Electric dipole moment

The electric dipole moment of a given charge distribution is a measure of the separation between the positive and negative charges in the system. The dipole moment is the first term in a multipole expansion. Electric dipole moment is measured in Coulomb-meters (C m) in the SI system. Electric dipole moment is a vector quantity. Given two point charges of equal magnitude q and separation d (the displacement vector pointing towards the positive charge), the electric dipole moment is equal to

$\mathbf{p} = q\mathbf{d}$

More generally, for a distribution of charge ρ, the electric dipole is equal to

$\mathbf{p}(\mathbf{r}) = \iiint\limits_{V} \rho(\mathbf{r})\, (\mathbf{r}-\mathbf{r}_0) \ d V,$

where r is the position vector and r₀ is any given point. For a collection of point charges, the dipole moment is equal to

$\mathbf{p}(\mathbf{r}) = \sum_{i=1}^N \, q_i \, (\mathbf{r}_i - \mathbf{r}_0 )$

An electric dipole placed in an electric field will experience a torque equal to

$\boldsymbol{\tau} = \bold{p} \times \bold{E}$

and will have potential energy

$U = \bold{p} \cdot \bold{E}$

The electric potential at a point at a distance r from a dipole will be

$V(\mathbf{r}) = \frac{1}{4 \pi \epsilon_0} \frac{\hat{\mathbf{r}} \cdot \mathbf{p}}{\mathbf{r}^2}$