Question #43834

In a plasma fusion reactor, the fusion consumption rate per unit volume is modeled as $\Phi(n) = A n \exp(-n_c / n)$, where $n$ is the local particle density, $n_c = 1.0 \times 10^{20} \text{ m}^{-3}$ is a critical density parameter, and $A = 3.0 e \text{ s}^{-1}$ ($e$ being the base of the natural logarithm). The reactor is maintained at a steady-state baseline density $n_0 = n_c$ by a constant fuel injection. To probe the reactor's dynamic response, a small oscillating fuel injection rate $\Delta S(t) = S_m \cos(\omega t)$ is superimposed, where $S_m = 1.0 \times 10^{18} \text{ m}^{-3}\text{s}^{-1}$ and $\omega = 8.0 \text{ rad s}^{-1}$. Assuming the density fluctuations $\Delta n(t)$ remain small ($\Delta n \ll n_0$), what is the amplitude of the resulting steady-state fluctuation in the fusion consumption rate, $\Delta \Phi_{\text{max}}$?