Animated Solution for Physics - Optics: Some laws/processes are given in Column I. Match these with the physical phenomena given in Column II.

The problem asks us to match various optical properties of lenses and telescopes with the physical parameters they depend on. Let's break down each property one by one to understand the underlying physics.

The primary function of a lens, especially in telescopes, is to gather light. The amount of light a lens can collect is directly proportional to its exposed surface area.

For a circular lens aperture with radius $R$, the area is given by:

Therefore, the intensity of the light received is directly proportional to the square of the radius of the aperture. This means that to see fainter objects, astronomers use telescopes with larger objective lenses or mirrors. Thus, Intensity of light matches with the radius of aperture ($R$).

When we use an astronomical telescope in normal adjustment (where the final image is formed at infinity for relaxed viewing), the angular magnification $m$ is defined as the ratio of the angle subtended by the image to the angle subtended by the object.

Mathematically, it is given by the ratio of the focal length of the objective lens ($f_o$) to the focal length of the eyepiece ($f_e$):

Clearly, the magnifying power depends entirely on the focal lengths of the two lenses. Thus, Angular magnification matches with focal length $f_o,f_e$.

The tube length of a telescope is the physical distance between the objective lens and the eyepiece. In normal adjustment, the objective lens forms a real image of a distant object at its focal point. For the eyepiece to form the final image at infinity, this intermediate image must lie exactly at the focal point of the eyepiece.

Therefore, the total length $L$ of the telescope is simply the sum of their focal lengths:

This shows that the physical length of the telescope is dictated by the focal lengths of its lenses. Thus, Length of telescope matches with focal length $f_o,f_e$.

The sharpness or clarity of an image formed by a lens is degraded by several optical defects and physical limits:

1. Diffraction: Due to the wave nature of light, a point object is imaged as a diffraction pattern (Airy disk). The resolving power, which dictates how sharp the image is, depends on the radius of the aperture ($R$). A larger aperture reduces diffraction blurring. 2. Chromatic Aberration: Different colors of light travel at slightly different speeds in glass, meaning the refractive index varies with wavelength. This property is called dispersion. Because of dispersion, a lens has different focal lengths for different colors, causing color fringing and reducing sharpness. 3. Spherical Aberration: A spherical lens surface does not perfectly focus all parallel rays to a single point. Rays striking the edges of the lens (marginal rays) are bent more strongly and focus closer to the lens than rays passing near the center (paraxial rays). This spread of focal points blurs the image.

Therefore, the sharpness of an image depends on the radius of aperture, the dispersion of the lens, and spherical aberration.

By analyzing the physical dependencies, we arrive at the following matches: - (A) Intensity of light $\to$ (p) radius of aperture - (B) Angular magnification $\to$ (r) focal length - (C) Length of telescope $\to$ (r) focal length - (D) Sharpness of image $\to$ (p) radius of aperture, (q) dispersion, (s) spherical aberration