Download Maxwell`s equations

TIME VARYING FIELDS AND MAXWELL’S EQUATION
PREPARED AND COMPILED BY:
NAME: KHUSHBOO .I. RAJPUROHIT
ENROLLMENT NO.: 130020111029
SEMESTER: 5th
BRANCH: ELECTRONICS AND COMMUNICATION
GUIDED BY:
Asst.Prof. CHIRAG PARMAR
TIME VARYING FIELDS

Time varying electric field can be produced by time varying magnetic field
and

Time varying magnetic field can be produced by time varying electric field.

The equations describing the relation between changing electric and
magnetic fields are known as MAXWELL’S EQUATION


Maxwell’s equation are extensions of the known work of Gauss ,Faradays
and Ampere.
There are two forms of Maxwell’s equation namely:
1. Integral form.
2. Differential or point form.
Faraday's Law

Any change in the magnetic environment of a coil of wire will cause a
voltage (emf) to be "induced" in the coil. No matter how the change is
produced, the voltage will be generated. The change could be produced by
changing the magnetic field strength, moving a magnet toward or away from
the coil, moving the coil into or out of the magnetic field, rotating the coil
relative to the magnet, etc.

Faraday's law is a fundamental relationship which comes from Maxwell's
equations. It serves as a succinct summary of the ways a voltage may be
generated by a changing magnetic environment. The induced emf in a coil
is equal to the negative of the rate of change of magnetic flux times the
number of turns in the coil. It involves the interaction of charge with
magnetic field.
Lenz’s Law

When an emf is generated by a change in magnetic flux according to
Faraday's Law, the polarity of the induced emf is such that it produces a
current whose magnetic field opposes the change which produces it. The
induced magnetic field inside any loop of wire always acts to keep the
magnetic flux in the loop constant. In the examples below, if the B field is
increasing, the induced field acts in opposition to it. If it is decreasing, the
induced field acts in the direction of the applied field to try to keep it
constant.
Faraday's Law
A nonzero value of emf may result from of the situations:
1 – A time-changing flux linking a stationary closed path
2 – Relative motion between a steady flux and a closed path
3 – A combination of the two

d 
emf   
 dt 
emf
 N
emf



d

dt
d 


dt 
E_dot_ d L
B_dot_ d S

d
emf
E_dot_ d L

B_dot_ d S
 dt

Applying Stokes' Theorem






( Del  E)_dot_ d S




d
B_dot_ d S
dt
Removing the intregals - assuming same surface
( Del  E)_dot_dS
Del  E
 d B

 dt 

 d B  _dot_dS

 dt 


emf
B y  d
d
 
dt
 d y d

 dt 
B 
B v  d
Consider example using concept of motional emf
F
Qv  B
F
v B
Q
Em
v B
The force per unit charge is called
the motional electric field intensity
Subject every portion of the moving conductor
Displacement Current & Maxwell’s Equations

Displacement current
 Maxwell’s equations: Gauss’s law
 Maxwell’s equations: Gauss’ law for magnetism
 Maxwell’s equations: Faraday's law
 Maxwell’s equations: Ampere’s law
Poynting’s Theorem

It is frequently needed to determine the direction in which the power
is flowing. The Poynting’s Theorem is the tool for such tasks.
We consider an arbitrary
shaped volume:
D
 H  J 
t
B
 E  
t
We take the scalar product of E and subtract it from the scalar product
of H.
B
H   E  E   H  H
E
t
D 

J 


t


Application of divergence theorem and the Ohm’s law lead to the PT:

s (E  H ) ds   t
Here
1
2
2
2

H


E
dv


E


v 2
v dv
S  EH
is the Poynting vector – the power density and
the direction of the radiated EM fields in W/m2.
Poynting’s Theorem

The Poynting’s Theorem states that the power that leaves a region
is equal to the temporal decay in the energy that is stored within the
volume minus the power that is dissipated as heat within it – energy
conservation.
1
w    H 2   E 2 
EM energy density is
2
Power loss density is
pL   E 2
The differential form of the Poynting’s Theorem:
w
 S
  pL
t
SUMMARY
REFERENCES
• https://www.google.co.in/search?q=maxwells+equation&oq=m
axwells+equation&aqs=chrome..69i57.6875j0j7&sourceid=chr
ome&es_sm=93&ie=UTF-8
• http://hyperphysics.phyastr.gsu.edu/hbase/electric/maxeq.html
• http://cms.szu.edu.cn/emc/PPT/7Timevaring%20Fields%20and%20Maxwell%20equations.pdf
• http://www.slideshare.net/jayaraju_2002/time-varying-fieldsand-maxwells-equations
• http://www.rpi.edu/dept/phys/Courses/ppd1050/Lecture23.ppt