Resonant frequencies in an electromagnetic eccentric spherical cavity

The interior boundary-value electromagnetic (vector) problem in the region between two perfectly conducting spheres of radii ${R_1}$, ${R_2}$ and distance $d$ between their centers is considered. Surface singular integral equations are used to formulate the problem. Use of spherical vector wave functions and related addition theorems reduces the solution of the integral equations to the problem of solving an infinite set of linear equations. Their determinant is evaluated in powers of $kd = 2\pi d/\lambda$ to a few terms. It is then specialized to the axially symmetric case and set equal to zero. This yields closed-form expressions for the coefficients ${g_{ns}}$ in the resulting relations ${\omega _{ns}}\left ( {kd} \right ) = {\omega _{ns}}\left ( 0 \right )\left [ {1 + {g_{ns}}{{\left ( {kd} \right )}^2} \\ + \cdot \cdot \cdot } \right ]$ for the natural frequencies of the cavity. Numerical results, comparisons and possible generalizations are also included.