1. ELEC 3105 Basic EM and Power Engineering
Electric dipole
Force / torque / work on electric dipole Z

2. The Electric Dipole z P(x, z)
Consider electric field and potential produced by 2 charges (+q, -q) separated by a distance d.
P(x, z) +q d x -q

3. The Electric Dipole z P(x, z)
The dipole is represented by a vector of magnitude qd and pointing from –q to +q.
P(x, z) +q d x -q
Note: small letter p
Units
{p} dipole moment; Coulomb meter {Cm}

4. Potential at P(x, y) due to charge +q.
The Electric Dipole z
P(x, z)
Potential at P(x, y) due to charge +q. +q d x -q
Units
{p} dipole moment; Coulomb meter {Cm}

5. Potential at P(x,y) due to charge -q.
The Electric Dipole z
P(x,z)
Potential at P(x,y) due to charge -q. +q d x -q
Units
{p} dipole moment; Coulomb meter {Cm}

6. The Electric Dipole Suppose (x,y) >>> d P(x, z) z +q d -q x
Can be rewritten and the expression for the potential simplified.
Then:

7. Suppose (x, z) >>> d
The Electric Dipole
Suppose (x, z) >>> d
Binomial expansion

8. Suppose (x, z) >>> d
The Electric Dipole
Suppose (x, z) >>> d
Binomial expansion

9. Suppose (x,y) >>> d
The Electric Dipole
Suppose (x,y) >>> d z
P(x, z) +q d -q x

10. The Electric Dipole Suppose (x, z) >>> d
Potential produced by the dipole

11. Suppose (x, z) >>> d
The Electric Dipole
Suppose (x, z) >>> d z
P(x, z) +q d -q x

12. The Electric Dipole Suppose (x, z) >>> d P(x, z) z
Cartesian coordinates (x, z) +q d -q x

13. The Electric Dipole Suppose (x, z) >>> d P(r, , ) z
Spherical coordinates (r, , ) +q d -q x

14. The Electric Dipole z x V Drops off as 1/r2 for a dipole
P(r, , ) x
V Drops off as 1/r2 for a dipole
V Drops off as 1/r for a point charge

15. The Electric Dipole Now to compute the electric field expression
P(x, z))
Cartesian coordinates (x, z) z
P(r, , )
Spherical coordinates (r, , ) +q d -q x

16. Now to compute the electric field expression
The Electric Dipole
Now to compute the electric field expression

17. Spherical coordinates (r, , )
The Electric Dipole
Spherical coordinates (r, , )
No 𝜙 dependence

18. The Electric Dipole

19. Here consider dipole as a rigid charge distribution
Force on a dipole in a uniform electric field
Here consider dipole as a rigid charge distribution +q d
No net translation
since -q
Opposite
direction

20. Here consider dipole as a rigid charge distribution
Force on a dipole in a non-uniform electric field
Here consider dipole as a rigid charge distribution +q d
net translation since -q
And / Or

21. Force on a dipole in a non-uniform electric fieldy
Force on a dipole in a non-uniform electric field +q
Manipulate expression to get simple useful form d -q x

22. Force on a dipole in a non-uniform electric fieldy
Force on a dipole in a non-uniform electric field
After the manipulations end we get: +q d -q x
We will obtain this expression using a different technique.

23. Here consider dipole as a rigid charge distribution
Torque on a dipole
Here consider dipole as a rigid charge distribution +q
d/2
d/2 -q
The torque components + and  - act in the same rotational direction trying to rotate the dipole in the electric field.

24. Review of the concept of torque
Torque on a dipole
Torque:
Pivot
Moment arm length
Force
Angle between vectors r and F

25. For simplicity consider the dipole in a uniform electric field
Torque on a dipole
Act in same
direction +q d -q
Also valid for small dipoles in a non-uniform electric field.

26. Consider work dW required to rotate dipole through an angle d
Work on a dipole
By definition
When you have rotation
If we integrate over
some angle range then +q d -q

27. Work on a dipole
For  = 90 degrees
W = 0. Thus  = 90 degrees is reference orientation for the dipole. It corresponds to the zero of the systems potential energy as well. U=W

28. Work on a dipole
For  = 0 degrees
W = -pE. Thus  = 0 degrees is the minimum in energy and corresponds to having the dipole moment aligned with the electric field.

29. Work on a dipole
For  = 180 degrees
W = pE. Thus  = 180 degrees is the maximum in energy and corresponds to having the dipole moment anti-aligned with the electric field.

30. Force on a dipole +q d -q Work Force
After the manipulations end we get: +q d
We will obtain this expression using a different technique.
Work -q
Force
Recall principle of virtual work and force

31. Exam question: once upon a time
Stator dipole
+q -q +Q 2R 2r
Rotor dipole
E on +q
F on +q
 on +q
 on rotor -Q
(0,0)

32. Exam question: Once upon a time
D>>R 2R
e)  on dipole -Q

33. Polarization Atom Negative electron cloud Positive nucleus
No external electric field
With an external electric field
Charge polarization occurs in the presence of electric field

34. Polarization Negative electron cloud Positive nucleus
Each atom acquires a small dipole moment For low intensity electric fields the polarization is expected to be proportional to the field intensity:
With an external electric field
Atomic polarizability of the atom

35. Atomic Polarizability And Ionization Potential

36. Polarization With an external electric field
No external electric field
If the density of particles per cubic meter is N, the net polarization is:
has units of {C/m2}

37. Some molecules have built in dipole moments due to ionic bonds.
Polarization
Some molecules have built in dipole moments due to ionic bonds.
Negative region
O2- H+ 𝑝 𝑝
Positive region
Water

38. Polarization O2- Negative region 𝑝 𝑝 H+ Positive region
For low intensity applied fields this polarization of the material is again proportional to the field intensity so a more general expression for polarization is:
With the electric susceptibility of the material

39. Materials where is proportional to are called DIELECTRIC materials.
Polarization
For low intensity applied fields this polarization of the material is again proportional to the field intensity so a more general expression for polarization is:
Materials where is proportional to are called DIELECTRIC materials.

40. Dielectrics
Why is the dielectric constant in the medium different than the dielectric constant in vacuum?
The answer is contained in the nature of the material being placed in the electric field.

41. Dielectrics d
Consider the slab of material “dielectric” immersed in an external field
Endface Area A
The molecules inside will be polarized due to the presence of the electric field.

42. Dielectrics + -
dipole
Consider the slab of material “dielectric” immersed in an external field + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + + +
The molecules inside will be polarized due to the presence of the electric field. +
SLAB + + + + + + + + + + + + + + + +
Area A
Polarization induced surface charge density

43. Dielectrics - + For the single dipole, the dipole moment is: dipole
For the dipoles along one line between the endfaces, the dipole moment is: d

44. Dielectrics The total dipole moment of the slab is then .+
The total dipole moment of the slab is then
Q: Total charge on one face of the slab.

45. Dielectrics d
The polarization or dipole moment per unit volume of the slab is then. +
End face Area A

46. FLUX CAPACITOR MEETS THE FUSOR
Electric flux Density
𝐷 =𝜀 𝐸
FLUX CAPACITOR MEETS THE FUSOR

47. Dielectrics d Electric field is shown normal to the surface.
Endface Area A
The electric flux density vectors Do = Dd . We are treating only normal components here.

48. Dielectrics
These two charge sheets will produce an electric field directed from the positive sheet towards the negative sheet.
Polarization bound charge sheets of each endface.
Original slab in external electric field
In order to determine the magnitude of the electric field, a parallel plate capacitor analysis can be applied to this charge configuration.

49. Dielectrics
Induced electric field due to the polarization effect in the dielectric.
Through Gaussian analysis:
Polarization bound charge sheets of each end face.
Note: The external field Eo and the induced field Ei are in opposite directions.
If electrons where free in the dielectric, then the magnitudes of Eo and Ei would end up the same, their directions would be opposite and the net electric field inside the medium would be zero. Such a material is called a metal due to the free electrons.

50. Dielectrics d Endface Area A
Further manipulations of equations required to obtain desired result. Recall that desired result is: Why mediums have a different dielectric constant from that of vacuum?
Endface Area A

51. Dielectrics
Endface Area A d
Vector diagram
inside dielectric

52. Dielectrics
Endface Area A d
Since:
and
Then
Vector diagram

53. Dielectrics
Endface Area A d
And:
with
Vector diagram
Then

54. Dielectrics Endface Area A d
The dielectric constant can now be obtain ed using :
Vector diagram

55. Answer: Polarization and orientation of internal dipoles.
Dielectrics
With the electric susceptibility of the material.
Why is the dielectric constant in the medium different than the dielectric constant in vacuum?
Answer: Polarization and orientation of internal dipoles.

56. ELEC 3105 Basic EM and Power Engineering
Next slides: forget me not

57. Dielectric Materials

58. Polarization Field
P = electric flux density induced by E

59. Electric Breakdown
Electric Breakdown

60. Electric flux Density
𝐷 =𝜀 𝐸

61. From other definitions of flux we can obtain other useful expressions for electrostatics
Divergence theorem

62. Divergence theorem Integrands must be the same for all dV
Point function
Gauss’s law in differential form
Medium dependence

63. Divergence theorem Integrands must be the same for all dV
Point function
Gauss’s law in differential form
No dependence on the dielectric constant

64. Boundary conditions Normal Component of
ELECTROSTATICS
Boundary conditions Normal Component of
Gaussian surface on metal interface encloses a real net charge s.
Gaussian surface on dielectric interface encloses a bound surface charge sp , but also encloses the other half of the dipole as well. As a result Gaussian surface encloses no net surface charge.
Air Dielectric
Gaussian Surface

65. Top Ten
What is the definition of a shock absorber? A careless electrician.
Do you know how an electrician tells if he's working
with AC or DC power? If it's AC, his teeth chatter
when he grabs the conductors. If it's DC, they just
clamp together.
What did the light bulb say to the generator? I really get a charge out of you.
What Thomas Edison's mother might have said to her son: Of course I'm proud that you invented the electric light bulb. Now turn it off and get to bed.
Sign on the side of the electrician's van: Let Us Get Rid of Your Shorts.
Two atoms were walking down the street one day, when one of them exclaimed, 'Oh no - I've lost an electron!' 'Are you sure?' the other one asked. 'Yes,' replied the first one, 'I'm positive.'
Why are electricians always up to date? Because they are "current" specialists.
What would you call a power failure?  A current event.
Did you hear about the silly gardener? He planted a light bulb and thought he
would get a power plant.
How do you pick out a dead battery
from a pile of good ones? It's got no spark.