Question

Let E:=Exi+Eyj+Ezk  and H:=Hxi+Hyj+Hzk be two vectors assumed to have continuous partial derivatives (of second order at least) with respect to position and time. Suppose further that E and H satisfy the equations:

∇⋅E=0,∇⋅H=0,∇×E=−1/c. ∂H/∂t, ∇×H=1/c. ∂E/∂t

prove that E and H satisfy the equation

∇^2Ei =1/c^2. ∂^2Ei/∂t^2  and ∇^2Hi=1/c^2. ∂^2Hi/∂t^2

Here, i=x,y  or z.

Hint: Use the fact that∇×(∇×V)=∇(∇⋅V)−∇^V.

Answer 1

$∇×(∇×E)=-\frac{1}{c}\frac{\partial (∇×H)}{\partial t}=-\frac{1}{c^2}\frac{\partial E^2}{\partial t^2}$

$∇×(∇×E)=∇(∇⋅E)−∇^2E=−∇^2E$

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

$∇^2E_i=\frac{1}{c^2}\frac{\partial E_i^2}{\partial t^2}$

$∇×(∇×H)=\frac{1}{c}\frac{\partial (∇×E)}{\partial t}=-\frac{1}{c^2}\frac{\partial H^2}{\partial t^2}$

$∇×(∇×H)=∇(∇⋅H)−∇^2H=−∇^2H$

$∇^2H=\frac{1}{c^2}\frac{\partial H^2}{\partial t^2}$

$∇^2H_i=\frac{1}{c^2}\frac{\partial H_i^2}{\partial t^2}$