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=−1c∂H∂t,∇×H=1c∂E∂tprove that E and H satisfy the equation∇2Ei=1c2∂2Ei∂t2 and ∇2Hi=1c2∂2Hi∂t2Here, i=x,y or z.

Answer 1

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

$∇×(∇×E)=∇(∇⋅E)−∇^2E=−∇^2E$

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

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

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

$∇×(∇×H)=∇(∇⋅H)−∇^2H=−∇^2H$

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

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