Electric dipole potential and field

Electric dipole

An electric dipole is a pair of equal and opposite point charges +Q and −Q separated by a small distance d. It is a source of an electric field that falls off faster (1/r³) than that of a single charge.

Potential at a far point P

Let the charges lie on the z-axis at z = ±d/2, with +Q above. P is at distance r from the centre, at angle θ from the dipole axis, and r ≫ d.

             P
            ╱│╲
        r₂ ╱ │ ╲ r₁         r₁ = distance from +Q
          ╱  │r ╲           r₂ = distance from −Q
        −Q───●───+Q
          ←──d──→

V = (Q/4πε₀) (1/r₁ − 1/r₂) = (Q/4πε₀) (r₂ − r₁)/(r₁ r₂)

For r ≫ d the lines from the charges to P are almost parallel, so

r₂ − r₁ ≈ d cosθ,      r₁ r₂ ≈ r²

V = Q d cosθ / (4πε₀ r²)

With the dipole moment p = Qd (magnitude Qd, directed from −Q to +Q), p · a_r = Qd cosθ, so

V = p · a_r / (4πε₀ r²)  =  p cosθ / (4πε₀ r²)

Electric field

Use E = −∇V in spherical coordinates:

E_r = −∂V/∂r = 2 p cosθ / (4πε₀ r³)
E_θ = −(1/r) ∂V/∂θ = p sinθ / (4πε₀ r³)
E_φ = −(1/(r sinθ)) ∂V/∂φ = 0

E = ( p / (4πε₀ r³) ) ( 2cosθ a_r + sinθ a_θ )

Special cases: on the axis (θ = 0), E = 2p / (4πε₀ r³) along a_r. On the equatorial plane (θ = 90°), E = p / (4πε₀ r³) along a_θ, i.e. opposite to p.

Dipole moment

The electric dipole moment p = Qd is a vector of magnitude charge × separation, directed from the negative to the positive charge. Unit: C·m. It measures the strength of the dipole. The potential and field depend only on the product Qd, not on Q and d separately. In an external field E, a dipole experiences a torque T = p × E.