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Maxwell General & Wave Equation
We use both MKS and Gaussian units
Interaction of EM radiation with Matter
We will be dealing with the index of refraction:
0 0 μ , ε , μ, ε parameters of material
Permittivity: κ =ε /ε 0
Permeability: κ m =μ /μ 0 (usually = 1)
Velocity ( ) ( ) ( ) ( )
-1/ 2 1/ 2 -1/2 -1/
v = με = μ ε 0 0 κ κ m =c κ κ m
-1/
c / v ≡ n = κ κ m κ m ≅ 1
1/ η = κ Index of Refraction
This is only true for non-polar; otherwise we will deal with high frequency terms (for η).
Displacement
Now D^ =^ ε 0 E+
G G
P
G
P = electric dipole moment per volume
G
D = εE
G G
so ( 0 ) 0 0
0
P E E 1 E
G G G G
Electric susceptibility: so,
0
; P = α E ; αis a complex number
G G
For nonlinear properties, we expand this in a power series in E
G
Also: B^ =^ μ 0 ( H+ )
G G
M
G
B = μ 0 H
G G
in vacuum ; B = μH
G G
in general
Maxwell’s four equations relate charge densities ( ρ ), current densities ( ), and field
quantities to their time and space derivatives.
J
G
Gauss’ Law (electric flux):
0 0
E (^) n
S n V
ds q ρ dV ε ε
G G
i
v
n n
∑ q = net charge inside of S.
FIRST M AXWELL EQUATION Æ ∇^ D=^ ρ
G G
i (^) differential form
Biot-Savart Law:
0 3
r B J( ) (^4) r V
r dV
μ
π
= ×
G
G G
0 J( )
B
4 r
r dV
μ
π
= ∇ ×
G
G G
Thus, ∇^ B^ =^0
G G
i (^) M AXWELL’ S 2 ND LAW
Ampere’s Law:
∫ B^ d =^ μ^0 I
G G
i A
v
Differential Form: (^0)
D
B J( ) v t
μ
∇ × = ⎜ + ⎟
G
G G G
; M AXWELL’ S 3
RD LAW
Faraday’s Law: E - m d t
G G
i A
v
Φ m = magnetic flux passing through area defined by the path (d A).
M AXWELL’ S 4
TH LAW
B
E -
t
∇ × =
G
G G
We will be dealing with EB waves.
In a Vacuum :
∇ E = 0
G G
i ∇ H = 0
G G
i
0
E
H
t
∇ × =
G
G G
0
H
E -
t
∇ × =
G
G G
We replaced Bwith
G
μ 0 H
G
because m = 0
G
Take curl of equations: ( )
0 0
H
H
E - -
t t
⎛ ⎞ ∂ ∇ ×
∇ × ∇ × = ∇ × ⎜ ⎟=
⎝ ∂^ ⎠ ∂
G^ G^ G
2
(^0 0 )
E
E -
t
∇ × ∇ × =
G
G
and ( )
2
(^0 0 )
H
H -
t
∇ × ∇ × =
G
G
These are MAXWELL’ S WAVE EQUATIONS
Group Velocity: v g
d
dk
If the medium is not dispersive (^) v v φ = g
v g ≡ velocity at which information can be sent
** General Solution of the wave equation **:
( )
A x, y, z, t A (^0)
i k r t e
− ω
G G G G (^) i k is in the direction of travel of the plane wave.
G
So, for instance
E E 0
i k r t e
− ω
G G G G (^) i is a solution for the Electric Field
Note, in free space ∇ E = 0 , so,
G G
i ∇ E = i k E = 0
G G G G
i i. That is, E k = 0
G G
i transverse wave
Also, H k = 0.
G^ G
i
** i e. , E and H are ⊥ to k.**.
G G G
** There are two orthogonal components to E 0
G
- ( ) - ( ) E E (^) 0x i E0y j
i k zz t i k zz t e e
− ω − ω = +
G G G^ G G
These are the two polarizations! Note that the φ factor is the same for both x and y
components in an isotropic medium.
FLOW OF E NERGY:
Poynting’s Theorem: S = E ×H
G G G
time dependent
MKS unites of
2 ω/ m
Magnitude and direction of the radiation
Average Value: (^0 )
S E H
= ×
G G
