1. Electric Dipole and Equipotential Surfaces
Electric field lines
Equipotential surfaces

2. The Electric Potential of a Dipolez P +q y -q x
Find: Potential V at point P anywhere.

3. Solution

4. Things to note
The potential for a point charge (monopole) varies as 1/r
and the electric field as 1/r2 The potential for two charges (dipole) varies as 1/r2
and the electric field as 1/r3 (see Assignment 3)
For higher order multi-poles, the potentials will vary as: /r3, 1/r4, 1/r5… and the corresponding electric fields at: /r4, 1/r5, 1/r6…

5. Fields and Equipotential Lines/Surfaces (extra)
Q: How can we obtain the lines that describe a field, or
the equipotential lines associated with it? A: we need one or the other to start with, and use some
algebra/calculus.

6. Example For the field intensity E=Ex i + Ey j, how would the flux
lines look like? How would the equipotential lines look like? - at any point, dy/dx = Ey/Ex for the field (flux lines) - at any point, the equipotentials are perpendicular, hence
they must have a slope:
dy/dx = -1/(slope of flux lines)

7. Example For the field intensity E=y i + x j, how would the flux
lines look like? What about the equipotential lines?

8. Solution

9. In General, for 3D in any coords:

10. E and V for a Dipole The equipotential lines are the dashed blue lines
The electric field lines are the brown lines
The equipotential lines are everywhere perpendicular to the field lines
These lines are a cross section of surfaces !