Download E 1 - WordPress.com

Electromagnetic Waves
Parag Bhattacharya
Department of Basic Sciences
School of Engineering and Technology
Assam Don Bosco University
Maxwell's Equations
In free space, in the absence of any charge or currents (i.e. ρ = 0,
and J = 0)
1) Gauss Law for electric fields
1) Gauss Law for electric fields
2) Gauss Law for magnetic fields
2) Gauss Law for magnetic fields
3) Faraday's Law
3) Faraday's Law
4) Ampere's Law
4) Ampere's Law
Integral form
Differential form
Consider the differential form of the Maxwell's Equations
(1)
(2)
(3)
(4)
(3)
Taking the curl on both sides of (3), we get
Therefore,
(A)
(A)
(A) implies:
Each component of the electric field
obeys the scalar differential wave
equation.
Thus, any disturbance in the
electric field propagates as a wave.
(4)
Similarly, taking the curl on both sides of (4), we get
Therefore,
(B)
(B)
(B) implies:
Each component of the magnetic
field obeys the scalar differential
wave equation.
Thus, any disturbance in the
magnetic field also propagates as a
wave.
(A)
(B)
Therefore, equations (A) and (B) implies that:
Any disturbance in the electromagnetic field will
propagate on its own, independent of the medium,
in the form of an electromagnetic wave, commonly
known as “light”.
Comparing (A) and (B) with the differential wave equation:
where 'c': speed of light in free space
Plane EM wave:
Spherical EM wave:
Superposition of electromagnetic
waves
Consider a 1D EM wave propagating along the positive X-axis:
We can separate the space and time parts of the phase by
introducing a new quantity as:
Thus,
Simplest case:
We study the superposition of two harmonic waves with same
wavelength and frequency.
By Supersposition Principle, the resultant wave is:
We introduce E0 and α such that:
Conversely, knowing E01, E02, α1 and α2, we can find E0, α as:
Therefore,
Thus, the composite wave is harmonic and is of the same frequency
as the components, though of different amplitude and phase.
The intensity of a wave is proportional to the square of its amplitude.
Thus,
(for 1st wave)
(for 2nd wave)
(for resultant wave)
where β is a constant
We have,
Therefore,
Responsible for the variation of intensity
due to superposition of waves
Interference Term:
where the total phase difference is:
Thus,
Therefore, the total phase difference
between the waves may arise due
to:
 A difference in the path lengths
traversed
 A difference in the initial phase angle
Coherence:
Waves for which ϵ2 – ϵ1 is constant, regardless of its value, are said
to be coherent.
The above results can be obtained using the complex representation:
The resultant is:
E0
Im(E)
E02
α2
α
α1
0
Here,
E01
Re(E)
Superposition of multiple scalar EM waves
Consider N harmonic waves with the same frequency and travelling
in the same direction.
The ith wave is:
or,
or,
where, i = 0, 1, 2, ... , N
The resultant wave is:
or,
Superposition of multiple scalar EM waves
Thus,
where,
(A)
and
(B)
(A)
Note: Intensity is proportional to the time-average of the magnitude
of the field squared (taken over a long interval of time)
If the sources are incoherent:
The 2nd summation in (A) will average out to zero because of the
random rapid nature of phase changes
Thus,
If each source emits waves of the same amplitude E01
(A)
If the sources are coherent:
In the 2nd summation in (A), say the phase is same for both waves
at all times, i.e.,
And,
Thus,
If each source emits waves of the same amplitude E01
Assignment:
Two harmonic scalar waves E1 and E2 with the same frequency and
amplitude travel in the same direction separated by a distance Δx.
E1 E2
Δx
E
Apply the superposition principle to finally arrive at the resultant
wave E.
Obtain the amplitude and phase of the resultant wave in terms of the
path difference Δx.
Use both trigonometric and complex methods to solve the above
problem.
Superposition of two harmonic scalar waves E1 and E2 with the same
frequency travel in the same direction separated by a distance Δx.
E1 E2
Δx
Assuming the two waves to be travelling in the same medium, both
waves have the same speed, and hence the same wavelength.
Thus, the angular wave number k is same for both waves.
Therefore, we can represent the two waves as:
E1 E2
Δx
Thus,
By superposition principle, the resultant is:
(1)
where,
(2)
and,
(3)
E1 E2
Δx
E
Since the amplitudes are equal, i.e., E01 = E02, equations (2) and (3)
become:
How???
Prove these results.
Therefore,
If
If
Then,
Then,
Thus,
Thus,
Constructive Interference
Destructive Interference
Thank You!
Assam Don Bosco University