electromagnetic waves through a conducting media.The distance over which a plane wave is attenuated (reduce in amplitude) by a factor of 1/e is called skin depth. E = 𝐄_0 𝒆^(−𝜷𝒛).𝑒^(𝑖(𝛼𝑧−𝜔𝑡)) This represent an attenuated wave The attenuation is due to the Joule –loss equation(15) represent a plane wave traveling along z-direction .the imaginary part of k results in an attenuation of the wave.

2. Electricity and Magnetism-2
Electromagnetic waves in
conducting media

3. Plane Electromagnetic waves in
conducting media
Maxwell’s equations are;
𝛻. 𝐷 = 𝜌
𝛻. 𝐵 = 0
𝛻 × 𝐸 = −
𝜕𝐵
𝜕𝑡
𝛻 × 𝐻 = 𝐽 +
𝜕𝐷
𝜕𝑡
Consider a conducting medium with permeability 𝜇
,permittivity 𝜀 and conductivity 𝜎.

4. 1.Now Maxwell 1st
equation becomes
Charge density= 𝜌=0
𝛻. 𝐷 = 0
As we know that
𝐷 = 𝜀𝐸
𝛻 . 𝜀𝐸 =0
𝜀 𝛻 . 𝐸 =0
Where 𝜀 is not zero.
𝛻. 𝐸 = 0
2. 2nd Maxwell’s
equation is
𝛻. 𝐵 = 0
𝐵 =𝜇𝐻
𝛻 . 𝜇𝐻 =0
𝜇 𝛻 . 𝐻 =0
Where 𝜇 is not zero
𝛻. 𝐻 = 0
3. 3rd Maxwell’s equation
becomes
𝛻 × 𝐸 = −
𝜕𝐵
𝜕𝑡
where 𝐵
=𝜇𝐻
𝛻 × 𝐸 = −
𝜕𝜇𝐻
𝜕𝑡
𝛻 × 𝐸 = −𝜇
𝜕𝐻
𝜕𝑡

5. The 4th Maxwell’s equation is
𝛻 × 𝐻 = 𝐽 +
𝜕𝐷
𝜕𝑡
where 𝐽 = 𝜎 𝐸 and 𝐷 = 𝜀𝐸 .so,
𝛻 × 𝐻 = 𝜎𝐸 +
𝜕𝜀𝐸
𝜕𝑡
𝛻 × 𝐻 = 𝜎𝐸 +𝜀
𝜕𝐸
𝜕𝑡
So we get
𝛻 × 𝐸 = −𝜇
𝜕𝐻
𝜕𝑡
…………(1)
𝛻 × 𝐻 = 𝜎𝐸 +𝜀
𝜕𝐸
𝜕𝑡
……….(2)
𝛻. 𝐸 = 0 ………………......(3)
𝛻. 𝐻 = 0 …………………(4)

8. 𝛻2
E − 𝜇𝜎
𝜕𝐄
𝜕𝑡
−𝜇𝜖
𝜕2
𝜕2
𝐄
t
=0 ……..(7)
𝛻2𝐇 − 𝜇𝜎
𝜕𝑯
𝜕𝑡
−𝜇𝜖
𝜕2
𝜕2
𝑯
t
=0 ……..(8)
Equation(7) and(8)are modified wave equations for
E
and H . So equation (7) is admits plane wave
solution
E = 𝐄0𝑒𝑖(𝑘𝑥−𝜔𝑡) _________-(9)
This can be easily checked by putting Eq.(9) in
Eq.(7)
-𝑘2+ i𝜔𝜇𝜎 + 𝜔2𝜖𝜇 = 0 ------------------------(10)
Which show that wave numbers k must be complex

9. Let k = 𝛼 +iβ.then from equation (10) we get
−[𝛼+i β] 2
+ i𝜔𝜇𝜎 + 𝜔2𝜖𝜇 = 0
− 𝛼2 + −1 𝛽2 + 2iα𝛽 + i𝜔𝜇𝜎 + 𝜔2𝜖𝜇 = 0
−𝛼2 + 𝛽2 − 2iα𝛽 + i𝜔𝜇𝜎 + 𝜔2𝜖𝜇 = 0
𝛼2 + 𝛽2 +(2𝛼𝛽 − 𝜔𝜇𝜎)𝑖 − 𝜔2𝜖𝜇 = 0
𝛼2
− 𝛽2
+(2𝛼𝛽 − 𝜔𝜇𝜎)𝑖 = 𝜔2
𝜖𝜇 + 0𝑖

10. Equating real and imaginary parts, we get
𝛼2
− 𝛽2
= 𝜔2
𝜖𝜇 __________(11)
2𝛼𝛽 − 𝜔𝜇𝜎 = 0_________(12)
Solve the equation 12 for value of 𝛽
2𝛼𝛽 =ωμσ
𝛽 =
𝜔𝜇𝜎
2𝛼
___________(13)
Put value of 𝛽 𝑖𝑛 𝑒𝑞𝑢𝑎𝑡𝑖𝑜𝑛 11 𝑡𝑜 𝑔𝑒𝑡 𝑣𝑎𝑙𝑢𝑒 𝑜𝑓 𝛼

11. 𝛼 = 𝜔 𝜀𝜇[
1
2
±
1
2
1 +
𝜎2
𝜔2𝜀2
1
2
]1/2
The negative sign would make 𝛼 complex so, we
Must choose the positive sign .so
𝛼 = 𝜔 𝜀𝜇[
1
2
+
1
2
1 +
𝜎2
𝜔2𝜀2
1
2
]1/2…….(14)
Now, when k is positive equation (9) becomes
E = 𝐄0𝑒 𝑖𝛼+𝑖2𝛽 𝑧−𝑖𝜔𝑡)
E = 𝐄0𝑒 (𝑖𝛼+(−1)𝛽 𝑧−𝑖𝜔𝑡)
E = 𝐄0𝑒 𝑖𝛼𝑧−𝛽𝑧 −𝑖𝜔𝑡)

12. E = 𝐄0𝒆−𝜷𝒛
.𝑒𝑖(𝛼𝑧−𝜔𝑡)
___________(15)
This represent an attenuated wave The attenuation is
due to the Joule –loss equation(15) represent a plane
wave traveling along z-direction .the imaginary part
of k results in an attenuation of the wave.
Skin depth:
The distance over which a plane wave is attenuated
(reduce in amplitude) by a factor of 1/e is called skin
depth.
𝛿 =
1
𝛽

13. It is measure to depth to which an electromagnetic
wave can penetrate the conductor.
Meanwhile, the real part of k determine the wave
length, the propagation speed and the index of
refraction in the usual way:
𝜆 =
2𝜋
𝛼
𝑣 =
𝜔
𝛼
𝑛 =
𝑐𝛼
𝜔

14. The quantity
𝑐𝛼
𝜔
is called complex index of
refraction
We call the material a poor conductor if
𝜎
𝜔𝜀
≪ 1.
In this case
𝛼 ≈ 𝜀𝜇
𝛼 ≈
𝜎
2
𝜇
𝜔
And skin depth is independent of frequency

15. For a good conductor,
𝜎
𝜔𝜀
≫ 1.
So we get
𝛼 ≈ 𝛽 ≈
𝜔𝜇𝜎
2
The skin depth decrease with increase in frequency .
The skin depth at optical frequencies (𝜔 ≈ 1015𝑠−1)
Is roughly 10−8𝑚, which explain why metals are
opaque.