potential, electric field and torque of electric dipole

1. Course: Electromagnetic Theory
paper code: EI 503
Course Coordinator: Arpan Deyasi
Department of Electronics and Communication Engineering
RCC Institute of Information Technology
Kolkata, India
Topic: Electrostatics – Electric Dipole
03-11-2021 Arpan Deyasi, EM Theory 1
Arpan Deyasi
Electromagnetic
Theory

2. Electric Dipole
03-11-2021 Arpan Deyasi, EM Theory 2
l
+q
-q
If two equal and opposite point charges are separated by infinitesimal
small distance, then the system is termed as electric dipole
Dipole moment is defined as the product of magnitude of
any one charge with the separating distance between them
p ql
=
Arpan Deyasi
Electromagnetic
Theory

3. 03-11-2021 Arpan Deyasi, EM Theory 3
Electrostatic Potential due to Dipole
A
P
B
A’
B’
O
r
d θ
‘d’ is the dipole length
‘r’ is the distance between point P to the center of the dipole O
r d
 
' '
AP A P OP OA
= = +
cos( )
OP OA 
= +
0.5 cos( )
r d 
= +
According to the figure -q +q
Arpan Deyasi
Electromagnetic
Theory

4. 03-11-2021 Arpan Deyasi, EM Theory 4
Electrostatic Potential due to Dipole
A
P
B
A’
B’
O
r
d θ
' '
BP B P OP OB
= = −
cos( )
OP OB 
= −
0.5 cos( )
r d 
= −
-q +q
Arpan Deyasi
Electromagnetic
Theory

5. 03-11-2021 Arpan Deyasi, EM Theory 5
Electrostatic Potential due to Dipole
A
P
B
A’
B’
O
r
d θ
Potential at P due to ‘-q’ is
1
.
4
AP
q
AP


= −
 
1
.
4 0.5 cos( )
AP
q
r d

 
= −
+
-q +q
Arpan Deyasi
Electromagnetic
Theory

6. 03-11-2021 Arpan Deyasi, EM Theory 6
Electrostatic Potential due to Dipole
A
P
B
A’
B’
O
r
d θ
Potential at P due to ‘+q’ is
1
.
4
BP
q
BP


=
 
1
.
4 0.5 cos( )
BP
q
r d

 
=
−
-q +q
Arpan Deyasi
Electromagnetic
Theory

7. 03-11-2021 Arpan Deyasi, EM Theory 7
Electrostatic Potential due to Dipole
A
P
B
A’
B’
O
r
d θ
Potential at P due to dipole is
P AP BP
  
= +
 
 
1
0.5 cos( )
1
4
0.5 cos( )
P
r d
q
r d




 
 
−
 
=
 
−
 
 
+
 
-q +q
Arpan Deyasi
Electromagnetic
Theory

8. 03-11-2021 Arpan Deyasi, EM Theory 8
Electrostatic Potential due to Dipole
A
P
B
A’
B’
O
r
d θ
2
cos( )
4
P
q d
r



 
=  
 
2
cos( )
4
P
p
r



=
2
2
cos( )
4
cos( )
4
P
q d
d
r




 
 
=  
 
−
 
 
-q +q
Arpan Deyasi
Electromagnetic
Theory

9. 03-11-2021 Arpan Deyasi, EM Theory 9
Problem 1
An electric dipole of dipole moment 100 pC/m is located at the origin.
Find the potential at (0,0,10).
k̂
3
.
4
P
p r
r


=
Soln
ˆ
10
r k
=
12
9
3
ˆ ˆ
100 10 .10
9 10
10
P
k k
V

−

=  
9
P mV
 =
Arpan Deyasi
Electromagnetic
Theory

10. 03-11-2021 Arpan Deyasi, EM Theory 10
Electric Field due to Dipole (Cartesian Coordinate)
2 2
ˆ
cos( ) .
4 4
P
p p r
r r


 
= =
E 
= − 2
1
4
p
r

 
= −  
 
3
1 .
4
p r
r

 
= −  
 
Arpan Deyasi
Electromagnetic
Theory

11. 03-11-2021 Arpan Deyasi, EM Theory 11
( )
3 3
1
( . ) ( . ) ( )
4
r p r p r r

− −
= −  + 
( )
3 5
1
( . )( 3 )
4
r p p r r r

− −
= − + −
( )
3 5
1
3( . )( )
4
r p p r r r

− −
= − −
Electric Field due to Dipole (Cartesian Coordinate)
Arpan Deyasi
Electromagnetic
Theory

12. 03-11-2021 Arpan Deyasi, EM Theory 12
Electric Field due to Dipole (Cartesian Coordinate)
5 3
1 3( . )
4
p r r p
r r

 
= −
 
 
( )
3
1
ˆ ˆ
3( . )
4
E p r r p
r

= −
Arpan Deyasi
Electromagnetic
Theory

13. 03-11-2021 Arpan Deyasi, EM Theory 13
Problem 2
An electric dipole of dipole moment 100 pC/m is located at the origin.
Find the electric field at (0,0,10).
k̂
Soln
ˆ
10
r k
=
( )
3
1
ˆ ˆ
3( . )
4
E p r r p
r

= −
9
12 12
9 10 ˆ ˆ ˆ ˆ
3(100 10 . ) ((100 10 ) /
1000
E k k k k V m
− −
  
=  − 
 
ˆ
1.8 /
E k mV m
=
Arpan Deyasi
Electromagnetic
Theory

14. 03-11-2021 Arpan Deyasi, EM Theory 14
Electric Field due to Dipole (Spherical Coordinate)
E = −
1 1
ˆ ˆ
ˆ
sin( )
r
r r r
 
  
 
  
= − + +
 
  
 
1 ˆ
r̂
r r r

 
 
= − +
 
 
 
Arpan Deyasi
Electromagnetic
Theory

15. 03-11-2021 Arpan Deyasi, EM Theory 15
Electric Field due to Dipole (Spherical Coordinate)
2 2
cos( ) 1 cos( )
ˆ
ˆ
4 4
p p
r
r r r r r
 

 
   
   
= − +
   
 
 
   
 
3 2
2 cos( ) 1 sin( )
ˆ
ˆ
4 4
p p
r
r r r
 

 
 − − 
   
= − +
   
 
   
 
3 3
2 cos( ) sin( )
ˆ
ˆ
4 4
p p
E r
r r
 

 
 
= +
 
 
Arpan Deyasi
Electromagnetic
Theory

16. 03-11-2021 Arpan Deyasi, EM Theory 16
Electric Field due to Dipole (Spherical Coordinate)
( )
1/2
2 2 2 2
3
1
4 cos ( ) sin ( )
4
E p p
r
 

= +
( )
1/2
2 2
3
3 cos ( ) 1
4
p
E p
r


= +
Arpan Deyasi
Electromagnetic
Theory

17. 03-11-2021 Arpan Deyasi, EM Theory 17
Torque due to Dipole
A
P
B
A’
B’
O
r
d θ
1
F qE
=
2
F qE
= −
Force on charge ‘+q’
Force on charge ‘-q’ -q
+q
Arpan Deyasi
Electromagnetic
Theory

18. 03-11-2021 Arpan Deyasi, EM Theory 18
Torque due to Dipole
A
P
B
A’
B’
O
r
d θ
Torque of F1 about O is
1 1
OB F

→
= 
1 OB qE

→
= 
1 q OB E

→
 
= 
 
 
-q +q
Arpan Deyasi
Electromagnetic
Theory

19. 03-11-2021 Arpan Deyasi, EM Theory 19
Torque due to Dipole
A
P
B
A’
B’
O
r
d θ
Torque of F2 about O is
2 2
OA F

→
= 
2 ( )
OA qE

→
=  −
2 q OA E

→
 
= − 
 
 
2 q AO E

→
 
= 
 
 
-q
+q
Arpan Deyasi
Electromagnetic
Theory

20. 03-11-2021 Arpan Deyasi, EM Theory 20
Torque due to Dipole
A
P
B
A’
B’
O
r
d θ
Net torque
1 2
  
= +
q OB E q AO E

→ →
   
=  + 
   
   
q AB E

→
 
= 
 
 
-q
+q
Arpan Deyasi
Electromagnetic
Theory

21. 03-11-2021 Arpan Deyasi, EM Theory 21
Torque due to Dipole
A
P
B
A’
B’
O
r
d θ
q d E
  
= 
 
p E
 = 
-q
+q
Arpan Deyasi
Electromagnetic
Theory