Question #43940

A Michelson interferometer is used to monitor the rise of a liquid in a vertical glass capillary tube of inner radius $r$. The liquid perfectly wets the glass (contact angle $\theta = 0^\circ$), forming a dynamic hemispherical meniscus. A broad monochromatic plane wave of wavelength $\lambda$ (where $\lambda \ll r$) is incident vertically downwards, illuminating the entire cross-section of the meniscus. The light reflected from the meniscus interferes with a reference plane wave of equal uniform intensity $I_0$. A detector collects the entire interference pattern over the circular area of the tube and outputs a total power signal $P(t)$. As the liquid rises with a steady velocity $v$, the detector records an oscillating beat signal. If a student incorrectly assumes the meniscus is a perfectly flat mirror of radius $r$, they predict a beat signal amplitude $A_{\text{flat}}$. What is the ratio of the actual beat signal amplitude $A_{\text{actual}}$ to the predicted amplitude $A_{\text{flat}}$? (Neglect reflections from the tube walls and assume the meniscus maintains its hemispherical shape).