A neutral dark matter particle may possess an electric dipole moment (EDM) or a magnetic dipole moment (MDM), so that its scattering with nuclei is governed by electromagnetic interactions. If the moments are associated with relevant operators of dimension-5, they may be detectable in direct search experiments. We calculate complete expressions of the scattering cross sections and the recoil energy spectra for dark matter with these attributes. We also provide useful formulae pertinent to dark

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... atter that interacts via an electric charge form factor (CFF) which is related to the charge radius defined by an effective dimension-6 operator. We show that a 7 GeV dark matter particle with an EDM, MDM or CFF easily reproduces the CoGeNT excess while remaining consistent with null searches.