Along the axis of a point dipole with moment p at distance r, what is the leading-order expression for the far-field electric field?

Along the axis of a point dipole with moment p at distance r, what is the leading-order expression for the far-field electric field?

• A. E ≈ (1/4π ε0) (2 p) / r^3 along the axis
• B. E ≈ (1/4π ε0) p / r^2
• C. E ≈ (1/4π ε0) p / r^3
• D. E ≈ (1/4π ε0) (2 p) / r^2

Answer: (A)

The leading far-field of a point dipole is dipolar in nature and scales as 1/r^3. The general far-field expression is E(r) = (1/4πϵ0 r^3)[3(p·r̂) r̂ − p]. Along the axis of the dipole, r̂ is aligned with the dipole moment, so p·r̂ = p, and the bracket becomes 3p r̂ − p = 2p r̂. This gives E ≈ (1/4πϵ0) (2p)/r^3 in the direction of the axis. The factor of 2 arises from the vector subtraction along that axis. The 1/r^3 dependence is why the other options, which propose 1/r^2 or missing the factor of 2, are not correct for the far field.

The leading far-field of a point dipole is dipolar in nature and scales as 1/r^3. The general far-field expression is E(r) = (1/4πϵ0 r^3)[3(p·r̂) r̂ − p]. Along the axis of the dipole, r̂ is aligned with the dipole moment, so p·r̂ = p, and the bracket becomes 3p r̂ − p = 2p r̂. This gives E ≈ (1/4πϵ0) (2p)/r^3 in the direction of the axis. The factor of 2 arises from the vector subtraction along that axis. The 1/r^3 dependence is why the other options, which propose 1/r^2 or missing the factor of 2, are not correct for the far field.