  The intriguing superradiant amplification phenomenon allows an orbiting
scalar field to extract rotational energy from a spinning Kerr black hole.
Interestingly, the energy extraction rate can grow exponentially in time if the
black-hole-field system is placed inside a reflecting mirror which prevents the
field from radiating its energy to infinity. This composed
Kerr-black-hole-scalar-field-mirror system, first designed by Press and
Teukolsky, has attracted the attention of physicists over the last four
decades. Previous numerical studies of this spinning {\it black-hole bomb} have
revealed the interesting fact that the superradiant instability shuts down if
the reflecting mirror is placed too close to the black-hole horizon. In the
present study we use analytical techniques to explore the superradiant
instability regime of this composed
Kerr-black-hole-linearized-scalar-field-mirror system. In particular, it is
proved that the lower bound ${{r_{\text{m}}}\over{r_+}}>{1\over
2}\Big(\sqrt{1+{{8M}\over{r_-}}}-1\Big)$ provides a necessary condition for the
development of the exponentially growing superradiant instabilities in this
composed physical system, where $r_{\text{m}}$ is the radius of the confining
mirror and $r_{\pm}$ are the horizon radii of the spinning Kerr black hole. We
further show that, in the linearized regime, this {\it analytically} derived
lower bound on the radius of the confining mirror agrees with direct {\it
numerical} computations of the superradiant instability spectrum which
characterizes the spinning black-hole-mirror bomb.
