It is shown that dark matter axions cause an oscillating electric current to flow along magnetic field lines. The oscillating current induced in a strong magnetic field B0 produces a small magnetic field Ba. We propose to amplify and detect Ba using a cooled LC circuit and a very sensitive magnetometer. This appears to be a suitable approach to searching for axion dark matter in the 10 −7 to 10 −9 eV mass range.PACS numbers: 95.35.+d Shortly after the Standard Model of elementary particles was established, the axion was postulated [1] to explain why the strong interactions conserve the discrete symmetries P and CP. Further motivation for the existence of such a particle came from the realization that cold axions are abundantly produced during the QCD phase transition in the early universe and that they may constitute the dark matter [2]. Moreover, it has been claimed recently that axions are the dark matter, at least in part, [3][4][5] because axions form a Bose-Einstein condensate and this property explains the occurrence of caustic rings in galactic halos. The evidence for caustic rings with the properties predicted by axion BEC is summarized in ref. [6]. In supersymmetric extensions of the Standard Model, the dark matter may be a mixture of axions and supersymmetric dark matter candidates [7].Axion properties depend mainly on a single parameter f a , called the axion decay constant. In particular the axion mass (h = c = 1) m a ≃ 6 · 10 −6 eV 10 12 GeV f aand its coupling to two photonswith g = g γ α πfa . Here a(x) is the axion field, E(x) and B(x) the electric and magnetic fields, α the fine structure constant, and g γ a model-dependent coefficient of order one. g γ ≃ −0.97 in the KSVZ model [8] whereas g γ ≃ 0.36 in the DFSZ model [9]. Cold axions are produced during the QCD phase transition, when the axion mass turns on and the axion field begins to oscillate in response. The resulting axion cosmological energy density is proportional to (f a ) 7 6 and, in the simplest case, reaches the critical energy density for closing the universe when f a is of order 10 12 GeV [2]. This suggests that the most promising mass range in axion searches is near 10 −5 eV. This happens to be approximately where the cavity axion detection technique [10] is most feasible and where the ADMX experiment [11] is searching at present.However, it is desirable to search for axion dark matter over the widest possible mass range because the axion mass is, in reality, poorly constrained. In particular, it has been argued that if there is no inflation after the Peccei-Quinn phase transition, the contribution of axion strings to the axion cosmological energy density [12] implies that the preferred mass for dark matter axions is in the 10 −3 to 10 −4 eV mass range [13]. On the other hand, if there is inflation after the Peccei-Quinn phase transition, the axion field gets homogenized during inflation and the homogenized field may accidentally lie close to the minimum of its effective potential [14], in which case axions may be the dark ma...