Question #43130

A point light source is placed on the principal axis at a distance $u = -30 \text{ cm}$ from an equiconvex lens. The lens has an aperture diameter $D = 2.0 \text{ cm}$ and a radius of curvature $R = 20 \text{ cm}$ for both surfaces. A screen is fixed exactly at the mean image position $v_0$. The light source is powered by a tunable mechanism controlled by a series LCR circuit with $L = 0.1 \text{ H}$, $C = 10 \mu\text{F}$, and resistance $R_0 = 10 \Omega$. The circuit is driven by an AC voltage source $V(t) = 10 \sin(1000 t) \text{ V}$. The wavelength $\lambda$ of the emitted light varies instantaneously with the voltage $V_C(t)$ across the capacitor as $\lambda(t) = \lambda_0 + \beta V_C(t)$, where $\beta = 0.5 \text{ nm/V}$. The refractive index of the lens material varies with wavelength as $n(\lambda) = n_0 - \gamma (\lambda - \lambda_0)$, where the mean refractive index is $n_0 = 1.5$ and $\gamma = 2 \times 10^{-4} \text{ nm}^{-1}$. Assuming the paraxial approximation holds for the focal shift, what is the maximum diameter of the image spot formed on the screen?