Solution To Maxwell Equation Derivation Problem

 Problem: 

Starting with the Maxwell vector equation:

Ñ x E  =  - ¶B / ¶ t

Derive the wave equation in terms of E.

Solution:

We can start with: Ñ x E  =  =  - ¶B / ¶ t 

Take  the curl of both sides::

Ñ x  Ñ x E  =      Ñ x  (-¶B / ¶ t)

And:

-Ñ x  (¶B / ¶ t) =  -Ñ X   (m ¶H / ¶ t) = - m(Ñ x  ¶H / ¶ t)

But by Maxwell’s first equation: Ñ X H  =  ¶D / ¶ t 

Therefore:

Ñ x (¶H / ¶ t)  = ¶² D / ¶ t²  =  e ¶² E / ¶ t² 

Then:

-Ñ x  (¶B / ¶ t) =  - me ¶² E / ¶ t² 

 Whence:

Ñ x  Ñ x E  =   - me ¶² E / ¶ t² 

But by a vector identity:

Ñ x  Ñ x E  =    Ñ · Ñ ·E -   Ñ ² E

But:   Ñ ·E =    1/ m_o (Ñ ·D) = 0

Since: (Ñ ·D) = 0  (No charges)

Then: Ñ · Ñ ·E  = 0 

So:   -   Ñ ² E   =    -m ε  (¶² E / ¶ t²) 

Or: Ñ ² E   =    m_o ε_o (¶² E / ¶ t²) 

Writing out all component wave eqns.:

¶ ²E _x  / ¶ x²  =   m_oε_o  ¶ ² E _x / ¶ t²

¶ ²E _y  / ¶ x²  =   m_oε_o  ¶² E _y / ¶ t²

¶ ²E _z  / ¶ x²  =   m_oε_o  ¶² E _z / ¶ t²