Question #43112

A transparent liquid is contained in a large tank. Let $y$ represent the depth measured downwards from the top surface ($y=0$). The temperature $T$ of the liquid varies with depth as $T(y) = T_0 - \beta y$, where $\beta > 0$ is a constant. The density $\rho$ of the liquid depends on temperature as $\rho(T) = \rho_0 [1 - \gamma (T - T_0)]$, where $\gamma$ is the coefficient of volume expansion and $\rho_0$ is the density at the surface. The refractive index $n$ of the liquid is related to its density by the Gladstone-Dale relation $n - 1 = C \rho$, where $C$ is a positive constant. A narrow laser beam is injected horizontally into the liquid at the origin $(x=0, y=0)$ in the $+x$ direction. Assuming $\gamma \beta y \ll 1$, which of the following represents the equation of the trajectory $y(x)$ of the beam? (Let $n_0 = 1 + C\rho_0$ be the refractive index at $y=0$)