Electromagnetic Wave Theory

1. Series: EMF Theory
Lecture: #1.31
Dr R S Rao
Professor, ECE
ELECTROSTATICS
Passionate
Teaching
Joyful
Learning
Electric scalar potential from field intensity and from source distribution,
infinite straight line charge, infinite plane sheet charge, spherical shell charge and
solid spherical charge.

2. • The scalar potential can be computed in two ways: either from
potential-field relation or from potential-charge relation.
• The first one requires the field expression on hand whereas the
second one uses source distribution.
• Both of them involve integration. All the time it may not be possible
to integrate. Hence, only for certain field or charge configurations,
these two methods are useful.
• One can opt for numerical methods for integration also but they
have their own limitations, like errors, accuracy etc.
• Here, both the methods are illustrated by considering various
standard charge distributions.
Electric Scalar Potential, V
Electrostatics
Electric
Scalar
Potential
2

3. Electric Scalar Potential, V
Solution procedure
Electrostatics
Electric
Scalar
Potential
• Infinite straight line charge
• Infinite plane sheet charge
• Hollow spherical shell-inside and outside
• Solid spherical charge-inside and outside
3
Four types of charge distributions are
considered:

4. S.No. Charge distribution
Electric scalar
potential
Electric field
intensity
1.
Infinite line charge
λ
ln
2
r
V

 
 
  
 
ˆ
2



E ρ
2. Infinite plane sheet
charge
σ
2
z
V

 
σ
ˆ
2

E n
3.
Spherical
shell
Outside
1
4
Q
V
r

 2
1
ˆ
4
Q
r


E r
Inside
1
4
Q
V
R

 0
4.
Solid sphere
Outside
1
4
Q
V
r

 2
1
ˆ
4
Q
r


E r
Inside
2
2
1
3
4 2
o
Q r
V
R R

 
 
 
 
2
1
ˆ
4
r
Q
r


E r
Electric Scalar Potential, V
Electrostatics
Electric
Scalar
Potential
4

5. λ
ˆ
2

E 
   
 
λ 1 ˆ
ˆ ˆ ˆ
2
λ λ
ln ln
2 2
V d d dz
d


   
 


  


   
     


z
  
Infinite straight line charge:
Absolute potential exists not :
Relative potential is:
Electric Scalar Potential, V
 
λ λ
ln
2 2
r
r
d
V





   
 
    
 

Electrostatics
Electric
Scalar
Potential
5
( )
P
V P d

  E l

6. Two infinite straight line charges of opposite polarity: The potential at P
with respect to reference O is sum of potentials due to individual charges.
Electric Scalar Potential, V
Electrostatics
Electric
Scalar
Potential
  ln ln ln ln
2 2 2 2
a a a
V P
a

   
      
   
   

   
  ln
2
V P


 



As the reference point moves towards infinity, the ratio a+/a− tends to
6
Thus, the potential of the given parallel wire conductor system, with reference at
infinity is,
one, resulting in a value of zero to second term on the right hand side

7. Infinite sheet charge:
Absolute potential exists not wrt z→∞:
Relative potential exist, it is:
σ
ˆ
2

E z
   
σ ˆ
ˆ
ˆ ˆ
2
z
V d d dz
   
 
   
z z
 
 
σ
2 2
z
dz z

 

     

0
σ σ
2 2
z
V dz z
 
   

 
σ σ
2 2
r
r
z
zz r
z
V dz z z
 
   

Electric Scalar Potential, V
Electrostatics
Electric
Scalar
Potential
7
Absolute potential exists wrt z=0:
( )
P
V P d

  E l

8. Charged spherical shell:
Outside, it is:
2
1
ˆ outside
4
Q
r

 
E r
2
4
P
Q dr
r
 
  
 
2
1 ˆ ˆ
ˆ ˆ sin
4
r r θ
P
Q
V dr rd r d
r
  
 
    
 
ˆ
d
 
1
V
4 4
o
h
o
o
Q Q
V r h a
r h
 


    
Inside, it is:
  2
1 1
0 V
4 4
i
h
a
i
a
Q Q
V r h a dr dr
r a
 

     
 
Electric Scalar Potential, V
Electrostatics
Electric
Scalar
Potential
8
( )
P
V P d

  E l

9. Solid Charged Sphere:
Outside, it is:
2
1
ˆ outside
4
Q
r

 
E r
2
4
P
Q dr
r
 
  
 
2
1 ˆ ˆ
ˆ ˆ sin
4
r r θ
P
Q
V dr rd r d
r
  
 
    
 
ˆ
d
 
1 1
V
4 4
o
h
o
o
Q Q
V r h a
r h
 


    
Inside, it is:
 
2 2 2
2 3 3 2
( )
1 1 1
3 V
4 4 4 2 4 2
i
h
a
i i
a
h a h
Q Q r Q Q
V r dr dr
r a a a a a
   

   

      
   
   
 
Electric Scalar Potential, V
Electrostatics
Electric
Scalar
Potential
9
( )
P
V P d

  E l

10. Electric Scalar Potential, V
Solution procedure
Electrostatics
Electric
Scalar
Potential
The solution procedure involves:
• Selecting appropriate coordinate system: If the source exhibits
cylindrical symmetry, like infinite straight line, cylindrical
coordinate system is best suited. The spherical coordinate system
requires to be used when the source exhibits spherical symmetry.
• Locating the source over the coordinate system: When it is placed
properly, the solution becomes incredibly simple and easy.
• Selecting appropriate potential formula: Depending upon the
source distribution, one of the formulas, available for line charge,
surface charge and volume charges, is to be selected.
10

11. Coordinate system→ Cylindrical system
Location→ charge over z-axis with field point over xy-plane
Electric Scalar Potential, V
Infinite straight line charge
Formula→
Electrostatics
Electric
Scalar
Potential
1 λ
4
dl
V
R
 
=
2 2
, ( )
dl dz R z

  
11

12.  
2 2 1 2
2 2 1 2
0
2 2
0
1 λ
4 ( )
2λ
4 ( )
λ
ln Indetrminate
2
o
o
dz
V
z
dz
z
z z
 
 









   


=
Electric Scalar Potential, V
Infinite straight line charge
Electrostatics
Electric
Scalar
Potential

13. Coordinate system→ Cylindrical system
Location→ charge over xy-plane with field point
over z-axis
Formula→
Electric Scalar Potential, V
Infinite sheet charge
1 σ
4
da
V
R
 
=
2 2
, ( )
da d d R z
   
  
Electrostatics
Electric
Scalar
Potential
13

14. Electric Scalar Potential, V
Infinite sheet charge
Electrostatics
Electric
Scalar
Potential
2
2 2 2 2
0 0 0
1 σ 1 σ
( )
4 2
a a
d d d
V P
h h

    
 
 
 
 
  
 
2 2 2 2
0
σ σ
2 2
a
h a h h

 
    
 
2 2
σ
indeterminate
2
a
a
V a h h



   

15. Coordinate system→ Spherical system
Location→ Center of charge over the origin
Formula→
Electric Scalar Potential, V
Spherical shell charge
1 σ
4
da
V
R
 
=
2
2 2 1 2
sin
( 2 cos )
da a d d
R h a ha
  

  

  
=
Electrostatics
Electric
Scalar
Potential
15

16. Electric Scalar Potential, V
Spherical shell charge
Electrostatics
Electric
Scalar
Potential
2 2
σ sin σ σ
4 4 4
o o
a d d a d RdR a
dV d dR
R R h a h
   

  
  
= = =
2 2
σ σ 2σ 4 σ 1 1
Outside
2 2 2 4 4 4
o
o
o
o
h a
h a
h a
o o o o o
h a
a a a a Q Q
V dR R V
h h h h h r

     




       

2
σ σ σ σ 4 σ 1 1
2 Inside
2 2 2 4 4
i
i
i
i
a h
a h
i
a h
i i i
a h
a a a a a Q
V dR R h
h h h a a

     




      


17. Coordinate system→ Spherical system
Location→ Center of shell right over the origin
Electric Scalar Potential, V
Solid spherical charge
Electrostatics
Electric
Scalar
Potential
17
The solid sphere is divided into thin shells of thickness dr. The potential
due to each shell is found and potentials due to all the shells are then
added by integration to get potential due to the entire solid sphere.

18. Electric Scalar Potential, V
Solid spherical charge
Electrostatics
Electric
Scalar
Potential
3 3
2
0
4 ρ 4 ρ 1 4 ρ 1 1
Outside
4 4 3 4 3 4 4
a
o o o o
a a Q Q
r dr V
h h h h r
  
    
      

2
0 0 0
1 1
ρ4
4 4
a a a
dr
dr
o o
dQ
V dV r dr
h h

 
  
  
1
4
dr
dr
o
dQ
dV
h


2
ρ4
dr
dQ r dr

=
2 2 2
0 3 2 2
1 3 1
1 3 Inside
4 8 4 2
i i
i i i
h h a
Qh h h
Q Q
V V V
a a a a a
  
 
   
       
   
   

19. ENOUGH
FOR
TODAY
ENOUGH
FOR
TODAY
ENOUGH
FOR
TODAY
ENOUGH
FOR
TODAY
ENOUGH
FOR
TODAY
19