We show that the magnetic component of the photon field produced by dark matter axions via the two-photon coupling mechanism in a Sikivie Haloscope is an important parameter passed over in previous analysis and experiments. The interaction of the produced photons will be resonantly enhanced as long as they couple to the electric or magnetic mode structure of the Haloscope cavity. For typical Haloscope experiments the electric and magnetic coupling is the same and implicitly assumed in past

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... tivity calculations. However, for future planned searches such as those at high frequency, which synchronize multiple cavities, the sensitivity will be altered due to different magnetic and electric couplings. We define the complete electromagnetic form factor and discuss its implications for current and future high and low mass axion searches, including some effects which have been overlooked, due to the assumption that the two couplings are the same.