JEE Physics Ray Optics and Optical Instruments Notes Set 03

Reflection and Plane Mirrors

Light travels in straight lines in a uniform medium. At a smooth surface the angle of reflection equals the angle of incidence, both measured from the normal, and the incident ray, the reflected ray and the normal lie in one plane.
A plane mirror deviates a ray by \( 180^\circ-2i \). Its image is virtual, erect, laterally inverted, the same size as the object and as far behind the mirror as the object is in front.
If the mirror turns through an angle \( \theta \) with the incident ray fixed, the reflected ray turns through \( 2\theta \). For an object moving with velocity \( \vec v_O \) and a mirror moving with \( \vec v_M \), the image velocity has the same component as the object along the mirror and \( 2v_{M\perp}-v_{O\perp} \) perpendicular to it.
Two plane mirrors inclined at \( \theta \) form \( \frac{360^\circ}{\theta}-1 \) images when \( \frac{360^\circ}{\theta} \) is an even integer (for any position of the object) or when the object is on the bisector. A ray reflected once from each of the two mirrors is deviated by \( 360^\circ-2\theta \), and by \( 180^\circ \) (emerging antiparallel) when the mirrors are at right angles.
To see his full image, a person needs a mirror at least half his height, placed with its lower edge at half the height of his eye above the ground, whatever his distance from the mirror. A clock seen in a mirror shows the time 12:00:00 minus the actual time.

Spherical Mirrors

Sign convention: distances are measured from the pole, positive along the direction of the incident light. A concave mirror has negative f and R; a convex mirror has positive f and R. The focal length is half the radius of curvature, \( f=\frac R2 \), for rays close to the axis.
Mirror formula: \( \frac1v+\frac1u=\frac1f \). Lateral magnification: \( m=-\frac vu=\frac{f}{f-u} \). A negative m means an inverted (real) image.
A convex mirror always gives a virtual, erect, diminished image of a real object, lying between the pole and the focus, so its image distance can never exceed f. A concave mirror gives a real inverted image for u beyond f, and a virtual, erect, magnified image for an object between the pole and the focus.
A small object along the axis has longitudinal magnification \( \frac{dv}{du}=-m^2 \) in magnitude \( m^2 \). The velocity of the image along the axis is \( v_I=-m^2v_O \).

Refraction and Total Internal Reflection

Snell's law: \( n_1\sin i=n_2\sin r \). The refractive index is \( n=\frac cv \). When light enters another medium its frequency stays the same; speed and wavelength change in proportion to \( \frac1n \).
Apparent depth for an object in a denser medium, viewed normally from the rarer medium: \( d_{app}=\frac{d}{n} \). A slab of thickness t shifts the image towards the observer by \( t\left(1-\frac1n\right) \). A ray through a parallel slab emerges parallel to the incident ray, displaced sideways by \( \frac{t\sin(i-r)}{\cos r} \).
Total internal reflection needs light going from the denser to the rarer medium at an angle of incidence greater than the critical angle \( C \), where \( \sin C=\frac{n_2}{n_1} \). Because violet has the highest index, its critical angle is the smallest.
In a stack of parallel layers the product \( n\sin\theta \) is the same in every layer, so the emergent angle depends only on the first and last media.

Prism and Dispersion

For a prism of angle A: \( r_1+r_2=A \) and the deviation is \( \delta=i+e-A \). The same deviation occurs for two angles of incidence, which are interchanged for the emergent ray; their sum is \( A+\delta \).
At minimum deviation \( i=e \), \( r_1=r_2=\frac A2 \) and \( \mu=\frac{\sin\frac{A+\delta_m}{2}}{\sin\frac A2} \). Any other passage has a larger deviation. For a thin prism \( \delta=(\mu-1)A \).
A ray incident on the second face at the critical angle, or one that retraces its path from a silvered face, needs \( r_2=C \) or \( r_2=0 \) respectively.
Dispersion is the splitting of white light because \( \mu \) depends on wavelength. Angular dispersion \( =(\mu_v-\mu_r)A \); dispersive power \( \omega=\frac{\mu_v-\mu_r}{\mu_y-1} \), which depends only on the material. Two thin prisms give deviation without dispersion when \( \omega_1\delta_1+\omega_2\delta_2=0 \) (achromatic combination), and dispersion without deviation when \( (\mu_1-1)A_1=(\mu_2-1)A_2 \) with opposite orientation.

Refraction at a Spherical Surface and Thin Lenses

Single spherical surface: \( \frac{n_2}{v}-\frac{n_1}{u}=\frac{n_2-n_1}{R} \), with the same sign convention as for mirrors.
Lens maker's formula: \( \frac1f=(\mu-1)\left(\frac1{R_1}-\frac1{R_2}\right) \) for a lens in air; in a liquid of index \( \mu_0 \) replace \( \mu-1 \) by \( \frac{\mu}{\mu_0}-1 \). A lens that is optically denser than its surroundings and convex is converging.
Thin lens formula: \( \frac1v-\frac1u=\frac1f \), with \( m=\frac vu \). Power \( P=\frac1f \) (in dioptres with f in metres). For thin lenses in contact \( P=P_1+P_2 \), and for lenses separated by d, \( P=P_1+P_2-dP_1P_2 \).
Cutting a lens by a plane through the axis leaves the focal length unchanged but halves the brightness of the image; cutting it by a plane perpendicular to the axis gives a plano-convex lens of focal length 2f.
Displacement method: with object and screen D apart (D at least 4f), there are two lens positions giving a sharp image, separated by \( d=\sqrt{D^2-4Df} \), and \( f=\frac{D^2-d^2}{4D} \).

Eye and Optical Instruments

A short-sighted (myopic) eye is corrected by a concave lens whose focal length equals the far point. A long-sighted (hypermetropic) eye is corrected by a convex lens that forms a virtual image at the near point of an object placed at 25 cm.
Simple microscope: magnifying power \( 1+\frac Df \) for the image at the near point (D = 25 cm) and \( \frac Df \) for the image at infinity.
Compound microscope: the objective forms a real, inverted, magnified intermediate image, which the eyepiece magnifies. Final image at infinity: \( m\approx\frac{L}{f_o}\cdot\frac{D}{f_e} \), where L is the distance between the objective's second focus and the eyepiece's first focus. A longer tube and shorter focal lengths give a larger magnification.
Astronomical telescope: \( m=-\frac{f_o}{f_e} \) in normal adjustment, with tube length \( f_o+f_e \); a larger \( f_o \) gives more magnification. Galilean (terrestrial) telescope: \( m=\frac{f_o}{f_e} \), tube length \( f_o-f_e \), erect image.
The limit of resolution of a telescope of aperture a is an angle \( \frac{1.22\lambda}{a} \); a larger aperture resolves finer detail and also collects more light.

Worked Examples

Worked Example 1. A concave mirror has a focal length of 15 cm. Where is the image of an object placed 25 cm from it, and what is the magnification?
Answer:
With \( u=-25 \) and \( f=-15 \): \( \frac1v=\frac1f-\frac1u=-\frac1{15}+\frac1{25}=-\frac{2}{75} \), so \( v=-37.5\ \text{cm} \) (real image, 37.5 cm in front of the mirror).
\( m=-\frac vu=-\frac{-37.5}{-25}=-1.5 \): the image is inverted and 1.5 times the size of the object.

Worked Example 2. A prism of angle 60° and refractive index \( \sqrt2 \) is used. Find the angle of minimum deviation.
Answer:
\( \sin\frac{A+\delta_m}{2}=\mu\sin\frac A2=\sqrt2\times\frac12=\frac1{\sqrt2} \), so \( \frac{60^\circ+\delta_m}{2}=45^\circ \).
\( \delta_m=30^\circ \).

Worked Example 3. A coin lies at the bottom of a vessel with 12 cm of water (\( \mu=\frac43 \)). At what depth does it appear when viewed from above?
Answer:
\( d_{app}=\frac{12}{4/3}=9\ \text{cm} \); the coin appears 3 cm higher than its real position.

Worked Example 4. A double convex lens has equal radii of 20 cm and \( \mu=1.5 \). Find its focal length and power.
Answer:
\( \frac1f=(1.5-1)\left(\frac1{20}+\frac1{20}\right)=\frac1{20} \), so \( f=20\ \text{cm}=0.2\ \text{m} \).
Power \( =\frac1{0.2}=+5\ \text{D} \).

Worked Example 5. A convex lens of focal length 10 cm is in contact with a concave lens of focal length 20 cm. Find the focal length of the combination.
Answer:
\( P=\frac{100}{10}-\frac{100}{20}=10-5=+5\ \text{D} \).
\( f=\frac{100}{5}=20\ \text{cm} \), and the combination is converging.