \times (\nabla \times \underline{E})##, use the above and an appropriate vector identity to deduce $$\left(\frac{1}{c^2} \frac{\partial^2}{\partial t^2} - \nabla^2\right) \underline{E} = 0.$$
Therby deduce an expression for ##c##. Find a similar eqn for B.
(Note: This question is from a Calculus course)
The Attempt at a Solution
So, considering ##\nabla \times (\nabla \times \underline{E})## I rewrote this as (which I believe is the 'appropriate vector identity' : ##\nabla(\nabla \cdot \underline{E}) - (\nabla^2) \underline{E}. ## However, I believe both these terms vanish because of the equations given. The other thing I tried more directly was subbing in what we have for ##\nabla \times \underline{E}##, to get ##\nabla \times \left(-\frac{\partial \underline{B}}{\partial t}\right).## It looks like I could then use the eqn above involving del cross B, but I am not sure whether I can just move the ∂/∂t around. (Probably not)
Many thanks for any advice.