Question #42530

A gas of hydrogen-like atoms is in thermal equilibrium inside a cavity at temperature $T_0$. The peak emission wavelength $\lambda_{\text{max}}$ of the cavity's blackbody radiation exactly matches the wavelength of a specific atomic transition. This transition occurs when an electron (charge $-e$, mass $m$) drops from a circular orbit of radius $r$ and speed $v$ to a ground state with zero orbital angular momentum. A uniform magnetic field $B$ is then slowly established perpendicular to the plane of the excited orbit, in a direction that increases the electron's orbital speed. To ensure the cavity's peak blackbody wavelength continues to exactly match the new transition wavelength, the cavity's temperature must be adjusted to $T$. Assuming Wien's displacement constant is $b$ and Planck's constant is $h$, what is the required change in temperature $\Delta T = T - T_0$? (Assume the orbital radius $r$ remains constant and ignore electron spin).