emf-1-unit (1).ppt

1
UNIT - 1
ELECTROSTATICS
ELECTROMAGNETIC
FIELDS THEORY

CONTENTS
• Vector Calculus
• Coordinate Systems
• Differential elements of the coordinate
systems
• Electric field intensity
• Coulomb’s law
• Electric flux density
• Gauss’s law and its applications
• Divergence and divergence theorem
Chapter 1 EE1005 ELECTROMAGNETIC FIELDS THEORY 2

Syllabus
UNIT –1 - “ ELECTROSTATICS”
Introduction to various co ordinate
systems – Coulomb’s law – Electric field
intensity – Electric fields due to point,
line , surface and volume charge
distributions – Electric flux density –
Gauss’s law and its applications –
Electric potential – Potential gradient –
Divergence – divergence theorem.

Coordinate systems
The various coordinate systems used often in
electromagnetic fields are,
i) Cartesian coordinate system,
ii) Cylindrical coordinate system,
iii) Spherical coordinate system

Cartesian coordinates
• Very convenient when dealing with the problems
having rectangular symmetry.
• This system has three coordinate axes represented
by (x,y,z) which are mutually at right angles to each
other.
• These three axes intersect at a common point called
origin of the system.

Cartesian coordinates

Cartesian coordinate
• In this system, x = 0 indicates 2 dimensional y - z
plane, y = 0 indicates 2 dimensional x - z plane and
z = 0 indicates 2 dimensional x - y plane.
• A vector in Cartesian coordinates can be written
as (Ax, Ay, Az) or Axax + Ayay + Azaz.
• The ranges of (x,y,z) are as follows:
-∞ < x < ∞
-∞ < y < ∞
-∞ < z < ∞

Cylindrical Coordinates
• Very convenient when dealing with problems having
cylindrical symmetry.
• A point P in cylindrical coordinates is represented as
(r, Φ, z) where
– r: is the radius of the cylinder; radial displacement
from the z-axis
– Φ: azimuthal angle or the angular displacement
from x-axis
– z : vertical displacement z from the origin (as in the
Cartesian system).

Cylindrical coordinates

• The range of the variables are
0 ≤ r < ∞, 0 ≤ Φ < 2π , -∞ < z < ∞
• vector in cylindrical coordinates can be
written as (Ar, Aφ, Az) or Arar + Aφaφ+ Azaz
• The magnitude of is
2
2
2
|
| z
r A
A
A
A 

 


Relationships Between Variables
• The relationships between the variables (x,y,z)
of the Cartesian coordinate system and the
cylindrical system (ρ, φ , z) are obtained as
• So a point P (3, 4, 5) in Cartesian coordinate is
the same as?
z
z
x
y
y
x
r





/
tan 1
2
2

z
z
r
y
r
x





sin
cos

Relationships Between Variables
• So a point P (3, 4, 5) in Cartesian coordinate is
the same as P ( 5, 0.927,5) in cylindrical
coordinate.
5
927
.
0
3
/
4
tan
5
4
3
1
2
2







z
rad
r


Spherical Coordinates
• The spherical coordinate system is used
dealing with problems having a degree of
spherical symmetry.
• Point P represented as (r,θ,φ) where
– r : the distance from the origin,
– θ : called the colatitude is the angle between z-axis
and vector of P,
– Φ : azimuthal angle or the angular displacement
from x-axis (the same azimuthal angle in
cylindrical coordinates).

2
2
2
|
| 
 A
A
A
A r 




Relation between cartesian coordinate
system:
x
y
z
y
x
z
y
x
r
1
2
2
1
2
2
2
tan
)
(
tan















cos
sin
sin
cos
sin
r
z
r
y
r
x




Point transformation
Point transformation between cylinder and
spherical coordinate is given by
2
2
z
r 
  
 

 

 sin
r
 
cos
r
z 

Differential Elements
• In vector calculus the differential elements
in length, area and volume are useful.
• They are defined in the Cartesian,
cylindrical and spherical coordinate

Cartesian Coordinates
z
y
x dza
dya
dxa
l
d 



ax
ay
az

z
y
x
dxdy
S
d
dxdz
S
d
dydz
S
d
a
a
a






0

Chapter 1 EE 1005 ELECTROMAGNETIC FIELDS THEORY 24
z
y
x dz
dy
dx
l
d a
a
a 



z
y
x
dxdy
S
d
dxdz
S
d
dydz
S
d
a
a
a






dxdydz
dv 

0
z
dz
d
d
l
d a
a
a 

 
 



P

z
a
d
d
S
d
dza
d
S
d
dza
d
S
d















z
dza
a
d
a
d
l
d 

 
 



z
a
d
d
S
d
dza
d
S
d
dza
d
S
d














dz
d
d
dv 




Chapter 1 BEE 3113ELECTROMAGNETIC FIELDS THEORY 28



 d
r
d sin


 

 a
d
r
a
rd
dra
l
d r sin













a
a
a
rdrd
S
d
drd
r
S
d
d
d
r
S
d r






sin
sin
2


 

 a
d
r
a
rd
dra
l
d r sin












a
rdrd
S
d
a
drd
r
S
d
a
d
d
r
S
d r






sin
sin
2


 d
drd
r
dv sin
2


Electrostatics
• Electrostatics is the branch of
electromagnetics dealing with the effects
of electric charges at rest.
• The fundamental law of electrostatics is
Coulomb’s law.

Electric Charge
• Electrical phenomena caused by friction
are part of our everyday lives, and can be
understood in terms of electrical charge.
• The effects of electrical charge can be
observed in the attraction/repulsion of
various objects when “charged.”
• Charge comes in two varieties called
“positive” and “negative.”

Electric Charge
• Objects carrying a net positive charge
attract those carrying a net negative
charge and repel those carrying a net
positive charge.
• Objects carrying a net negative charge
attract those carrying a net positive charge
and repel those carrying a net negative
charge.
• On an atomic scale, electrons are
negatively charged and nuclei are

Electric Charge
• Electric charge is the physical property of
the matter that causes it to experience a
force when placed in an electromagnetic
field.
e = 1.602  10-19 C.

Coulomb’s Law
• Coulomb’s law is the “law of action”
between charged bodies.
• The coulombs law states that the force
between the two point charges Q1 and Q2 ,
o Acts along the line joining the two points,
o Is directly proportional to the product of
the two charges,
o Is inversely proportional to the square of
the distance between them.

Coulomb’s law
Point Charge:
A point charge is a electric charge that
occupies a region of space which is
negligibly small compared to the distance
between the point charge and any other
object.

Coulomb’s law
Chapter 1
BEE 3113ELECTROMAGNETIC FIELDS THEORY 37
• Consider the two point charges Q1 and Q2,
• Mathematically the force between the
charges can be explained as,
Where,
• Q1Q2 - the product of the charges,
• Distance between the two charges.
R

Coulomb’s law
It also states that this force depends on
the medium in which the point charges are
located.
where,
K – Constant of proportionality

Coulomb’s law
where,
•ε - Permittivity of the medium in which
charges are located in Farad/metre
(absolute permittivity)
•ε0 – Permittivity of free space or vacuum
•εr - Relative permittivity or dielectric
constant of the medium

Vector form

Coulomb’s law
• The force on Q1 due to Q2 is equal in
magnitude but opposite in direction to the
force on Q2 due to Q1.
1
2 F
F 


Electric Field Intensity
• Electric field intensity is defined as
the strength of an electric field at any
point.
• It is equal to the force per unit charge
experienced by a test charge placed at
that point.
• The unit of measurement is the volt/meter.

Charge distributions
• Point Charge:-
If the dimensions of a surface carrying
charge is very very small when compare to
the surface surrounding it, then it is treated
as a point. Such a charge is called as
point charge.

• Line Charge:-
If the charge may be spreaded all along a
line which may be finite or infinite, then
such a charge is called as line charge.
Line Charges

• Surface charge:-
If the charge is uniformly distributed along a
two dimensional surface then it is called as
surface charge or a sheet of charge.
Surface charges

• Volume Charge:-
If the charge is distributed uniformly over a
volume, it is called volume charge.
Volume charge

E due to line charge
Line charge distribution

E due to line charge
• Differential charge:
• Differential field strength:

E due to surface charge
Surface Charge Distribution

E due to surface charge

E due to volume charge
Volume Charge Distribution

E due to volume charge

E due to infinite line charge

• Consider an infinitely straight line carrying uniform line
charge having density ρL C/m.
• Let this line lies along z axis from -∞ to ∞ and hence
called infinite line charge.
• Let point P is on Y axis at which electric field intensity is
to be determined.
• The distance of point P from the origin is ‘r’.
• Consider a small differential length dl carrying a charge
dQ along the line.
• It is along Z axis hence, dl = dz.

Symmetry of charges in z direction

• Therefore, dQ = ρL dl = ρL dz.
• The coordinates of dQ are (0,0,z) while
the coordinates of point P are (0,r,0).
• Hence the distance vector R can be
written as,

Therefore, we get the differential field intensity,
The field intensity of the infinite line charge is given by,

• For the integration, use the following
substitution,
•
• Here, r is not the variable of integration.
• By changing the limits,

• Therefore,
• Hence the result can be expressed as,

E due to charged circular ring
Charged circular Ring

• Consider a charged circular ring of radius r
placed in XY plane with Centre as origin,
carrying a charge uniformly along its
circumference.
• The charge density is ρL C/m.
• The point P is at a perpendicular distance
z from the ring as shown in the figure.

• Consider a small differential length dl on
this ring.
• The charge on it is dQ.
where,
• R – Distance of point P from dl.

• Consider the cylindrical coordinate
system.
• For dl, we are moving in ϕ direction,

• Here, R can be obtained from it two
components, in cylindrical system.
• They are,
• i) Distance r in the direction of - ar, radially
inward. i.e., -r. ar
• ii) Distance z in the direction of az vector.
i.e., z az.

• Therefore,

• By substituting the values, we get,
Now,

• Therefore,

• Therefore,
where,
• r – Radius of the ring,
• z – Perpendicular distance of point P from
the ring.
• This is the electric field at a point P (0,0,z)
due to the circular ring of radius r placed in

Electric Field due to infinite sheet of
charge
• Consider an infinite sheet of charge having
uniform charge density ρS in Coulomb per
metre square. It is placed in XY plane.
• Let us use cylindrical coordinates.
• The point P at which E to be calculated is
on z axis.
• Consider the differential surface area ds
carrying a charge dQ.
• The normal direction to dS is z direction
hence dS normal to z direction is r dr dϕ .

Electric Field due to infinite
sheet of charge

sheet of charge
• The distance vector R has two
components.
• i) The radial component r along – ar .
i.e., -r ar .

• With these two components R can be
obtained from the differential area towards
point P as,
sheet of charge

sheet of charge
• The dE can be written as,
• For a infinite sheet in XY plane, r varies
from 0 to ∞ while ϕ varies from 0 to 2π.

sheet of charge
• As there is symmetry about z axis from all
radial direction, all ar components of E are
going to cancel each other and the net E
will not have any radial component.

sheet of charge
• Therefore the E can be written as,
• By changing the limits, we get,

sheet of charge
• Therefore on integration,

sheet of charge
• It can be written as,
where,
• an – direction normal to the surface charge
• For the points below XY plane, an = - az
• Therefore,

Electric Flux
• The total number of lines of force in any
particular electric field is called electric
flux.
• It is represented by the symbol ψ.
• The unit of flux is Coulomb.
Chapter 1 EE 1005 ELECTROMAGNETIC FIELDS THEORY 79
Electric flux lines

Properties of electric flux
• The flux lines start from positive charge
and terminate at negative charge.
• If the negative charge is absent, then the
flux lines terminate at infinity.
• There are more number of lines. i.e.,
crowding of lines if the electric field is
stronger.
• These lines are parallel and never cross
each other.
• The lines are independent of the medium

Electric Flux Density
• Electric flux density or displacement
density is defined as the electric flux per
unit area.
• D = ψ/S Coulomb / metre square.
where, Ψ – Total Flux, S – Surface Area
• For a sphere surface area,
A = 4 Π r2
D = Q / 4 Π r2
But, E = Q / 4 Π ε0 r2

Electric flux Density
• In vector form,
• where,
• dψ – total flux lines crossing normal to the
surface area dS.
• dS – differential surface area
• an – unit vector direction normal to the
differential surface area.

D due to a line charge
• Consider a line charge having density ρL C
/ m.
• Total charge along the line is given by,
• If the line charge is infinite,

D due to surface charge
• Consider a surface charge having density
of
ρS C / m 2.
• Total charge along the surface is given by,

D due to surface charge
• If the sheet of charge is infinite then,

D due to volume charge
• Consider a charge enclosed by a volume,
with a uniform charge density ρv C / m 3
• Then the total charge enclosed by the
volume is given by,

Divergence
• Divergence is the outflow of flux from a
small closed surface area (per unit
volume) as volume shrinks to zero.

Divergence
Example for divergence:-
•Water leaving a bathtub (incompressible)
•Air leaving a punctured tire (positive
divergence)

Divergence
• Mathematical expression for divergence,
• Surface integral as the volume element
approaches zero
• D is the vector flux density
div D
 
0
v
S
D
v




d
lim


Divergence
• For Cartesian coordinates,
• For cylindrical coordinates,
z
z
y
y
x
x A
A
A










 A
z
A
A
A
z


















1
)
(
1
A

Divergence
• For spherical coordinates,
• Divergence is a scalar.
• Dot product of two vectors will give the
resultant as a scalar as divergence.


















A
r
A
r
A
r
r
r
A r
sin
1
)
sin
(
sin
1
)
(
1 2
2

Del Operator
Written as is the vector differential
operator. Also known as the gradient
operator. The operator in useful in defining:
1.The gradient of the scalar V, written as V
2.The divergence of a vector A, written as
3.The curl of a vector A, written as
4.The laplacian of a scalar V, written as


A


A


V
2


Divergence Theorem
• The volume integral of the divergence of
a vector field over a volume is equal to the
surface integral of the normal component
of this vector over the surface bounding
the volume.
div
S E
d dV
 
 
F S F

Divergence theorem
From the gauss law we can write,
• Q = ∫∫ D. dS ------ (1)
While the charge enclosed in a volume is
given by,
• Q = ∫∫∫ ρv dv ------- (2)
But according to Gauss’s law in the point
form,
• D = ρv ------- (3)
Using equation (2),

Divergence theorem
Equating equations (1) and (4),
∫∫ D. dS = ∫∫∫ ( D) dv
The above equation is called divergence
theorem. It is also called as Gauss -
Ostrogradsky theorem.

Gauss’s Law
• “The electric flux passing through any
closed surface is equal to the total charge
enclosed by that surface.”

Gauss’s Law
Closed Surface

Gauss’s Law
• Consider a small element of area ds in a
plane surface having charge Q and P be a
point in the element.
• At every point on the surface the electric
flux density D will have value Ds.
• Let Ds make an angle θ with ds.
• The flux crossing ds is the product of the
normal component of Ds and ds.

Gauss’s law
• Therefore,
dψ = Ds normal .ds
= Ds cosθ ds
dψ = Ds. ds …. (dot
product)
• The flux passing through the closed
surface is given by,
ψ = ∫ dψ = ∫∫ Ds . ds
ψ = Q

Point form of Gauss’s Law
• The divergence of electric flux density in a
medium at a point (differential volume
shrinking to zero), is equal to the volume
charge density (charge per unit volume) at
the same point.
• This point form is also called as differential
form.
• Therefore,

Applications of Gauss’s Law
Point Charge:-
•Let a point charge Q be located at the
origin.
•To determine D and to apply Gauss’s law,
consider a spherical surface around Q with
Centre as origin.
•This spherical surface is Gaussian surface
and it satisfies the required condition.
•The D is always directed radially outwards
along ar which is normal to the spherical

Point Charge

• Consider a differential surface area ds.
• The direction normal to the surface ds is ar
(Spherical coordinate system).
• The radius of the sphere is (r = a).
• Then,

• The D due to a point charge is given by,
• Note that, θ’ is the angle between D and
ds.
Where,

• The normal to ds is ar, while D also acting
along ar, hence angle between D and ds,
i.e., 0 degree.
• Therefore,

• Therefore,
• This proves the Gauss’s law.

Infinite line of charge:-
•Consider an infinite line charge of density ρl
C/m lying along z axis from - ∞ to ∞.
•Consider the Gaussian surface as a right
circular cylinder with z – axis as its axis and
radius r.
•The length of the cylinder is L.
•The flux density on any point on the surface
is directed radially outwards. i.e., in the ar
direction according to the cylindrical

Infinite line
charge

• Consider differential surface area ds
which is at a radial distance r from the line
charge.
• The direction normal to dS is ar.
• As the line charge is along z axis, there
cannot be any component of D in z
direction.
• So D has only radial component.
• Now,

• The integration is to be evaluated for side
surface, top surface and bottom surface.
• Therefore,
• Now,
• Now, Dr is constant over the side surface.

• As D has only radial component and no
component along az and – az, hence
integrations over top and bottom surfaces
is zero.
• Therefore,

• Therefore, we get,
• The results are same as obtained from the
.. Due to infinite charge

• Co-axial Cable:-
• Consider the two co-axial cylindrical
conductors forming a co-axial cable.
• The radius of the inner conductor is ‘a’
while the radius of the outer conductor is
‘b’.
• The length of the cable is L.
• The charge distribution on the outer
surface of the inner conductor is having
density ρs C/m.

Co-axial cables

• Hence,ρs can be expressed in terms of ρL.
• Therefore,
• Thus the line charge density of inner
conductor is ρL C/m.
• Consider the right circular cylinder of
length L as the Gaussian surface. Due to

Application of Gauss’s Law
• Now, we can write,
• The total charge on the inner conductor is
to be obtained by evaluating the surface
integral of the surface charge distribution.
• Now,
Where,
…… (a)

• Therefore,
• By equating (a) and (b),
• Therefore,
…….. (b)

• But,
• This is same as obtained for infinite line
charge.
• Every flux line starting from the positive
charge on the inner cylinder must
terminate on the negative charge on the

• Hence the total charge on the inner
surface of the outer cylinder is,
• But,
• Therefore,

Infinite Sheet of Charge:-
•Consider the infinite sheet of charge density
ρs C/m square, lying in the z = 0 plane. i.e.,
XY plane.
•Consider a rectangular box as a Gaussian
surface which is cut by the sheet of charge
to give ds = dx dy.

Infinite sheet of charge

• But,
• The results are same as obtained by the
coulomb’s law for infinite sheet of charge.

• Differential Volume element:-
Differential Volume element

• According to Gauss’s Law,
• The total surface integral is to be
evaluated over six surfaces front, back,
left, right , top and bottom.
• Therefore,

• Consider the front surface of the
differential element.
• Though D is varying with distance, for
small surface like front surface it can be
assumed constant.
• Now,

Applications of Gauss’s law
• It has been mentioned that Dx , front is
changing in X direction. At P, it is Dx0 while
on the front surface it will change and
given by,
• Therefore,

• Consider the integral over the back
surface,
• Therefore,
where,

• Now,

• But,
• The charge enclosed in a volume is given
by,
• This result leads to the concept of
divergence.

Electric Potential
The electric potential may be illustrated with
the help of the following concepts.
•Work done
•Potential Difference
•Absolute potential

Work done
• The work is said to be done when the test
charge is moved against the electric field.
• The potential energy of the test charge is
nothing but the work done in moving the
charge against the electric field. i.e., work
done in moving the test charge against the
direction of electric field over the distance
from initial position to final position.
• It is measured in joules.

Differential Work done
• If a charge Q is moved from initial position
to final position, against the direction of
electric field E then the total work done is
obtained by integrating the differential
work done over a distance from initial
position to the final position.

Line Integral
• Consider that the charge is moved from
initial position B to the final position A,
against the electric field E the work done is
given by,
• This is called line integral.

Potential Difference
• The work done in moving unit charge from
point B to A in the field of E is called
Potential difference between the points B
and A.
• It is denoted by V and its unit is J / C or V.
• If B is the initial point and A is the final
point then the P.D is denoted by VAB.

Potential due to Point charge
• Consider a point charge, located at the
origin of a spherical co ordinate system,
producing E radially in all the directions .

Potential due to Point charge
• Assuming free space ,the field E due to a
point charge Q at a point having radial
distance r from origin is given by,
• Consider a unit charge which is placed at
a point B which is at a radial distance of rB
from the origin.
• The point A is at a radial distance of rA
from the origin.

Potential due to point charge
• The differential length in spherical system
is,
• Hence the potential difference VAB
between points A and B is given by,
•

Potential due to a point charge
• Therefore we get,

Absolute potential
• The absolute potentials are measured with
respect to a specified reference positions.
• Such reference position is assumed to be
at zero potential.
SHIELDED CABLE

Absolute Potential
• Consider potential difference VAB due to
movement of unit charge from B to A in a
field of a point charge Q.
• Now let the charge is moved from infinity
to the point A . i.e., rB = infinity.
• Hence,

Absolute Potential
• Therefore,
• The potential of point A or absolute
potential of point A is ,
• The potential of point B or absolute
potential of point B is,

Absolute potential
• The potential difference can be expressed
as the difference between the absolute
potentials of two points.
• Hence the absolute potential at any point
which is at a distance r from the origin of a
spherical system is given by,

Potential due to point charge
not at origin
• If the point charge is not located at the
origin of a spherical system then obtain
the position vector where Q is located.
• The absolute potential at a point A located
at a distance r from the origin is given by,
•

Potential due to a line charge
Line charge distribution

Potential due to a line charge

Potential due to surface charge
Surface charge distribution

Potential due to surface charge

Potential due to volume charge
Volume charge distribution

Potential due to volume charge

Potential Gradient
• The rate of change of potential with
respect to the distance is called the
potential gradient.

Gradient of a scalar
• In Cartesian coordinates,
• In Cylindrical coordinates,
• In Spherical coordinates,
z
y
x a
z
V
a
y
V
a
x
V
V
V
grad











z
a
z
V
a
V
1
a
V
V












 















 a
V
sin
r
1
a
V
r
1
a
r
V
V r

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