  A number of proposals have been put forward for detecting axion dark matter
(DM) with grand unification scale decay constants that rely on the conversion
of coherent DM axions to oscillating magnetic fields in the presence of static,
laboratory magnetic fields. Crucially, such experiments $\unicode{x2013}$
including ABRACADABRA $\unicode{x2013}$ have to-date worked in the limit that
the axion Compton wavelength is larger than the size of the experiment, which
allows one to take a magnetoquasistatic (MQS) approach to modeling the axion
signal. We use finite element methods to solve the coupled
axion-electromagnetism equations of motion without assuming the MQS
approximation. We show that the MQS approximation becomes a poor approximation
at frequencies two orders of magnitude lower than the naive MQS limit.
Radiation losses diminish the quality factor of an otherwise high-$Q$ resonant
readout circuit, though this may be mitigated through shielding and minimizing
lossy materials. Additionally, self-resonances associated with the detector
geometry change the reactive properties of the pickup system, leading to two
generic features beyond MQS: there are frequencies that require an inductive
rather than capacitive tuning to maintain resonance, and the detector itself
becomes a multi-pole resonator at high frequencies. Accounting for these
features, competitive sensitivity to the axion-photon coupling may be extended
well beyond the naive MQS limit.
