Question #43224

A laser beam is incident horizontally at the origin $(0,0)$ into a non-homogeneous transparent medium occupying the region $x > 0$. The refractive index of the medium varies with height $y$ as $n(y) = n_0 \sqrt{1 + \alpha y}$, where $n_0$ is a dimensionless constant and $\alpha$ is a positive constant. The beam strikes a vertical screen placed at a horizontal distance $x = L = (20.0 \pm 0.1) \text{ cm}$. The height $y$ of the laser spot on the screen is measured using a vertical scale with a least count of $0.01 \text{ cm}$. The measured reading for the spot's height is $12.50 \text{ cm}$. However, the scale has a known zero error of $+10.50 \text{ cm}$ (the scale reads $10.50 \text{ cm}$ when the actual height is zero). Calculate the value of the medium parameter $\alpha$ along with its absolute error $\Delta \alpha$, in SI units ($\text{m}^{-1}$). Assume the error in $y$ is solely due to the least count of the scale.