Question #43587

Two identical uniform metal rods, each of length $L$, cross-sectional area $A$, Young's modulus $Y$, and coefficient of linear thermal expansion $\alpha$, are aligned horizontally. Their outer ends are rigidly clamped to two immovable vertical walls. The inner ends of the rods face each other, leaving a small gap of width $a_0$ ($a_0 \ll L$). This gap acts as a single slit for a diffraction experiment. The inner ends are also connected by an ideal spring of stiffness $k$ (placed outside the path of the light) which has negligible thermal expansion. Initially, the spring is in its relaxed state. A monochromatic light of wavelength $\lambda$ is incident normally on the slit, producing a diffraction pattern on a distant screen. The temperature of the rods is uniformly increased by $\Delta T$. As a result, the angular width of the central diffraction maximum is observed to double. Assuming the angles involved are small, which of the following expressions gives the required temperature increase $\Delta T$?