full information on electodynamics chapter.

1. ELECTRODYNAMICS
LECTURE 1

2. Gradient of a vector field:
Rate of change of something with respect to change in position is gradient
In the above two images the value of the functions are represented in black and white,
black representing higher values and its corresponding gradient is represented by blue
arrows.

3. Like the derivatives, the gradient represents the slope of the tangent of the graph
of the function.
Gradient of a Scalar field
Let ϕ be a differentiable scalar field defined on R. The gradient of ϕ is defined
by
grad ϕ=𝛁ϕ= 𝒊
𝝏ϕ
𝝏𝒙
+j
𝝏ϕ
𝝏𝒚
+k
𝝏ϕ
𝝏𝒛
where 𝛁 (read nabla or del) denotes the vector differential operator
𝛁= i
𝝏
𝝏𝒙
+j
𝝏
𝝏𝒚
+k
𝝏
𝝏𝒛

4. (i) Gradient of a scalar quantity is a Vector quantity.
(ii) Magnitude of that vector quantity is equal to the Maximum
rate of change of that scalar quantity.
(iii) Change of scalar quantity does not depend only on the
coordinate of the point, but also on the direction along which the
change is shown.

5. DIVERGENCE

6. • Divergence (div) is “flux density”—the amount
of flux entering or leaving a point.
• Divergence is a single number, like density.
• Divergence and flux are closely related – if a volume
encloses a positive divergence (a source of flux), it
will have positive flux.
• “Diverge” means to move away from, which may help
you remember that divergence is the rate of flux
expansion (positive div) or contraction (negative div).

7. Total flux change = (field change in X direction) + (field
change in Y direction) + (field change in Z direction)
Mathematically:
div F= 𝛁.F=
𝝏F
𝝏𝒙
+
𝝏F
𝝏𝒚
+
𝝏F
𝝏𝒛

8. CURL
The curl of a vector field measures the tendency for the
vector field to swirl around. Imagine that the vector field
represents the velocity vectors of water in a lake. If the vector
field swirls around, then when we stick a paddle wheel into
the water, it will tend to spin.

9. What would happen if a stick of
wood is placed in the flow
Activity

10. Mathematically
Curl F =

11. The above image is taken from Bing images and shows Hurricane
outbreak. Red color shows that a lot of Curl is going down there and
blue color is where curl is of smaller amount.

12. Gauss Div Theorem:
The Volume Integral of the divergence of a vector field over any volume V is
equal t o the surface integral of that field taken over the closed surface
enclosing the volume V that is
𝜵. 𝑬 𝒅𝒗 = 𝑬. 𝒅𝒔
Stokes Theorem:
The integral of the tangential component of a vector field is equal to the surface
integral of curl of that vector field, that is
(𝜵𝑿𝑽) 𝒅𝒔 = 𝑽. 𝒅𝒓

13. Maxwell’s Equation:
Name Integral Equations Differential Equation Meaning
Gauss’s Law
𝑬. 𝒅𝒔 =
𝟏
𝜺 𝟎
𝝆𝒅𝒗 𝜵. 𝑬 =
𝝆
𝜺 𝟎
The electric
flux leaving a
volume is
proportional
to the charge
inside.
Gauss’s law for
magnetism 𝑩. 𝒅𝒔 = 𝟎
𝜵. 𝑩 =0 There are
no monopoles
; the total
magnetic flux
through a
closed surface
is zero.

14. Name Integral Equations Differential
Equation
Maxwell-Faraday
Equation 𝑬. 𝒅𝒍 = −
𝒅
𝒅𝒕
𝑩. 𝒅𝒔 𝜵 X E = -
𝝏𝑩
𝝏𝒕
The voltage
induced in a
closed loop is
proportional
to the rate of
change of the
magnetic flux
that the loop
encloses.
Ampere’s circuital
law
𝑩. 𝒅𝒍 = 𝝁 𝟎 𝑱. 𝒅𝒔 + 𝝁 𝟎 𝜺 𝟎
𝒅
𝒅𝒕
𝑩. 𝒅𝒔 𝜵 X B =
𝝁 𝟎 (J+𝜺 𝟎
𝝏𝑬
𝝏𝒕
)
The magnetic
field induced
around a
closed loop is
proportional
to the electric
current plus
displacement
current

15. Wave Equation:
Maxwell’s four equation
𝜵. 𝑬 =
𝝆
𝜺 𝟎
(1)
𝜵. 𝑩 =0 (2)
𝜵 X E = -
𝝏𝑩
𝝏𝒕
(3)
𝜵 X B = 𝝁 𝟎 (J+𝜺 𝟎
𝝏𝑬
𝝏𝒕
) 𝟒
Taking curl of eq.(3)
𝜵X(𝜵XE)=𝜵X(-
𝝏𝑩
𝝏𝒕
)
𝜵(𝜵.E)-𝜵2E = -(𝜵 𝑿
𝝏𝑩
𝝏𝒕
)

17. Taking curl of Eq (4)
𝜵𝑿(𝜵 X B) = 𝜵X 𝝁 𝟎 𝜺 𝟎
𝝏𝑬
𝝏𝒕
𝜵(𝜵.B)-𝜵2B = 𝝁 𝟎 𝜺 𝟎
𝝏
𝝏𝒕
(𝜵𝑿 E)
We can use the vector identity
𝜵X(𝜵XV)=𝜵(𝜵.V)-𝜵2V
𝜵(𝜵.V)=𝜵2V
where V is any vector function of space. And
𝜵.E=0
𝜵.B=0
then the first term on the right in the identity vanishes and we obtain the
wave equations:
𝜵2B = −𝝁 𝟎 𝜺 𝟎
𝝏
𝝏𝒕
(𝜵𝑿 E)

18. OPTICAL FIBRE
•An optical fibre is made up of transparent material
•Very thin hair like structure
• Used to transmit light from one end of the fiber to the
other
•They are the guiding Channels through which light
energy propagates
•The light entering into the fibre must satisfy the
necessary condition of TIR

22. Acceptance Angle: The maximum angle of incidence that a light ray can
make with the axis of the fibre to propagate within the fibre Let us consider
an optical fibre into which light is incident
R.I. of core = μ1
R.I of cladding= μ2
R.I of the medium from which light is entering = μ0
A light wave enters the fibre at angle θi and refracted at an angle θr
For TIR μ1> μ2

23. Applying Snell’s law Sini/Sinr=μ1/ μ0
As I increases r increases and therefore r=(90- θc decreases.
sin r=Sin(90-θ)= cosθ
Therefore Sinθi=( μ1/ μ0i)cosθ
If i is maximum ,θ becomes θc
Sin θimax=( μ1/ μ0 i)cosθc
But sinθc= μ2/ μ1
cosθ= 1 −
μ22
μ12
Sinθi=( μ1/ μ0 ) 1 −
μ22
μ12=
Θi= Acceptance angle

24. Acceptance cone:

25. • Optical fibers are classified based on
Material
Number of modes and
Refractive index profile

26. On the basis of refractive index:
1. Step Index Fibre
2. Graded Index Fibre

28. Step index fiber :
In a step index fiber, the refractive index
changes in a step fashion, from the centre of
the fiber, the core, to the outer shell, the
cladding.
It is high in the core and lower in the
cladding. The light in the fiber propagates by
bouncing back and forth from core-cladding
interface.
The step index fibers propagate both single
and multimode signals within the fiber core.
The light rays propagating through it are in
the form of meridional rays which will cross
the fiber core axis during every reflection at
the core – cladding boundary and are
propagating in a zig – zag manner.

29. Graded index fibers :
A graded index fiber is shown in Fig. Here, the
refractive index n in the core varies as we move
away from the centre.
The refractive index of the core is made to vary
in the form of parabolic manner such that the
maximum refractive index is present at the centre
of the core.

30. Sr.
No.
Step index fiber Graded Index Fiber
1 The refractive index of the core is uniform and
step or abrupt change in refractive index takes
place at the interface of core and cladding in step
index fibers.
The refractive index of core is non-uniform,
the refractive index of core decreases
parabolically from the axis of the fiber to its
surface.
2 The light rays propagate in zig-zag manner inside
the core. The rays travel in the fiber as meridional
rays and they cross the fiber axis for every
reflection.
The light rays,propagate in the form of skew
rays or helical rays. They will not cross
the,fiber axis.
3 Step index fiber is of two types viz; mono mode
fiber and multi mode fiber.
Graded index fiber is of only one type,that is
multi mode fiber.
4 The refractive index of the core of step index fiber
is constant throughout the core.
The refractive index of the core of the
graded index fiber is maximum at
center,core and then it decreases towards
core-Cladding interface.
5 The refractive index profile may be defined as: The refractive index profile may be defined
as:
6 Number of modes for step index
fiber Ms=V22Ms=V22. V number can be less that
2.405 or more that 2.405 for step index
Number of modes for graded index fiber
is Mg=v24Mg=v24.
7 Modal dispersion affects signal quality in step
index fiber.
Graded index fiber,provides zero dispersion
as the velocity of modes is changed by
changing R.I,such that time taken by all
modes is same.

31. The advantages of fiber optic over wire
cable
• Thinner
• Higher carrying capacity
• Less signal degradation
• Light signal
• Low power
• Flexible
• Non-flammable
• Lightweight

32. Disadvantage of fiber optic over copper
wire cable
• Optical fiber is more expensive per meter than
copper
• Optical fiber can not be join together as easily
as copper cable. It requires training and
expensive splicing and measurement
equipment.

33. The loss of fiber optic
• Material absorption
• Material Scattering
• Waveguide scattering
• Fiber bending
• Fiber coupling loss

36. A Light Sources
LED (Light emitting diode) ILD (injection laser diode)

37. Detectors
•Detector is the receiving end of a fiber optic link.
There are two kinds of Detectors
1. PIN (Positive Intrinsic Negative)
2. APD (Avalanche photo diodes)
PIN
APD

38. A BUNDLE OF PRECISELY ALIGNED FLEXIBLE
OPTICAL FIBERS IS USED SOME FOR USE AS AN
IMAGE GUIDE AND THE OTHER AS A LIGHT
GUIDE
Coherent and Incoherent Bundle