$(a)$ If a person wears a lens of power $-6\, D$ for distant vision and for correcting his near vision he needs a lens of $+2\, D$. Determine the focal length of the lenses in both the cases.
$(b)$ Give reasons for the following natural phenomena:
$(i)$ Stars twinkle.
$(ii)$ Planets do not twinkle.
$(iii)$ Stars appear raised in the sky.

(A) $(i)$ Focal length for distant vision: $f = \frac{1}{P} = \frac{1}{-6} = -0.1667\, m = -16.67\, cm$.
$(ii)$ Focal length for near vision: $f = \frac{1}{P} = \frac{1}{2} = +0.5\, m = +50\, cm$.
$(b)$ $(i)$ Stars twinkle due to atmospheric refraction. The Earth's atmosphere has layers of varying densities and temperatures,causing the refractive index to change continuously. Light from distant stars (point sources) undergoes continuous refraction,causing the apparent position and intensity of the star to fluctuate,which we perceive as twinkling.
$(ii)$ Planets are much closer to Earth than stars and appear as extended sources of light rather than point sources. $A$ planet can be considered as a collection of a large number of point-sized sources. The total variation in the amount of light entering our eye from all these point-sized sources averages out to zero,resulting in a steady light,so they do not twinkle.
$(iii)$ Stars appear raised due to atmospheric refraction. As starlight enters the Earth's atmosphere,it travels from a rarer medium to a denser medium,bending towards the normal at each layer. This continuous bending makes the star appear at a higher position than its actual location.