Question #43135

A thin biconvex lens of initial focal length $f$ is made of a semiconductor material with an initial refractive index $n$. A point object is placed on the principal axis at a distance $x$ from the first principal focus, producing a real image at a distance $x'$ from the second principal focus. The temperature of the lens is then increased by $\Delta T$. The material has a coefficient of linear thermal expansion $\alpha$. The refractive index of the semiconductor depends on its energy band gap $E_g$ according to the relation $n = n_0 - k E_g$, where $n_0$ and $k$ are positive constants. The band gap narrows with temperature linearly as $\Delta E_g = -\beta \Delta T$ ($\beta > 0$). If the object is continuously adjusted to maintain the exact same distance $x$ from the newly shifted first principal focus, which of the following expressions gives the shift in the image position $\Delta x'$ relative to the new second principal focus? (Assume $\alpha \Delta T \ll 1$ and $k \beta \Delta T \ll n-1$, and all quantities are in standard SI units)