  The dipole approximation is employed to describe interactions between atoms
and radiation. It essentially consists of neglecting the spatial variation of
the external field over the atom. Heuristically, this is justified by arguing
that the wavelength is considerably larger than the atomic length scale, which
holds under usual experimental conditions. We prove the dipole approximation in
the limit of infinite wavelengths compared to the atomic length scale and
estimate the rate of convergence. Our results include N-body Coulomb potentials
and experimentally relevant electromagnetic fields such as plane waves and
laser pulses.
