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 the 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

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. Electric flux Density extra topic (1)𝐷
𝐷 =𝜀 𝐸

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

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

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