Physics, Electromagnetic wave

1. ELECTROMAGNETIC WAVE
Presented by-
ACHUYT BHARALEE
Roll No- PHP22108
Dept of Physics
Tezpur University

2. Electromagnetic Wave:
 A time varying magnetic field can produce a space varying electric field.
 A time varying electric field can produce a space varying magnetic field.
 Experimentally Hertz prove that EM wave can produce by variation
of current.

3. EM Wave Equation:
Let the EM wave travelling in a region of space
where there is no charge.
If the charge on one end oscillates sinusoidally then electric & magnetic field also
oscillates same manner.
𝐸 𝑥, 𝑡 = 𝐸0𝑆𝑖𝑛(𝑘𝑥 − 𝜔𝑡) 𝐵 𝑥, 𝑡 = 𝐵0𝑆𝑖𝑛(𝑘𝑥 − 𝜔𝑡)
Along top & bottom sections of the rectangle
But for vertical sides-
𝐸. 𝑑𝑙 = 0
𝐸. 𝑑𝑙 = 𝐸 + 𝑑𝐸 ∆𝑦 − 𝐸∆𝑦 = 𝑑𝐸∆𝑦
−
𝑑𝜑𝐵
𝑑𝑡
= −
𝑑𝐵𝐴
𝑑𝑡
= −
𝑑𝐵
𝑑𝑡
𝑑𝑥∆𝑦
∴
𝜕𝐸
𝜕𝑥
= −
𝜕𝐵
𝜕𝑡 1
𝑬. 𝒅𝒍 = −
𝒅𝝋𝑩
𝒅𝒕
𝑩. 𝒅𝒍 = 𝝁𝟎𝝐𝟎
𝒅𝝋𝑬
𝒅𝒕

4. 𝐵. 𝑑𝑙 = 𝐵∆𝑧 + 0 − 𝐵 + 𝑑𝐵 ∆𝑧 − 0 = −𝑑𝐵∆𝑧
Again
𝜇0𝜖0
𝑑𝜑𝐸
𝑑𝑡
= 𝜇0𝜖0
𝑑𝐸
𝑑𝑡
𝑑𝑥∆𝑧
𝜕𝐵
𝜕𝑥
= −𝜇0𝜖0
𝜕𝐸
𝜕𝑡 2
Differentiating equation 1,2 w.r.t x and t we get-
𝝏𝟐𝑬
𝝏𝒙𝟐
= 𝝁𝟎𝝐𝟎
𝝏𝟐𝑬
𝝏𝒕𝟐
𝝏𝟐
𝑩
𝝏𝒙𝟐 = 𝝁𝟎𝝐𝟎
𝝏𝟐
𝑩
𝝏𝒕𝟐
And by calculating we can also get that -
𝑬
𝑩
=
𝑬𝟎
𝑩𝟎
= 𝒗 =
𝟏
(𝝁𝟎𝝐𝟎)𝟏 𝟐
= 𝒄

5. Properties of EM wave:
Electric and magnetic fields are always in phase.
They are transverse in nature.
Light is also a electromagnetic wave.
Electric field and magnetic field are always in perpendicular to each other.
Magnetic electric fields carry the same amount of energy as an electromagnetic
wave.

6. Energy in EM wave:
Just like mechanical waves, EM waves also carry energy with propagation.
EM wave consist of electric and magnetic fields.
The energy is stored in the electric and magnetic fields.
𝑈𝐸 =
1
2
𝜖0𝐸2
𝑈𝐵 =
1
2
𝐵2
𝜇0
Total energy will be - 𝑈 = 𝑈𝐸 + 𝑈𝐵 =
1
2
𝜖0𝐸2
+
1
2
𝐵2
𝜇0
By using 𝑐 =
𝐸
𝐵
=
1
(𝜇0𝜖0)1 2 the energy density in terms of E and B will be-
𝒖 = 𝝐𝟎𝑬𝟐 =
𝑩𝟐
𝝁𝟎
Here E and B is the field strength in particular time instant.

7. Poynting Vector:
 The energy carried by the EM wave per unit time per unit area is Poynting Vector (S).
𝑺 =
𝒆𝒏𝒆𝒓𝒈𝒚
𝒕𝒊𝒎𝒆.𝒂𝒓𝒆𝒂
 Units will be-
𝑱
𝒔𝒆𝒄. 𝒎𝟐 =
𝑾
𝒎𝟐
 The power transported by EM wave per unit area.
 S in the direction of propagation of wave.

8.  The energy (dU) contained in volume dV will be-
𝑑𝑈 = 𝑢𝑑𝑉 = (𝜖0𝐸2
)(𝐴𝑐𝑑𝑡)
Hence energy transported in time dt per area A is-
𝑆 =
𝑒𝑛𝑒𝑟𝑔𝑦
𝑡𝑖𝑚𝑒. 𝑎𝑟𝑒𝑎
= 𝝐𝟎𝒄𝑬𝟐
 In terms of B-
𝑺 =
𝒄𝑩𝟐
𝝁𝟎
 In terms of both E and B will be- 𝑺 =
𝑬𝑩
𝝁𝟎
 For general case- 𝑺 =
𝑬 × 𝑩
𝝁𝟎

9.  This shows that the direction of energy flow is perpendicular to the plane containing
E and B.
 This is time dependent, it reaches maximum when E & B does.
 Here E and B is instantaneous values.
 For any sinusoidally varying fields we have to consider average pointing vector.
𝑺𝒂𝒗𝒈 =
𝟏
𝝁𝟎
𝑬𝟎
𝟐
×
𝑩𝟎
𝟐

10. Poynting Theorem:
EM field carry energy.
Poynting Theorem is a statement of conservation of energy for
EM field.
It relates the energy stored in the EM field to the work done
on a charge distribution through energy flux.
It states that, the work done on the charges by the EM force is
equal to the decrease in energy remaining in the fields and less
the energy flowed out through the surface.
Where,

𝒅𝑾
𝒅𝒕
is the rate at which the fields do work on charges in the volume.
 The 2nd term is the rate of change of the energy density in the volume.
 The 3rd term is the energy flow out of the volume, i.e., Poynting vector S.
𝒅𝑾
𝒅𝒕
= −
𝒅
𝒅𝒕 𝒗
𝟏
𝟐
𝝐𝟎𝑬𝟐
+
𝑩𝟐
𝝁𝟎
𝒅𝝉 −
𝟏
𝝁𝟎 𝒔
𝑬 × 𝑩 . 𝒅𝒂
In a region of free space- 𝝏𝑼
𝝏𝒕
= −𝜵. 𝑺

11. THANK YOU