Electric and magnetic energy at axion haloscopes

We review a recent letter published in Phys. Rev. Lett. $\textbf{116}$, 161804 (2016) of which the main argument is that the mode dependent magnetic form factors at axion haloscopes depend on the position of the cavity inside the solenoid while the corresponding electric form factors do not. We, however, find no such dependence, which is also equivalent to the statement that the electric and corresponding magnetic energy stored in the cavity modes are the same regardless of the position of the cavity inside the solenoid. Furthermore, we extend the statement to the cases satisfying $\vec{\nabla}\times\vec{B}_{\rm external}=0$, where $\vec{B}_{\rm external}$ is a static magnetic field provided by a magnet at an axion haloscope. Two typical magnets, solenoidal and toroidal, satisfy $\vec{\nabla}\times\vec{B}_{\rm external}=0$, thus the electric and corresponding magnetic energy stored in the cavity modes are always the same in both cases. The energy, however, is independent of the position of the cavity in axion haloscopes with a solenoid, but depends on those with a toroidal magnet.