Far from a static electric dipole, what is the electric field on the dipole axis (θ = 0) at large distance r?

Far from a static electric dipole, what is the electric field on the dipole axis (θ = 0) at large distance r?

• A. (1/(4π ε0)) (p) / r^2 perpendicular to the dipole.
• B. (1/(4π ε0)) (3 p) / r^3 along the dipole direction.
• C. (1/(4π ε0)) (2 p) / r^3 along the dipole direction.
• D. (1/(4π ε0)) (p cos θ) / r^2 along the axis.

Answer: (C)

At large distances, the electric field from a static dipole falls off as 1/r^3 and points along the dipole moment when you’re on the axis (θ = 0). Using the standard far-field expression E(r) = (1/4πε0 r^3)[3(p·r̂) r̂ − p], and taking r̂ along the dipole axis so that p·r̂ = p and r̂ = p̂, you get E = (1/4πε0 r^3)(3p p̂ − p p̂) = (1/4πε0)(2p/r^3) p̂. So the field on the axis is along the dipole direction with magnitude 2p/r^3. This matches the option that gives (1/4πε0) (2 p) / r^3 along the dipole direction. The other forms either have the wrong distance dependence (1/r^2) or direction (perpendicular to the dipole) or a incorrect numerical factor.

At large distances, the electric field from a static dipole falls off as 1/r^3 and points along the dipole moment when you’re on the axis (θ = 0). Using the standard far-field expression E(r) = (1/4πε0 r^3)[3(p·r̂) r̂ − p], and taking r̂ along the dipole axis so that p·r̂ = p and r̂ = p̂, you get E = (1/4πε0 r^3)(3p p̂ − p p̂) = (1/4πε0)(2p/r^3) p̂. So the field on the axis is along the dipole direction with magnitude 2p/r^3.

This matches the option that gives (1/4πε0) (2 p) / r^3 along the dipole direction. The other forms either have the wrong distance dependence (1/r^2) or direction (perpendicular to the dipole) or a incorrect numerical factor.