JEE Physics Ray Optics and Optical Instruments Notes Set 01

Reflection at Plane Mirrors

1. The angle of incidence equals the angle of reflection, and both are measured from the normal. A ray incident at angle i is deviated by \( 180^\circ-2i \). The image in a plane mirror is virtual, erect, laterally inverted and as far behind the mirror as the object is in front.
2. Two mirrors at angle \( \theta \) form \( 360/\theta-1 \) images when \( 360/\theta \) is an integer. A ray reflected from two mirrors is deviated by \( 360^\circ-2\theta \). The minimum mirror length to see the full image of a person is half the person's height.

Spherical Mirrors

1. Mirror formula \( \dfrac1v+\dfrac1u=\dfrac1f \) with \( f=R/2 \), and magnification \( m=-v/u \). The focal length does not depend on the colour of light. A convex mirror always gives a virtual, erect, diminished image; a concave mirror gives a real inverted image beyond f.
2. For a small displacement, the longitudinal magnification is \( (v/u)^2 \) and the speed of the image is \( (v/u)^2 \) times that of the object.

Refraction and Total Internal Reflection

1. Snell's law \( \mu=\sin i/\sin r=c/v \). Frequency is unchanged on refraction, while wavelength and speed decrease in a denser medium. Apparent depth is real depth divided by \( \mu \), and a slab of thickness t shifts the object by \( t(1-1/\mu) \).
2. Total internal reflection occurs from a denser to a rarer medium beyond the critical angle, \( \sin C=\mu_{rarer}/\mu_{denser} \). It explains optical fibres, the brilliance of diamond and mirages. Violet has the smallest critical angle.

Prism and Dispersion

1. For a prism \( \delta=i+e-A \) and at minimum deviation \( \mu=\sin\frac{A+\delta_m}{2}/\sin\frac A2 \). For a thin prism \( \delta=(\mu-1)A \).
2. Dispersive power \( \omega=\dfrac{\mu_v-\mu_r}{\mu_y-1} \). Two thin prisms give dispersion without deviation when \( (\mu_1-1)A_1=(\mu_2-1)A_2 \). A rainbow is formed by refraction, dispersion and internal reflection in raindrops.

Lenses

1. Lens formula \( \dfrac1v-\dfrac1u=\dfrac1f \) and lens maker's formula \( \dfrac1f=(\mu-1)\left(\dfrac1{R_1}-\dfrac1{R_2}\right) \). Power in dioptres is \( 100/f \) with f in cm. Powers add for lenses in contact; for separated lenses \( 1/F=1/f_1+1/f_2-d/f_1f_2 \).
2. In a medium the focal length becomes \( f_m=f_a\dfrac{\mu_g-1}{\mu_g/\mu_m-1} \); a lens in a medium of its own index disappears. Cutting a lens along its axis keeps f, while cutting it perpendicular to the axis doubles f. Achromatic doublet: \( \omega_1/f_1+\omega_2/f_2=0 \).

Optical Instruments and the Eye

1. Simple microscope: \( M=1+D/f \) at the near point and \( D/f \) at infinity. Compound microscope: \( M=M_oM_e \). Astronomical telescope in normal adjustment: \( M=f_o/f_e \) with tube length \( f_o+f_e \); at the near point \( M=\dfrac{f_o}{f_e}\left(1+\dfrac{f_e}{D}\right) \).
2. The normal eye sees from 25 cm to infinity. A short-sighted eye needs a concave lens that forms the image at its far point; a long-sighted eye needs a convex lens that forms the image at its near point.