The document discusses the behavior of electromagnetic waves in conducting media using Maxwell's equations, focusing on wave equations for electric and magnetic fields. It derives expressions for complex wave numbers, attenuation of waves, and the concept of skin depth, which measures how far an electromagnetic wave can penetrate a conductor. Additionally, it distinguishes between poor and good conductors based on conductivity and frequency, highlighting how these factors affect wave propagation and attenuation.

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.