Which scaling describes the far-field axial electric field of a point dipole with moment p at distance r?

Which scaling describes the far-field axial electric field of a point dipole with moment p at distance r?

• A. Proportional to 1/r^3
• B. Proportional to 1/r^2
• C. Proportional to 1/r
• D. Independent of r

Answer: (B)

The far-field of a small electric dipole falls off with distance as 1/r^3. In electrostatics, the dipole potential is V = (1/4πε0) p cosθ / r^2. The electric field is the negative gradient of the potential, which brings in an extra 1/r factor, giving E ∝ p / r^3. Along the axis where θ = 0, this becomes a magnitude E = (1/4πε0) (2p) / r^3, pointing along the dipole moment. So the axial far-field scales as 1/r^3. The 1/r^2 behavior would correspond to a monopole-like field from a single charge, not a dipole. (If the dipole were oscillating and you looked at the radiated field, that field falls as 1/r, but it is transverse and has a different angular dependence.)

The far-field of a small electric dipole falls off with distance as 1/r^3. In electrostatics, the dipole potential is V = (1/4πε0) p cosθ / r^2. The electric field is the negative gradient of the potential, which brings in an extra 1/r factor, giving E ∝ p / r^3. Along the axis where θ = 0, this becomes a magnitude E = (1/4πε0) (2p) / r^3, pointing along the dipole moment. So the axial far-field scales as 1/r^3. The 1/r^2 behavior would correspond to a monopole-like field from a single charge, not a dipole. (If the dipole were oscillating and you looked at the radiated field, that field falls as 1/r, but it is transverse and has a different angular dependence.)