Obtaining The Electromagnetic Wave Equations From Maxwell’s Equations

 

EM wave propagating through empty space

In general, Maxwell’s equations will be expressed in vector form:

i)  Ñ X H  =  J    + ¶D / ¶ t     (A current density J arises from a magnetic field)

ii)               Ñ X E  =  - ¶B / ¶ t       (A magnetic field can arise from an electric field)

iii)             Ñ · B  =  0     (There are no magnetic monopoles)

iv)             Ñ · D  =  r            (Charges are conserved)

In addition, there are three “constitutive relations” that allow any of the above vectors to be re-cast in slightly different forms:

v)  D  =   e E

vi)  B =  m H

vii) J  = s E

In the equations above, H represents the magnetic field intensity, B is the magnetic induction, E the electric field intensity, D the displacement current, and J is the current density. The constants, e  and m, denote the permittivity and the magnetic permeability – each for media. In vacuo, the constants used are: e ₀  and m ₀ and the speed of light can be expressed:

 c =  1/ Öe ₀  Ö m ₀  .

An important aspect of Maxwell’s equations is being able to derive the assorted wave equations.  This can be accomplished in one of two ways:

a)from Maxwell’s differential equations (as given above)

Or b) from the integral equations:

1)  ∮ E d ℓ =   -  d F_m / d t   =  - ò ò_S  ¶B / ¶ t  dS

2)  ∮ B d ℓ =  m_oε_o  d F_E / d t  = m_oε_o  ò ò_S  ¶E / ¶ t  dS  

Where  F_m  is the magnetic flux (i.e.  BA)  and F_E is the electric flux (i.e. .   F_E = EA) . Differentiating each of the above for one dimension (e.g. x) yields):

1’)  ¶E / ¶ x       =   - ¶B / ¶ t 

2’)   ¶B  / ¶ x       =   -  m_oε_o  ¶E / ¶ t 

We then take the derivatives for (1’) and combine with those for (2’) and get:

¶ ²E / ¶ x² = - ¶/ ¶ t  (¶B / ¶ x) =

 - ¶  / ¶ t  (-  m_oε_o  ¶E / ¶ t )

And:

¶ ²B / ¶ x² =

 - ¶ / ¶ x  (¶B / ¶ t) =  - ¶  / ¶ t  (-  m_oε_o  ¶B / ¶ t )

From which two wave equations follow:

¶ ²E / ¶ x²  =   m_oε_o  ¶² E / ¶ t²

¶ ²B  / ¶ x²  =   m_oε_o  ¶² B / ¶ t²

If we compare the preceding to the generic wave equation, i.e. for propagation of transverse waves, say on a string, we find:

¶ ²y/ ¶ x²  =   1/ v²  ¶ ²y/ ¶ t²  

Where v is the wave velocity. For the Maxwell wave equations, however, we have v = c. And hence we can equate:

1/ c²    =  m_oε_o    

c ²   = 1/  m_oε_o    and: c =  1/ Ö( m_oε_o  )

So that c = 2.99792  x 10⁸    m/s 

Which is the velocity of light in vacuo.

One can also obtain the wave equations from the Maxwell differential (vector) equations. For example, take:

Ñ X H  =  J    + ¶D / ¶ t    

Take  the current free (J=0) case and we know: 

D  =   ε_o E    and     B =  m_o H

Take  the curl of both sides of the vector equation in H:

Ñ X  Ñ X H  =      Ñ X  (¶D / ¶ t)

And:

Ñ X  (¶D / ¶ t)  =   Ñ X (ε_o  ¶E / ¶ t )

 = ε_o  [Ñ X (¶E / ¶ t )]

But:

Ñ X (¶E / ¶ t)  = - ¶² B / ¶ t²

Where:

 - ¶² B / ¶ t²    =  -  m_o ¶² H / ¶ t²

Whence:

Ñ X  Ñ X H  =    ε_o (-  m_o ¶² H / ¶ t²) 

= -m_o ε_o (¶² H / ¶ t²) 

But by a vector identity:

Ñ X  Ñ X H  =    Ñ · Ñ ·H -   Ñ ² H

But:   Ñ ·H =    1/ m_o (Ñ ·B) = 0

So:   -   Ñ ² H   =    -m_o ε_o (¶² H / ¶ t²) 

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

Which is one of the wave equations in terms of H.

Writing all the component wave equations  out:

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

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

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

Suggested Problem:

Starting with the Maxwell vector equation:

Ñ X E  =  - ¶B / ¶ t

Derive the wave equation in E.