**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 d**S**

2) ∮ **B** d ℓ = m_{o}ε_{o } d F_{E }/ d t = m_{o}ε_{o } ò_{ }ò_{S}_{ }¶**E** / ¶ t d**S**

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_{o}ε_{o } ¶**E** / ¶ t

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

¶ ** ^{2}E** / ¶ x

**= - ¶/ ¶ t (¶**

^{2}**B**/ ¶ x) =

- ¶** ** / ¶ t (-
m_{o}ε_{o } ¶**E** / ¶ t )

And:

¶ ** ^{2}B** / ¶ x

**=**

^{2} - ¶** **/ ¶ x (¶**B **/ ¶ t) = - ¶** ** / ¶ t (-
m_{o}ε_{o } ¶**B** / ¶ t )

From which two wave equations follow:

¶ ^{2}**E**
/ ¶ x** ^{2}** =
m

_{o}ε

_{o }¶

^{2 }**E**/ ¶ t

^{2}¶ ^{2}**B** / ¶ x** ^{2}** = m

_{o}ε

_{o }¶

^{2 }**B**/ ¶ t

^{2}If we compare the preceding to the
generic wave equation, i.e. for propagation of transverse waves, say on a
string, we find:

¶ ^{2}y/ ¶ x^{2} = 1/ v^{2} ¶ ^{2}y/ ¶ t^{2}

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

1/ c^{2} = m_{o}ε_{o }_{}

c ^{2} = 1/ m_{o}ε_{o }and: c = 1/ Ö( m_{o}ε_{o })

So that c = 2.99792 x 10^{8 } 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** / ¶ t^{2}

*Where : *

** **- ¶

^{2 }**B**/ ¶ t

^{2}**m**

*= -*_{o}¶

^{2 }**H**/ ¶ t

^{2}Whence:

**Ñ**** X ****Ñ**** X
H **= ** ** ε_{o} (** - **m

_{o}¶

^{2 }**H**/ ¶ t

**)**

^{2}= -m_{o} ε_{o} (¶^{2 }**H** / ¶ t** ^{2}**)

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**/ ¶ t

**)**

^{2}**Or**: **Ñ**** ^{2 }H = **m

_{o}ε

_{o}(¶

^{2 }**H**/ ¶ t

**)**

^{2}Which is one of the wave equations in
terms of **H**.

Writing all the component wave
equations out:

¶ ^{2}H ** _{x}** / ¶ x

**= m**

^{2}_{o}ε

_{o }¶

^{2 }**H**

**/ ¶ t**

_{x}

^{2}¶ ^{2}H ** _{y}** / ¶ x

**= m**

^{2}_{o}ε

_{o }¶

^{2 }**H**

**/ ¶ t**

_{y}

^{2}

^{}¶ ^{2}H ** _{z}** / ¶ x

**= m**

^{2}_{o}ε

_{o }¶

^{2 }**H**

**/ ¶ t**

_{z}

^{2}__Suggested Problem:__

Starting with the Maxwell vector equation:

**Ñ**** X E **= **-** ¶**B** / ¶ t

Derive the wave equation in **E**.

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