The document discusses the concept of electric scalar potential in electrostatics. It defines scalar potential as the work done per unit charge to bring a charge from infinity to a point in an electric field. Scalar potential is a function of position and is related to the electric field intensity by the gradient of potential. The summary discusses: 1) Scalar potential is defined as the work done to move a unit positive charge from infinity to a point in an electric field. It obeys the superposition principle. 2) Absolute potential is defined with respect to a zero potential reference (e.g. infinity), while relative potential is defined between two finite points. 3) Expressions are provided for the scalar potential of

1. Series: EMF Theory
Lecture: #1.30
Dr R S Rao
Professor, ECE
ELECTROSTATICS
Passionate
Teaching
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Learning
Electric scalar potential, definition, absolute potential, relative potential,
scalar potential for line charge, surface charge and volume charge distributions.

2. • Electric scalar potential, like field intensity, displacement density is one
of the parameters useful to give an insight into the nature of
electrostatic fields.
• It has an added advantage of being a scalar function, easy to deal with
when compared to vector functions.
• It is a function of source distribution, and is related to field intensity. In
fact, it is defined with respect to field, E surrounding the source charge
distribution.
• Potential function obeys super position principle. It is continuous in the
field and also single valued at any point.
Dr. R S Rao 2
Introduction
Electrostatics
Electric
Scalar
Potential-I

3. To define the potential, consider an electric field, E, surrounding an arbitrary
charge distribution.
The scalar (absolute)potential, denoted by V, at an arbitrary point P in the
electric field, E is defined as the work done by any external agency i.e. other
than field, to bring a unit positive charge from infinity to the point under
consideration.
Dr. R S Rao 3
Definition
Electrostatics
Electric
Scalar
Potential-I
As the potential is work done per unit charge, its units are J/C, usually
called Volts, indicated by capital letter, V.

4. ( )
P
V P d

  E l
Dr. R S Rao 4
Definition
Electrostatics
Electric
Scalar
Potential-I
Mathematically,
Explanation: A point charge in field experiences Coulomb's force. Work done
in moving this charge is dot product between force and distance. Force is
equal to field intensity when charge is unit positive charge. Hence, integral
E.dl gives work. Negative sign is to signify the work is by external agency.
Lower and upper limits are initial and final positions of the charge. As initial
position of charge is at infinity, lower limit on the integral is infinity.
The lower limit at infinity is acceptable in all cases except when the source
charge distribution itself extends to infinity, like infinite straight line or
infinite sheet charge.

5. Dr. R S Rao 5
Definition
Electrostatics
Electric
Scalar
Potential-I
ˆ ˆ ˆ ˆ ˆ ˆ
( )
V V V
dV dx dy dz
x y z
V V V
dx dy dz
x y z
V d
  
  
  
 
  
    
 
  
 
 
x y z x y z
l
The corresponding differential relation is,
dV d
 E l
V

E = -
From the defining integral relation,
From integral calculus,
By comparison, the defining differential relation can be obtained.

6. Curl and Div
• From the defining differential relation of potential, the Curl of field can be
found as,
Hence, the field intensity- grad V equality implies, curl of electrostatic field is
zero.
• When the curl is zero, the field is said to be irrotational. From Stokes
theorem, zero curl implies, closed line integral of field is zero.
When the line integral of field around any closed path is zero, the field is said
to be conservative. It implies, energy of the charge-field system remains
constant during charges motion in the field.
• Hence, electrostatic fields are irrotational and also conservative
0
V
   
E
0
d 
 E l
Dr. R S Rao 6
Electrostatics
Electric
Scalar
Potential-I

7. Absolute/Relative Potential
There are two types of potentials: absolute potential and relative
potential.
• Absolute potential is one which is computed with respect to zero
potential reference point.
• For finite charge distributions, zero potential point exists at
infinity, and hence infinity is used as lower limit on the defining
integral of potential.
• For infinite straight line charge, zero potential point exists nowhere
and hence, absolute potential exists not.
• For infinite plane sheet charge, zero potential point exists at the
origin and hence, reference to find absolute potential is the origin.
Dr. R S Rao 7
Electrostatics
Electric
Scalar
Potential-I

8. Absolute/Relative Potential
• Relative potential is one computed with respect to a point where the potential
is finite non-zero. Mathematically,
Here, Vab is the expression for relative potential at a point a with respect to
point b in field E. V(a) and V(b) are absolute potentials at points a and b.
Changing the reference point amounts to addition/subtraction of a constant to
the potential.
Potential difference does not depend upon path of integration, but only on the
end points.
• Without exception, relative potential exists for all charge distributions, finite
or infinite
( ) ( )
a
ab b
V d V a V b
   
 E l
Dr. R S Rao 8
Electrostatics
Electric
Scalar
Potential-I

9.  
2
1 ˆ ˆ
ˆ ˆ sin
4
r r θ
P
Q
V dr rd r d
r
  
 
    
 
̂
2
1
4 4 4
P
P
Q dr Q Q
r r r
  



    

Potential, V for Point Charge
The absolute potential due to a point charge, Q located at the origin
at an arbitrary point P can be calculated.
The relation, however, can be extended to configurations in which the
source charge is at an arbitrary point.
1 ( )
( )
4
Q
V
R



r
r
Dr. R S Rao 9
Electrostatics
Electric
Scalar
Potential-I
( )
P
V P d

  E l

10. Potential obeys superposition principle and expression for a
single point charge can be extended to other distributions.
In case of group of point charges,
1 2
1 2
1 2
1
......
1
.....
4
1
4
n
n
n
n
k
k k
V V V V
Q
Q Q
R R R
Q
R

 
  
 
   
 
 
 
=
Potential for Discrete Charge Distributions
Electrostatics
Electric
Scalar
Potential-I

11. Potential for Continuous Charge Distributions
Dr. R S Rao 11
Electrostatics
Electric
Scalar
Potential-I
The potential expression for point charge can be extended to continuous charge
distributions.

12. 1 1 λ
4 4
dQ dl
V
R R
 
 
= =
1 1 σ
4 4
dQ da
V
R R
 
 
= =
1 1 ρ
4 4
dQ d
V
R R

 
 
= =
1 1 1
4 4 4
Q dQ dQ
V dV V dV
R R R
  
   
= = = =
In case of line charge, dQ = λdl , in case of surface charge, dQ = σda
and in case of volume charge, dQ = ρdτ
←Line charge
← Surface charge
← Volume charge
For continuous distributions,
Potential for Continuous Charge Distributions
Dr. R S Rao 12
Electrostatics
Electric
Scalar
Potential-I

13. S.No
Charge
distribution
Electric scalar
potential,  
V r
Electric field
Intensity,  
E r
1.
Point charge
 
1
4
Q
R


r  
2
1 ˆ
4
Q
R


r
R
2. Discrete charge
distribution
 
1
1
4
n
k k
k k
Q
R
 


r  
2
1
1 ˆ
4
n
k k
k
k k
Q
R
 

r
R
3.
Line charge
1 λ( )
4
dl
R

 

r
2
1 λ( ) ˆ
4
dl
R

 

r
R
4.
Surface charge
1 σ( )
4
da
R

 

r
2
1 σ( ) ˆ
4
da
R

 

r
R
5.
Volume charge
1 ρ( )
4
d
R


 

r
2
1 ρ( ) ˆ
4
d
R


 

r
R
Electric Scalar Potential, V
Dr. R S Rao 13
Electrostatics
Electric
Scalar
Potential-I

14. Equipotential Contours
Flux lines and equipotential contours of (a) point charge (b) large sheet charge
(c) large straight line charge (d) long coaxial cable.
Dr. R S Rao 14
Electrostatics
Electric
Scalar
Potential-I

15. ENOUGH
FOR
TODAY
ENOUGH
FOR
TODAY
ENOUGH
FOR
TODAY
ENOUGH
FOR
TODAY
ENOUGH
FOR
TODAY
Dr. R S Rao 15