explains about Poyntings theorem, boundary conditions, poissons and laplace equations and uniqueness theorem and its importance in Electromagnetic theory

1. ELECTROMAGNETIC THEORY

2. POYNTING’S THEOREM

3. Poynting’s Theorem
• Poynting's theorem is a
statement of
conservation of energy
for the electromagnetic
field, in the form of a
partial differential
equation, due to the
British physicist John
Henry Poynting.

4. Poynting’s Theorem
• Poynting's theorem is
analogous to the work-
energy theorem in classical
mechanics, and
mathematically similar to
the continuity equation,
because it relates the
energy stored in the
electromagnetic field to the
work done on a charge
distribution (i.e. an
electrically charged object),
through energy flux.
Continuity equation

5. Statement:
The rate of energy transfer (per unit volume) from a
region of space equals the rate of work done on a
charge distribution plus the energy flux leaving that
region.
Or
The decrease in the electromagnetic energy per
unit time in a certain volume is equal to the sum of
work done by the field forces and the net outward
flux per unit time"

6. Expression(general representation):
• ∇•S is the divergence of
the Poynting vector
(energy flow)
• J•E is the rate at which
the fields do work on a
charged object (J is the
current density
corresponding to the
motion of charge, E is
the electric field.
• u is the Energy Density

7. POYNTING THEOREM -
DERIVATION

8. Maxwell's Equations:
• Energy can be
transported from one
point (where a
transmitter is located)
to another point (with a
receiver) by means of
EM waves.
• The rate of such energy
transportation can be
obtained from
Maxwell's equations:

10. Poynting’s Theorem

11. POYNTING VECTOR

12. Poynting Vector
• The cross product E × H is known as the Poynting
vector, S,
• The direction of the vector S indicates the
direction of the instantaneous power flow at a
point
• Many of us think of the Poynting vector as a
“pointing” vector. This homonym, while
accidental, is correct.

13. Poynting Vector
• Because S is given by the cross product of E
and H, the direction of power flow at any
point is normal to both the E and H vectors.
• This certainly agrees with our experience with
the uniform plane wave, for propagation in
the +z direction was associated with an Ex and
Hy component,

14. Poynting Vector

15. Dipole radiation of a dipole vertically in the page showing electric field
strength (colour) and Poynting vector (arrows) in the plane of the page.

16. Across any plane P between the battery and resistor, the Poynting
flux is in the direction of the resistor. The magnitudes (lengths) of the
vectors are not shown accurately; only the directions are significant.

17. Poynting Theorem -
Applications

18. 1. Co-Axial Cable
• Poynting vector within the dielectric insulator of a
coaxial cable is nearly parallel to the wire axis
• Electrical energy delivered to the load is flowing
entirely through the dielectric between the conductors.
• Very little energy flows in the conductors themselves,
since the electric field strength is nearly zero.
• The energy flowing in the conductors flows radially into
the conductors and accounts for energy lost to resistive
heating of the conductor.
• No energy flows outside the cable, either, since there
the magnetic fields of inner and outer conductors
cancel to zero.

20. 2. Plane waves
• Propagating sinusoidal
linearly polarized
electromagnetic plane
wave of a fixed
frequency, the Poynting
vector always points in
the direction of
propagation while
oscillating in
magnitude.

21. 2. Plane waves
• The time-averaged magnitude of the Poynting
vector is
• Em is the complex amplitude of the electric
field and η is the characteristic impedance of
the transmission medium, or just η = 377Ω for
a plane wave in free space

22. 3. Radiation Pressure
• Radiation pressure is
the pressure exerted
upon any surface
exposed to
electromagnetic
radiation.
• c is the speed of light in
free space

23. 4. Static Fields
• Shows the relativistic
nature of the Maxwell
equations and allows a
better understanding of
the magnetic component
of the Lorentz force, q(v ×
B)
• Circulating energy flow
may seem nonsensical or
paradoxical, it is
necessary to maintain
conservation of
momentum.

24. POISSON’S AND LAPLACE’S
EQUATION

25. Poisson’s equation
• Poisson's equation is a partial differential
equation of elliptic type with broad utility in
mechanical engineering and theoretical
physics.
• It arises, for instance, to describe the potential
field caused by a given charge or mass density
distribution; with the potential field known,
one can then calculate gravitational or
electrostatic field.

26. Poisson’s equation
• For a homogeneous medium,
• Solving Poisson's equation for the potential
requires knowing the charge density
distribution
• It is a generalization of Laplace's equation

27. Laplace’s equation
• If the charge density is zero in Poisson’s
equation, then Laplace's equation results,
when pv = 0
• Laplace's equation is a second-order partial
differential equation named after Pierre-
Simon Laplace who first studied its properties

28. Laplace’s Equations

29. Boundary Conditions
• If the field exists in a region consisting of two
different media, the conditions that the field
must satisfy at the interface separating the
media are called boundary conditions.
• These conditions are helpful in determining
the field on one side of the boundary if the
field on the other side is known

30. UNIQUENESS THEOREM

31. Statement:
“If a solution of Laplace's equation can be found
that satisfies the boundary conditions, then the
solution is unique”
The theorem applies to any solution of Poisson's
or Laplace's equation in a given region or closed
surface.

32. UNIQUENESS THEOREM -
DERIVATION

35. Application
Before we begin to solve boundary-value problems,
we should bear in mind the three things that
uniquely describe a problem:
1. The appropriate differential equation (Laplace's or
Poisson's equation in this chapter)
2. The solution region
3. The prescribed boundary conditions
A problem does not have a unique solution and
cannot be solved completely if any of the three
items is missing.