Ray Optics

🟢 Lite — Quick Review (1h–1d)

Rapid summary for last-minute revision before your exam.

Ray Optics — Key Facts

Ray optics (geometrical optics) treats light as rays that travel in straight lines. Light rays reflect and refract at interfaces between media.

Laws of Reflection:

1. Angle of incidence (i) = Angle of reflection (r)
2. Incident ray, reflected ray, and normal all lie in the same plane

Laws of Refraction (Snell’s Law): $$n_1 \sin\theta_1 = n_2 \sin\theta_2$$

where n = refractive index of medium

Critical Angle and Total Internal Reflection (TIR): $$\sin\theta_c = \frac{n_2}{n_1} \quad (n_1 > n_2)$$

TIR occurs when θ_i > θ_c inside denser medium.

Spherical Mirrors:

Mirror formula (sign convention — New Cartesian): $$\frac{1}{f} = \frac{1}{v} + \frac{1}{u}$$

where f = focal length (f = R/2 for spherical mirror), v = image distance, u = object distance

Mirror Type	Focal Length	Image Characteristics
Concave	Negative (f < 0)	Real inverted (object beyond C); Virtual upright (object between F and mirror)
Convex	Positive (f > 0)	Always virtual, upright, diminished

Magnification: $m = \frac{h_i}{h_o} = \frac{-v}{u}$

⚡ JEE Exam Tip: For concave mirror with object between pole and focus (u < f), image is virtual, upright, and magnified. This is how shaving mirrors and dentist’s mirrors work.

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🟡 Standard — Regular Study (2d–2mo)

For students who want genuine understanding…

Thin Lenses:

Lens formula: $$\frac{1}{f} = \frac{1}{v} - \frac{1}{u}$$

Lens Maker’s Formula: $$\frac{1}{f} = (n-1)\left(\frac{1}{R_1} - \frac{1}{R_2}\right)$$

where R₁, R₂ are radii of curvature (positive if centre of curvature is on the outgoing side)

Power of Lens: $$P = \frac{1}{f} \text{ (in metres)} \Rightarrow \text{Unit: Dioptre (D)}$$

Convex lens: positive power (converging) Concave lens: negative power (diverging)

Combination of Thin Lenses:

Lenses in contact: $\frac{1}{F} = \frac{1}{f_1} + \frac{1}{f_2}$

Lenses separated by distance d: $$\frac{1}{F} = \frac{1}{f_1} + \frac{1}{f_2} - \frac{d}{f_1 f_2}$$

Refraction through Prism:

Deviation angle: $\delta = i_1 + i_2 - A$

Minimum deviation: $\delta_{min}$ occurs when $i_1 = i_2$

Refractive index: $n = \frac{\sin\frac{A+\delta_m}{2}}{\sin\frac{A}{2}}$

Dispersion:

White light splits into colours because refractive index depends on wavelength (Cauchy’s equation: $n = A + B/\lambda^2$).

Angular dispersion: difference in deviation for different colours.

⚡ JEE Exam Tip: For thin lenses, when two convex lenses are in contact, the combined focal length is always less than the smaller focal length. When separated, the combination can behave as a diverging lens if the separation exceeds the sum of focal lengths.

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🔴 Extended — Deep Study (3mo+)

Comprehensive coverage for students on a longer study timeline.

Optical Instruments:

Microscope:

• Objective lens: small focal length f_o
• Eyepiece: focal length f_e
• Normal adjustment: $D =$ least distance of distinct vision (25 cm)
• Magnification: $M = \frac{v_o}{u_o} \times \frac{D}{f_e} \approx \frac{L}{f_o} \times \frac{D}{f_e}$

Telescope:

• Astronomical (refracting): $M = -\frac{f_o}{f_e}$ (final image at infinity)
• terrestrial: uses erecting lens
• Reflecting telescope (Cassegrain): $M = -\frac{f_o}{f_e}$

Aberrations:

Spherical Aberration:

• Paraxial rays focus at different points from marginal rays
• Minimised by: stopping down aperture, using plano-convex lens with curved side toward parallel rays, or using parabolic mirrors

Chromatic Aberration:

• Different wavelengths focus at different points (because n depends on λ)
• Corrected by achromatic doublet: combination of converging and diverging lenses made of different glasses

For achromatic doublet: $\frac{\phi_1}{\phi_2} = -\frac{\omega_1}{\omega_2}$ where $\omega$ = dispersive power

Coma:

• Off-axis point source produces comet-shaped image
• Avoided by using field stop or coma-corrected optics

Astigmatism:

• Rays in different planes focus at different distances
• Corrected by using compound lenses

Optical Fibres:

Step-index fibre: core (n₁) surrounded by cladding (n₂, n₁ > n₂).

Numerical Aperture: $NA = \sqrt{n_1^2 - n_2^2}$

Maximum acceptance angle: $\sin\theta_a = NA$

Applications: telecommunications, endoscopy, decorative lighting.

⚡ JEE Advanced 2023 Analysis: Questions on lens maker’s formula, combination of lenses, and optical instruments appeared in 2023. For telescope magnification, remember it’s the ratio of angles subtended, not sizes. Compound microscope magnification uses object distance from objective, not focal length.

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