Summary: The well-known superradiant amplification mechanism allows a charged scalar field of proper mass \(\mu\) and electric charge \(q\) to extract the Coulomb energy of a charged Reissner-Nordström black hole. The rate of energy extraction can grow exponentially in time if the system is placed inside a reflecting cavity which prevents the charged scalar field from escaping to infinity. This composed black-hole-charged-scalar-field-mirror system is known as the charged black-hole bomb. Previous numerical studies of this composed physical system have shown that, in the linearized regime, the inequality \(q / \mu > 1\) provides a necessary condition for the development of the superradiant instability. In the present paper, we use analytical techniques to study the instability properties of the charged black-hole bomb in the regime of linearized scalar fields. In particular, we prove that the lower bound \(\frac{q}{\mu} > \sqrt{\frac{r_{\mathrm{m}} / r_- -1}{r_{\mathrm m} / r_+ - 1}}\) provides a necessary condition for the development of the superradiant instability in this composed physical system (here \(r_\pm\) are the horizon radii of the charged Reissner-Nordström black hole and \(r_{\mathrm m}\) is the radius of the confining mirror). This analytically derived lower bound on the superradiant instability regime of the composed black-hole-charged-scalar-field-mirror system is shown to agree with direct numerical computations of the instability spectrum.