Answer:
(a) Focal length of a convex mirror is about half of its radius of curvature and it has nothing to do with the external medium. It remains same in water. The focal length of a convex lens depends on refractive index of its material as follows: \[f\propto \frac{1}{\mu -1}\] As the refractive index of glass with respect to water is less than the refractive index of glass with respect to air i.e., \[^{w}{{\mu }_{g}}{{<}^{a}}{{\mu }_{g}},\], so the focal length increases when the convex lens is held in water. (b) The air layers closer to the ground are hotter than higher layers. Oblique rays coming from distant sky therefore travel from denser to rarer parts of the atmosphere and get more and more oblique. When the angle of incidence exceeds critical angle (for denser air-rarer air interface), rays get totally reflected and may enter the observer's eye. The observer therefore sees a reflected image of the distant parts of the sky. (c) Twinkling of stars. The light from stars undergoes refraction continuously before it reaches earth. So the apparent position of the star is slightly different than its actual position. Due to variation in atmospheric conditions, like change in temperature, density etc., this apparent position keeps on changing. The amount of light entering our eyes from a particular star increases and decreases randomly with time. Sometimes, the star appears brighter and other times, it appears fainter. This gives rise to the twinkling effect of stars. (d) Since the atmosphere bends starlight towards the normal, the apparent position of a star is slightly 'above its actual position. Thus even when the sun has actually set (i.e., gone below the horizon) its apparent position remains above the horizon for some time.
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