Question

Let E:=E

x

i

^

+E

y

j

^

+E

z

k

^

 and H:=H

x

i

^

+H

y

j

^

+H

z

k

^

 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

∇

2

E

i

=1

c

2

∂

2

E

i

∂t

2

 and ∇

2

H

i

=1

c

2

∂

2

H

i

∂t

2

Here, i=x,y

 or z

.

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

2

V.

Answer 1

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

$∇×(∇×E)=∇(∇⋅E)−∇^2E=−∇^2E$

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

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

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

$∇×(∇×H)=∇(∇⋅H)−∇^2H=−∇^2H$

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

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