1) The document discusses electrostatic potential and electric fields resulting from point charges and continuous charge distributions. It defines electric potential as the work required to move a test charge between two points in an electric field. 2) Equipotential surfaces are surfaces where the electric potential is constant. The electric field is always perpendicular to equipotential surfaces. 3) For a point charge, the electric potential is defined and the potential due to multiple point charges or continuous charge distributions is derived. 4) Dipole moment and the electric potential of an electric dipole are defined. The potential of a dipole is approximated using a Taylor series expansion.

1. The divergence of E
If the charge fills a volume, with charge per unit
volume .
'
dq d
 

Where d is an element of
volume.
For a volume charge:
'
'
2
0
1 ( )
ˆ
( )
4 v
r
E R rd
r



 
R

2. Thus:
3 ' ' '
0
1
. 4 ( ) ( )
4 v
E r r r d
  

  

0
1
. ( )
E r


 
Gauss’s law in
differential form.
3
2
ˆ
. 4 ( )
r
r
r

 
 
 
 
 
' '
2
0
ˆ
1
. . ( )
4 v
r
E r d
r
 

 
 
  
 
 


3. Spherical polar coordinates (r, , )
r: the distance from the origin
: the angle down from the z axis is called polar angle
: angle around from the x axis is called the azimuthal
angle
sin cos
x r  

sin sin
y r  

cos
z r 

ˆ ˆ ˆ ˆ
sin cos sin sin cos
r x y z
    
  
ˆ ˆ ˆ ˆ
cos cos cos sin sin
x y z
     
  
ˆ ˆ ˆ
sin cos
x y
  
   ˆ ˆ
ˆ sin
dl drr rd r d
  
  

4. The Curl of E
For a point charge situated at origin:
2
0
1
ˆ
4
q
E r
r


Line integral of the field from some point a to some
other point b:
In spherical polar coordinates,
ˆ ˆ
ˆ sin
dl drr rd r d
  
  
2
0
1
.
4
q
E dl dr
r



5. 2
0
1
.
4
b b
a a
q
E dl dr
r


 
0
1
4 a b
q q
r r

 
 
 
 
True for electrostatic
field.
. 0
E dl 

Apply stokes theorem:
0
E
 
The integral around a closed path:

6. Electric Potential
Basic concept:
The absence of closed lines is the property of vector
field whose curl is zero.
E is such a vector whose curl is zero.
Using this special kind of it’s property we can reduce a
vector problem: using V, we can get E very easily.
Vector whose curl is zero, is equal to the gradient of
some scalar function
E=0  the line integral of E around any closed loop
is zero (due to Stokes' theorem).
E V
 

7. otherwise you could go out along path (i) and return
along path (ii) and
Because the line integral is independent of path, we
can define a function
O is some standard reference point.
. 0
E dl 

Therefore the line integral of E from
point a to point b is the same for all
paths.
. 0
E dl 

( ) .
r
o
V r E dl
 
is called electric
potential

8. The potential difference between two points a and b:
( ) ( ) . .
b a
o o
V b V a E dl E dl
   
 
. .
b o
o a
E dl E dl
  
 
.
b
a
E dl
 
Using fundamental theorem for gradients:
( ) ( ) ( ).
b
a
V b V a V dl
  

So ( ). .
b b
a a
V dl E dl
  
  E V
 
Electric field is the gradient of a scalar potential.

9. Electric Potential at an arbitrary point
•Electric potential at a point is given as the work done
in moving the unit test charge (q0) from infinity (where
potential is taken as zero) to that point.
• Electric potential at any point P is
Note that Vp represents the potential difference dV
between the point P and a point at infinity.
S.I. unit J/C defined as a volt (V) and 1 V/m = 1 N/C
.
p
p
V E ds

 
0
p
W
V
q


10. Potential Difference in Uniform E field
• Electric field lines always point in the direction of
decreasing electric potential.
Example: Uniform field along –y
axis (E parallel to dl)
.
B B
B A
A A
V V V E dl Edl
      
 
B
A
V Edl Ed
    

• When the electric field E is directed downward,
point B is at a lower electric potential than point A. A
positive test charge that moves from point A to point
B loses electric potential energy.

11. Potential Diff. in Uniform E field
Charged particle moves from A to B in uniform E
field.

.
b
a
V E ds
 

12. Potential Diff. In Uniform E field
(Path independence)
Show that the potential difference between point A and
B by moving through path (1) and (2) are the same as
expected for a conservative force field.
By path (1), . cos
B
A
V E dl El 
    


13. path (2)
= 0 since E is  to dl
. cos
C
A
V E dl Eh El 
      

. .
C B
A C
V E dl E dl
    
 

14. Equipotential Surfaces (Contours)
VC = VB ( same potential)
In fact, points along this
line has the same
potential. We have an
equipotential line.
. 0
B
C
V E dl
   

No work is done in moving a test charge between any two
points on an equipotential surface.
The equipotential surfaces of a uniform electric field
consist of a family of planes that are all perpendicular to the
field.

15. Equipotential Surface
Equipotential Surfaces (dashed blue lines) and electric field lines
(orange lines) for (a) a uniform electric field produced by infinite
sheet of charge, (b) a point charge, and (c) an electric dipole. In all
cases, the equipotential surfaces are perpendicular to the electric
field lines at every point.

16. 16
Electrostatic Potential of a Point
Charge at the Origin
Q
P
r
 
 
 
'
2
'
0
2
'
0 0
4
4 4
r r
r
Q
V r E dl dr
r
Q dr Q
r
r

 
 

     
 
 


17. 17
Electrostatic Potential Resulting from
Multiple Point Charges
Q1
P(R,,)
r 1
R
1
r
O
Q2
2
r
 
1 0
4
n
k
k k
Q
V r
R


 
2
R

18. 18
Electrostatic Potential Resulting from
Continuous Charge Distributions
 
 
 
0
0
0
1
4
1
4
1
4
L
S
V
dl
V r
R
ds
V r
R
dv
V r
R












 line charge
 surface charge
 volume charge

19. 19
Charge Dipole
• An electric charge dipole consists of a pair of equal
and opposite point charges separated by a small
distance (i.e., much smaller than the distance at
which we observe the resulting field).
d
+Q -Q

20. Dipole Moment
• Dipole moment p is a measure of the strength of the
dipole and has its direction.
p Qd

+Q
-Q
d
p is in the direction from the
negative point charge to the
positive point charge

21. 21
Electrostatic Potential Due to Charge Dipole
observation
point
d/2
+Q
-Q
z
d/2

P
ˆz
p a Qd

R
R
r

22. 22
d/2
d/2



cos
)
2
/
(
cos
)
2
/
(
2
2
2
2
rd
d
r
R
rd
d
r
R








R
r
P
   
0 0
,
4 4
Q Q
V r V r
R R

 
 
  
R

23. • first order approximation from geometry:


cos
2
cos
2
d
r
R
d
r
R






d/2
d/2

lines approximately
parallel
R
R
r

24. 24
• Taylor series approximation:








































cos
2
1
1
1
cos
2
1
1
cos
2
1
1
cos
2
1
1
1
r
d
r
R
r
d
r
r
d
r
d
r
R
  1
,
1
1
:
Recall



 x
nx
x
n
 
2
0
0
4
cos
2
cos
1
2
cos
1
4
,
r
Qd
r
d
r
d
r
Q
r
V






























25. 25
• In terms of the dipole moment:
2
0
ˆ
4
1
r
a
p
V
r




26. Electric Potential Energy
of a System of Point Charges
1
0
1
4
q
V
r


2 ( )
W q V r

1 3 2 3
1 2
12 13 23
0 12 13 23
1
( )
4
q q q q
q q
W W W W
r r r

     
2
W F r q E r
   
q1
q2
2
b b
a a
W F dl q E dl
    
 
2[ ( ) ( )]
W q V b V a
 
2[ ( ) ( )]
W q V r V
  
and we know

27. The Energy of a Continuous Charge Distribution
For a volume charge density p,
1
2
W Vd
 
 
0 .E
 
 
Using Gauss’s Law:
0
( . )
2
W E Vd


 

So:
0
.( ) .
2
W E V d VE da


 
   
 
 
By doing integration by part:
and so,
V E
   2
0
.
2 v s
W E d VE da


 
 
 
 
 
If we take integral over all space:
2
0
2 allspace
W E d


 

28. Poisson’s and Laplace’s Equation
E V
 
The fundamental equations for E:
0
. ;
E


  0
E
 
2
. .( )
E V V
     
Gauss’s law then says that:
2
0
V


   This is known as
Poisson’s equation.
In regions where there is no charge: 0
 
Poisson’s equation reduces to Laplace’s equation.
2
0
V
 
This is known as
Laplace’s equation.