<P> As a first example relating dipole moment to polarization, consider a medium made up of a continuous charge density ρ (r) and a continuous dipole moment distribution p (r). The potential at a position r is: </P> <Dl> <Dd> φ (r) = 1 4 π ε 0 ∫ ρ (r 0) r − r 0 d 3 r 0 + 1 4 π ε 0 ∫ p (r 0) ⋅ (r − r 0) r − r 0 3 d 3 r 0, (\ displaystyle \ phi ((\ mathbf (r))) = (\ frac (1) (4 \ pi \ varepsilon _ (0))) \ int (\ frac (\ rho \ left ((\ mathbf (r)) _ (0) \ right)) (\ left (\ mathbf (r)) - (\ mathbf (r)) _ (0) \ right)) d ^ (3) (\ mathbf (r)) _ (0) \ + (\ frac (1) (4 \ pi \ varepsilon _ (0))) \ int (\ frac ((\ mathbf (p)) \ left ((\ mathbf (r)) _ (0) \ right) \ cdot \ left ((\ mathbf (r)) - (\ mathbf (r)) _ (0) \ right)) ((\ mathbf (r)) - (\ mathbf (r)) _ (0) ^ (3))) d ^ (3) (\ mathbf (r)) _ (0),) </Dd> </Dl> <Dd> φ (r) = 1 4 π ε 0 ∫ ρ (r 0) r − r 0 d 3 r 0 + 1 4 π ε 0 ∫ p (r 0) ⋅ (r − r 0) r − r 0 3 d 3 r 0, (\ displaystyle \ phi ((\ mathbf (r))) = (\ frac (1) (4 \ pi \ varepsilon _ (0))) \ int (\ frac (\ rho \ left ((\ mathbf (r)) _ (0) \ right)) (\ left (\ mathbf (r)) - (\ mathbf (r)) _ (0) \ right)) d ^ (3) (\ mathbf (r)) _ (0) \ + (\ frac (1) (4 \ pi \ varepsilon _ (0))) \ int (\ frac ((\ mathbf (p)) \ left ((\ mathbf (r)) _ (0) \ right) \ cdot \ left ((\ mathbf (r)) - (\ mathbf (r)) _ (0) \ right)) ((\ mathbf (r)) - (\ mathbf (r)) _ (0) ^ (3))) d ^ (3) (\ mathbf (r)) _ (0),) </Dd> <P> where ρ (r) is the unpaired charge density, and p (r) is the dipole moment density . Using an identity: </P>

An short electric dipole having two point charges