Electromagnetic Wave Theory

1. Series: EMF Theory
Lecture: #1.20
Dr R S Rao
Professor, ECE
ELECTROSTATICS
Passionate
Teaching
Joyful
Learning
Displacement, flux lines, Gauss’ Law in integral form, differential form,
importance in electrostatics, Applications and limitations.

2. •Electrostatic fields possess force, field intensity, displacement density, potential,
energy etc….
•In the present session, the focus is on displacement density, D. It is has certain
special significance in electrostatics, Gauss' law is given in terms of D.
•In olden days, people used to believe in the existence of flow, from positive charges
and into negative charges. This flow was termed as displacement and graphically
denoted by flux/field lines by Faraday.
•Displacement, flux, current all mean same, that is flow and were used interchangeably
but over a period of time displacement is confined to electric fields, flux is confined to
magnetic fields, current is used to refer to flow of charge.
Displacement
2
Dr. R S Rao
Electrostatics
Gauss'
Law-I

3. D. a
s
d
  


D E
3
Dr. R S Rao
Electrostatics
Gauss'
Law-I Gauss' Law
•Gauss' law is in terms of electric displacement with a symbol ψ and
units C.
•Displacement density, D is displacement ψ per unit normal area with
units, C/ m2 .
•Displacement and displacement density are related, by definition,
through,
•Displacement density, D is related to field intensity E through
This relation can be found easily by comparing E with D at an arbitrary point in the field
due to, let us say, a point charge.

4. 4
Dr. R S Rao
Electrostatics
Gauss'
Law-I Flux Lines
A brief on flux/field lines:
•A vector field can be described either by analytical means i.e. through a
vector function or by graphical means, using a set of field/flux lines.
•Field/flux line, by definition, is a curve drawn in the field/flux in such a way
that the field/flux is always tangential at all points on it.
•Field lines coincide with flux lines when the medium is isotropic.
•These lines exhibit certain features like they originate on positive charge and
terminate on negative charge/infinity and they never cross each other.

5. 5
Dr. R S Rao
Electrostatics
Gauss'
Law-I Flux Lines
Certain aspects worth remembering about flux lines:
While sketching, there is no restriction on number of lines but
it should be sufficiently large so that the detailed variations of
field to the required extent are exhibited.
Consistency should be maintained in the number of lines so
that number is proportional to charge.
Lines must be placed fairly i.e. spacing should be as uniform
as possible, as if they emanate from a point charge
symmetrically in all directions.

6. Flux-line representations of fields due to a point charge with (a) positive polarity
and(b)negative polarity. Flux lines due to two equal charges with (c) opposite polarity (d)
same(positive) polarity.
6
Dr. R S Rao
Electrostatics
Gauss'
Law-I Flux Lines

7. 7
Dr. R S Rao
Electrostatics
Gauss'
Law-I Flux Lines
x y z
dx dy dz
E E E
 
z
d d dz
E E E
 
  
 
sin
r
dr rd r d
E E E
 
  
 
←Cartesian system
← cylindrical system
← spherical system
Analytical relations that represent flux lines:

8. 8
Dr. R S Rao
Electrostatics
Gauss'
Law-I Gauss' Law
What is Gauss' law….
•It describes the electric displacement emanating from a charge
distribution….
•It connects displacement with charge….two totally different
concepts…
•This is one of the two basic laws in static electric fields…

9. D. a enc
s
d Q


9
Dr. R S Rao
Electrostatics
Gauss'
Law-I Gauss' Law
•Gauss' law has two forms, integral form and differential form.
•Statement of its integral form is,
'the net electric displacement through any closed surface s is
equal to the net charge enclosed by that surface'.
Mathematically,
•Statement of its differential form is,
'the divergence of electric displacement density at an arbitrary
point in the field is volume charge density at that point'.
ρ
D
 
Mathematically,

10. 10
Dr. R S Rao
Net charge:
Electrostatics
Gauss'
Law-I Gauss' Law

11. ρ
D
 
11
Dr. R S Rao
( Divergence Theorem
d d
  
 
D a D)
ρ
enc
Q d
 
Conversion from integral to differential form:
Electrostatics
Gauss'
Law-I
Here, ρ is volume charge density. If the density is due to a point charge
or line charge or surface charge, then what is divergence ????….
Gauss' Law

12. 12
Dr. R S Rao
Electrostatics
Gauss'
Law-I
Similarly, the volume charge density of a charge sheet lying over the
xy-plane, with a surface charge density of σ C/m2, can be represented
with a delta function as
ρ( ) ( )
o
Q
 
r r r
ρ( ) λ ( ) ( )
y z
 

r
ρ( ) σ ( )
z


r
A point charge QC located at r=ro can be represented by a volume
charge with a density given by,
Line and surface charge distributions can also be represented with
delta functions. For example, the volume charge density of a line
charge, with a density of λ C/m, lying along x-axis can be denoted by
Gauss' Law

13. 13
Dr. R S Rao
Electrostatics
Gauss'
Law-I Gauss' Law
•Integral form connects displacement with enclosed charge and differential
form gives divergence of electric field.
•From one to other form of this law, one can easily move using divergence
theorem of calculus.
•Differential form has theoretical importance where as integral form has
practical utility, to find intensity.
•Integral form is applicable for any charge distribution but differential form is
useful only with volume charge distributions.
•Even though Gauss′ law is enunciated for static fields, it is considered valid
even for dynamic fields.

14. 14
Dr. R S Rao
Electrostatics
Gauss'
Law-I
 
o
 
D E P
ρf
 
D D a enc
d Q
  

 
1
o o o e o e
     
     
D E P E E E
Within a dielectric,
In linear dielectrics, polarization is proportional to the
field, and hence
in Dielectrics
Gauss' Law
And hence, the Gauss’ law in dielectrics is

15. 15
Dr. R S Rao
Electrostatics
Gauss'
Law-I Gauss' Law Applications
•Gauss′ law in integral form has an extraordinary power of
computing fields quickly and easily when the charge distribution
exhibits certain high level symmetry i.e. planar/ cylindrical/spherical
symmetry.
•Depending upon the symmetry exhibited by the charge
distributions, closed surfaces known as Gaussian surfaces, over
which integration is performed, are selected and most of the times,
it is either a cylinder or a pill box or a spherical shell.

16. 16
Dr. R S Rao
Electrostatics
Gauss'
Law-I Gauss' Law Applications
•The criterion in selecting the surface is, the integration involved in the Gauss' law
must become least complicated.
•It is so when the field is either normal or tangential to the closed surface, in former
case, dot product becomes ordinary product and in later case, dot product results in
zero, in both the cases, there is a lot of simplification.
•It is also possible to apply Gauss' law to combinations of symmetric objects, even
when the arrangement as a whole is not symmetrical.
•In such case, the law can be applied to each individual objects separately, and the
resulting field intensities are added to find the net field intensity using superposition
principle.

17. 17
Dr. R S Rao
Electrostatics
Gauss'
Law-I Gauss' Law Applications

18. 18
Dr. R S Rao
Electrostatics
Gauss'
Law-I Gauss' Law Limitations
•In integral form, this law is very handy in finding field intensity, easily, just
by mere observation.
•Its usefulness is limited to certain special situations, like where the charge
distribution is highly symmetrical.
•Another limitation is, its application to find field intensity requires the prior
information on direction of the field.
•It gives only magnitude not direction of field.

19. ENOUGH
FOR
TODAY
ENOUGH
FOR
TODAY
ENOUGH
FOR
TODAY
ENOUGH
FOR
TODAY
ENOUGH
FOR
TODAY
19
Dr. R S Rao