Special Cases in Electrostatics
High-Yield Summary
- Equipotential lines connect points of equal electric potential and are always perpendicular to electric field lines.
- Electric dipole: two equal, opposite charges separated by distance d. Dipole moment p = qd, pointing from − to + charge.
- Far-field dipole potential: V = kqd·cos θ/r² — falls off as 1/r², faster than a single charge's 1/r.
- On the dipole's perpendicular bisector, V = 0 (cos 90° = 0), but the electric field is not zero: E = kp/r³, pointing opposite p.
- An external field exerts zero net force but a net torque on a dipole: τ = pE sin θ — max at θ = 90°, zero once aligned (θ = 0°).
Dipole Moment, Potential, Field, and Torque
p = qd | V = kqd·cos θ/r² | E = kp/r³ | τ = pE sin θ
- p = Dipole moment (vector, − to + charge)
- q = Magnitude of each charge
- d = Separation between the two charges
- θ = Angle between position vector (or field) and dipole axis
- r = Distance from the dipole's center
- τ = Torque on the dipole in an external field E
Equipotential Line Patterns
| System | Equipotential Line Shape |
|---|---|
| Constant field (parallel plates) | Parallel lines, evenly spaced |
| Point charge | Concentric circles, spacing increases with distance |
| Dipole | Mirror-symmetric; perpendicular bisector = zero potential |
Must Know
- Electric field lines and equipotential lines are always perpendicular — this holds in every configuration.
- Zero potential ≠ zero field: the dipole's perpendicular bisector has V = 0 but a nonzero E field (pointing opposite p).
- Dipole torque is maximal at θ = 90° (sin 90° = 1) and zero once the dipole aligns with the field (θ = 0°, rotational equilibrium).
Common MCAT Trap
- Don't assume zero potential means zero field, or vice versa — the dipole's perpendicular bisector is the textbook counterexample.
- A dipole in a uniform external field feels zero net force (translational equilibrium) but can still feel a nonzero torque — force and torque aren't the same question.
Quick Recall
Why is the electric potential zero everywhere on a dipole's perpendicular bisector?
At what dipole orientation is torque in an external field at a maximum?