Electric and magnetic energy at axion haloscopes

We review the electro-magnetic energy at axion haloscopes and find that the electric and the corresponding magnetic energy stored in the cavity modes or, equivalently, the mode dependent electric and magnetic form factors are the same regardless of the position of the cavity inside the solenoid. Furthermore, we extend our argument to the cases satisfying $\overrightarrow{\nabla }\times {\overrightarrow{B}}_{\text{external}}=0$, where ${\overrightarrow{B}}_{\text{external}}$ is a static magnetic field provided by a magnet at an axion haloscope. Two typical magnets, solenoidal and toroidal, satisfy $\overrightarrow{\nabla }\times {\overrightarrow{B}}_{\text{external}}=0$; thus, the electric and the 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, and depends on those with a toroidal magnet.

Figure 1

Line (Dots) is ratios of $U_{a,m}$ ($C_B$) to $U_{a,m}^{\text{center}}$ ($C_B^{\text{center}}$) as a function of offset over the cavity radius, where $U_{a,m}^{\text{center}}$ ($C_B^{\text{center}}$) is the magnetic energy (magnetic form factor) when the cavity is located in the center of the solenoid and offset is the distance between the solenoid center and the cavity center. The results are calculated with the ${\mathrm{TM}}_{010}$ mode only.