JEE Physics Ray Optics and Optical Instruments Notes Set 02

Reflection and Plane Mirrors

1. Laws of reflection: the angle of incidence equals the angle of reflection, and the incident ray, reflected ray and normal lie in one plane.
2. A plane mirror forms a virtual, erect image as far behind as the object is in front. The image speed along the normal is \( v_i = 2v_m - v_o \), while the components parallel to the mirror stay equal to the object's.
3. Two mirrors at angle \( \theta \) form \( \frac{360^\circ}{\theta} - 1 \) images when \( \frac{360^\circ}{\theta} \) is an integer. Two parallel mirrors form an endless series of images.
4. A ray reflected successively from two mirrors at angle \( \theta \) is turned through \( 2\theta \) in total.

Spherical Mirrors

1. Mirror formula \( \frac{1}{v} + \frac{1}{u} = \frac{1}{f} \), with \( f = \frac{R}{2} \) and magnification \( m = -\frac{v}{u} \). Use the Cartesian sign convention, with distances measured from the pole and positive along the incident light.
2. Concave mirror: f is negative; converging. Convex mirror: f is positive; always gives a virtual, erect, diminished image and the widest field of view (rear-view mirrors).
3. Longitudinal magnification of a short object along the axis is \( m^2 \).
4. For a real image from a concave mirror, \( \frac{v}{f} = \frac{u/f}{u/f - 1} \), a hyperbola in the first quadrant with asymptotes at 1.

Refraction at Plane Surfaces

1. Snell's law \( \mu_1\sin i = \mu_2\sin r \). Refractive index \( \mu = \frac{c}{v} = \frac{\lambda_{vac}}{\lambda} \); frequency does not change.
2. Apparent depth (normal view) \( = \frac{\text{real depth}}{\mu} \). For layers, add \( \frac{t_i}{\mu_i} \). A flat slab raises an object by \( t\left(1 - \frac{1}{\mu}\right) \), and blue is raised more than red.
3. Total internal reflection needs light going from a denser to a rarer medium at an angle beyond the critical angle \( C = \sin^{-1}\frac{\mu_{rarer}}{\mu_{denser}} \). Uses: optical fibres, endoscopes, diamonds, prisms.
4. A fish at depth h sees the outside world through a circle of radius \( \frac{h}{\sqrt{\mu^2 - 1}} \).

Prisms and Dispersion

1. \( \delta = i + e - A \) and \( r_1 + r_2 = A \). The deviation is minimum when the ray passes symmetrically, with \( \mu = \frac{\sin\frac{A + \delta_m}{2}}{\sin\frac{A}{2}} \).
2. For a thin prism \( \delta = (\mu - 1)A \). In a liquid, use \( \mu \) relative to the liquid.
3. Dispersion: refractive index is larger for violet than for red, so violet deviates most. Dispersive power \( \omega = \frac{\mu_v - \mu_r}{\mu - 1} \).
4. Achromatic combination of thin lenses: \( \frac{\omega_1}{f_1} + \frac{\omega_2}{f_2} = 0 \). Deviation without dispersion for two thin prisms: \( (\mu_v - \mu_r)A = (\mu_v' - \mu_r')A' \).

Refraction at Spherical Surfaces and Thin Lenses

1. Spherical surface: \( \frac{\mu_2}{v} - \frac{\mu_1}{u} = \frac{\mu_2 - \mu_1}{R} \).
2. Thin lens: \( \frac{1}{v} - \frac{1}{u} = \frac{1}{f} \) and \( \frac{1}{f} = \left(\frac{\mu_g}{\mu_m} - 1\right)\left(\frac{1}{R_1} - \frac{1}{R_2}\right) \). A lens immersed in a medium of the same index has infinite focal length.
3. Lenses in contact: \( P = P_1 + P_2 \). Separated by d: \( P = P_1 + P_2 - dP_1P_2 \). Power is in dioptre when f is in metres.
4. Displacement method: two lens positions give the image on a screen at distance D when \( f = \frac{D^2 - d^2}{4D} \).
5. A lens with its plane face silvered acts like a concave mirror of focal length \( \frac{f}{2} \), because light crosses the lens twice.
6. For an object at \( \frac{f}{2} \) from a convex lens the image is virtual, at distance f on the same side, and twice the size.

Optical Instruments

1. A microscope or telescope does not enlarge the object itself. It increases the visual angle formed at the eye.
2. Normal eye: near point 25 cm, far point at infinity. The eye lens changes its focal length (accommodation) between these limits.

Worked Examples

1. A coin lies at the bottom of a liquid 4 cm deep. A ray reaching the surface at 3 cm horizontally from the coin moves along the surface. Then \( \sin C = \frac{3}{5} \), \( \mu = \frac{5}{3} \), and the speed in the liquid is \( \frac{3\times10^8}{5/3} = 1.8\times10^8 \) m/s.
2. A screen 90 cm from an object receives a sharp image for two lens positions 20 cm apart. \( f = \frac{8100 - 400}{360} = 21.4 \) cm.
3. A telescope with \( f_o = 60 \) cm and \( f_e = 5 \) cm in normal adjustment has \( M = 12 \), so a \( 2^\circ \) object gives a \( 24^\circ \) image.
4. Lenses of powers 3 D and 6 D in contact give 9 D. Placed 20 cm apart they give \( 9 - 0.2\times18 = 5.4 \) D.