Question #81811

An ionised gas (plasma) is trapped between two parallel capacitor plates of area $A$ located at $x = 0$ and $x = d$. The plasma is maintained at a constant temperature $T$. Under steady-state confinement, the plasma behaves as a dielectric medium whose relative permittivity varies spatially as $\epsilon_r(x) = 1 + \alpha x$ (where $\alpha > 0$). A constant potential difference $V_0$ is maintained across the plates. The electrostatic force density acting on the plasma due to the dielectric gradient is given by $f = \frac{1}{2}\epsilon_0 E^2 \frac{d\epsilon_r}{dx}$. In mechanical equilibrium, this force density is balanced by the pressure gradient of the gas ($\frac{dP}{dx} = f$). Assuming the plasma behaves as an ideal gas, if the number density of particles at $x = 0$ is $n_0$, what is the number density $n(d)$ at $x = d$? (Here $k_B$ is the Boltzmann constant).