JEE Physics Ray Optics and Optical Instruments Notes Set 04

Ray Optics: Quick Revision

Reflection obeys two laws: the incident ray, reflected ray and normal lie in one plane, and the angle of incidence equals the angle of reflection. A mirror turned through \( \theta \) turns the reflected ray through \( 2\theta \). Two plane mirrors inclined at angle \( \theta \) form \( \frac{360^\circ}{\theta}-1 \) images when this ratio is an integer. For a spherical mirror, \( f=\frac R2 \) and \( \frac1v+\frac1u=\frac1f \) with the sign convention measured from the pole; \( m=-\frac vu \). A moving object has image speed \( v_I=\left(\frac vu\right)^2v_O \) along the axis. Focal length of a mirror does not depend on colour.

Refraction and Total Internal Reflection

Snell's law is \( \mu_1\sin i=\mu_2\sin r \). Frequency does not change on refraction, while speed and wavelength scale as \( \frac1\mu \). Apparent depth is \( \frac{d}{\mu} \), and a slab of thickness t shifts the image by \( t\left(1-\frac1\mu\right) \). Total internal reflection needs light going to a rarer medium at an angle above the critical angle, \( \sin C=\frac{\mu_{rare}}{\mu_{dense}} \). A source at depth h under water is visible only within a circle of radius \( h\tan C \). Violet has the smallest critical angle. Blue sky and red sunsets come from scattering, which varies as \( \frac1{\lambda^4} \); a rainbow combines refraction, dispersion and internal reflection.

Prisms

For a prism, \( r_1+r_2=A \) and \( \delta=i+e-A \). At minimum deviation \( i=e \), \( r=\frac A2 \), and \( \mu=\frac{\sin\frac{A+\delta_m}{2}}{\sin\frac A2} \). For a thin prism, \( \delta=(\mu-1)A \). Two thin prisms of opposite orientation give dispersion without deviation when \( (\mu_1-1)A_1=(\mu_2-1)A_2 \). A right-angled isosceles prism reflects totally at the hypotenuse if \( \mu\ge\sqrt2 \). If one face of a prism is silvered, the ray retraces its path when it meets the silvered face normally.

Lenses and Combinations

The lens maker formula is \( \frac1f=(\mu-1)\left(\frac1{R_1}-\frac1{R_2}\right) \). In a medium, replace \( \mu \) by \( \frac{\mu_g}{\mu_m} \); the lens becomes weaker, and diverges if \( \mu_m>\mu_g \). Lenses in contact add their powers, \( P=\frac1f \) in metres. A plano-convex lens silvered on its flat face acts as a concave mirror of focal length \( \frac{R}{2(\mu-1)} \). An achromatic doublet satisfies \( \frac{\omega_1}{f_1}+\frac{\omega_2}{f_2}=0 \), which needs one convex and one concave lens.

Eye and Optical Instruments

Myopia is corrected by a concave lens whose focal length equals the far point; hypermetropia by a convex lens that brings the reading distance to the near point. Measure distances from the spectacle lens when it is away from the eye. A simple microscope has \( M=1+\frac D{f} \). A compound microscope forms a real, inverted, magnified intermediate image. A telescope in normal adjustment has \( M=\frac{f_o}{f_e} \) and length \( f_o+f_e \). For a near object the objective image distance exceeds \( f_o \), so the tube length is larger. Resolving limit is \( \Delta\theta=\frac{1.22\lambda}{D} \), so a large aperture helps both magnification and resolution.