A water molecule (a permanent electric dipole with moment p⃗) is placed in a perfectly uniform external electric field E⃗, with p⃗ initially perpendicular to E⃗. What happens to the molecule?
AIt accelerates in the direction of E⃗, because the positive end is pulled toward lower potential
BIt experiences a torque that tends to align p⃗ with E⃗, but no net translational force
CIt experiences a net force away from regions of high field strength
DNothing happens, because the equal and opposite charges produce forces that cancel in all respects
In a uniform field, the force on the +q end is qE⃗ and the force on the −q end is −qE⃗. These are equal in magnitude and opposite in direction, so they sum to zero net force — the dipole does not translate. However, the forces act at different locations (separated by d), creating a torque τ⃗ = p⃗ × E⃗ that rotates the dipole toward alignment with E⃗. The common misconception is that a dipole accelerates toward one plate of a capacitor in a uniform field — it does not. Net forces on dipoles require non-uniform fields.
Question 2 Multiple Choice
A dipole with moment p⃗ is placed in an external electric field E⃗. In which orientation does it have the lowest potential energy (most stable equilibrium)?
Ap⃗ perpendicular to E⃗, where U = 0
Bp⃗ antiparallel to E⃗, where U = +pE (maximum energy)
Cp⃗ parallel to E⃗, where U = −pE (minimum energy)
DThe potential energy is the same in all orientations because the net force is always zero
The potential energy is U = −p⃗·E⃗ = −pE cos(θ). This is minimized when θ = 0 (p⃗ parallel to E⃗), giving U = −pE. Physically, alignment with the field is stable because any small perturbation creates a restoring torque back toward alignment. The antiparallel configuration (U = +pE) is an unstable equilibrium — any perturbation leads to rotation toward alignment, not back toward antiparallel. The perpendicular case (U = 0) is the reference point, not a stability criterion.
Question 3 True / False
The electric potential of a dipole falls off as 1/r² at large distances, more rapidly than the 1/r potential of a point charge.
TTrue
FFalse
Answer: True
Correct. A point charge produces V = kq/r (falls as 1/r). A dipole has equal and opposite charges that nearly cancel at large distance; the small residual goes as V = (1/4πε₀)(p cos θ)/r², which falls as 1/r². This faster falloff is the signature of the dipole: its net charge is zero, so the monopole term vanishes, and the leading contribution is the dipole term at 1/r². Higher-order charge distributions (quadrupoles, etc.) fall off even more steeply.
Question 4 True / False
A dipole placed in a non-uniform electric field experiences no net translational force — mainly a torque.
TTrue
FFalse
Answer: False
A net translational force on a dipole requires a non-uniform field. In a uniform field, the forces on +q and −q are equal and opposite, producing zero net force (but a torque). In a non-uniform field, the field strength at the +q position differs from the strength at the −q position, so the magnitudes of the two forces are unequal — the net force is non-zero. This is why polar molecules are attracted toward regions of stronger field in an inhomogeneous setup, and it underlies the physics of dielectrophoresis.
Question 5 Short Answer
Explain why a dipole in a uniform electric field experiences a torque but not a net translational force. Use the forces on each individual charge in your explanation.
Think about your answer, then reveal below.
Model answer: In a uniform field E⃗, every point in space has the same field vector. The force on the +q charge is F₊ = +qE⃗ (in the direction of E⃗), and the force on the −q charge is F₋ = −qE⃗ (opposite to E⃗). Since these forces are equal in magnitude but opposite in direction, their vector sum is zero — no net translational force. However, they act at different positions separated by the dipole distance d, so they form a couple: two equal and opposite forces with different lines of action. A couple produces a pure torque τ = qEd sin θ = pE sin θ, tending to rotate the dipole until p⃗ aligns with E⃗.
The key is that 'equal and opposite forces' cancels net force but not torque. Torque depends on both the force magnitude and the perpendicular distance between the force lines of action (the moment arm). In a non-uniform field, the force magnitudes on +q and −q differ because the field is different at each location, breaking the cancellation and producing a net translational force in addition to the torque.