Question #43783

This question tests the student's ability to derive orbital parameters from first principles rather than relying on memorized formulas for the hydrogen atom.

The primary trap lies in assuming a standard Coulombic potential ($V \propto 1/r$) or a uniform charge density ($\rho = \text{const}$), which lead to incorrect dependencies ($T \propto n^3$ or $T \propto n^0$ respectively).

By introducing a non-uniform charge density $\rho(r) \propto r$, the effective potential $V_{eff}(r) = \frac{L^2}{2mr^2} + U(r)$ is fundamentally altered.

Students must correctly apply Gauss's Law to find the new force law ($F \propto r^2$) and couple it with the angular momentum quantization condition to find the non-linear shift in the orbital radius and time period.