For a point dipole with moment p, the potential at distance r and angle θ between p and r̂ is what?

For a point dipole with moment p, the potential at distance r and angle θ between p and r̂ is what?

• A. (1/(4π ε0)) (p sin θ)/ r^2.
• B. (1/(4π ε0)) (p cos θ)/ r.
• C. (1/(4π ε0)) (p cos θ)/ r^3.
• D. (1/(4π ε0)) (p cos θ)/ r^2.

Answer: (D)

The potential from a point dipole depends on how far away you are and how well the observation direction lines up with the dipole moment. For a dipole moment p at the origin, the scalar potential at distance r and angle θ between p and the line to the point is V = (1/(4π ε0)) (p cos θ) / r^2. This comes from thinking of the dipole as two opposite charges separated by a small distance and taking the limit where the separation goes to zero while keeping p = qd fixed; the net charge term cancels and the leading term is proportional to p cos θ with a 1/r^2 falloff. The cos θ factor is the projection p · r̂, so the potential is largest along the dipole axis (θ = 0) and zero in the plane perpendicular to the dipole (θ = 90°). The sin θ dependence appears in the field components, not in the potential, and the 1/r^2 dependence, not 1/r or 1/r^3, is characteristic of a dipole’s potential. Therefore the correct expression is the one with p cos θ divided by r^2.

The potential from a point dipole depends on how far away you are and how well the observation direction lines up with the dipole moment. For a dipole moment p at the origin, the scalar potential at distance r and angle θ between p and the line to the point is V = (1/(4π ε0)) (p cos θ) / r^2. This comes from thinking of the dipole as two opposite charges separated by a small distance and taking the limit where the separation goes to zero while keeping p = qd fixed; the net charge term cancels and the leading term is proportional to p cos θ with a 1/r^2 falloff. The cos θ factor is the projection p · r̂, so the potential is largest along the dipole axis (θ = 0) and zero in the plane perpendicular to the dipole (θ = 90°). The sin θ dependence appears in the field components, not in the potential, and the 1/r^2 dependence, not 1/r or 1/r^3, is characteristic of a dipole’s potential. Therefore the correct expression is the one with p cos θ divided by r^2.