Ray & Wave Optics Cheat Sheet for JEE Main

Mirror & Lens

• Mirror formula: $1/v + 1/u = 1/f$. Sign convention: distances measured from pole; against incident ray = negative.
• Magnification (mirror): $m = -v/u$; (lens): $m = v/u$.
• Lens maker: $1/f = (n-1)(1/R_1 - 1/R_2)$.
• Power $P = 1/f$ (m). Combination in contact: $P = P_1 + P_2$.
• Two thin lenses separated by $d$: $1/f = 1/f_1 + 1/f_2 - d/(f_1 f_2)$.

Refraction

• Snell: $n_1 \sin i = n_2 \sin r$.
• Critical angle: $\sin\theta_c = n_2/n_1$ (from denser to rarer).
• Prism: $A + \delta = i + e$; $\sin((A+\delta_m)/2) = n \sin(A/2)$.
• Small angle prism: $\delta = (n-1)A$.
• Apparent depth: $d' = d/n$ (object in denser medium seen from rarer).

Interference (YDSE)

• Fringe width $\beta = \lambda D/d$.
• Constructive: path diff $= n\lambda$; destructive: $(n+\tfrac12)\lambda$.
• Intensity: $I = I_1 + I_2 + 2\sqrt{I_1 I_2}\cos\phi$; max/min ratio $= (\sqrt I_1+\sqrt I_2)^2/(\sqrt I_1-\sqrt I_2)^2$.
• Immersion in liquid of index $n$: $\beta \to \beta/n$.

Diffraction (Single slit)

• Position of minima: $a \sin\theta = n\lambda$.
• Angular width of central maximum $= 2\lambda/a$.

Polarisation

• Malus: $I = I_0 \cos^2\theta$.
• Brewster: $\tan i_p = n$; reflected ray is completely polarised ⊥ to incident plane.

Quick Tricks

• Silvered lens: treat as mirror with $1/f_{eq} = 2/f_l + 1/f_m$.
• Newton's formula: $x_1 x_2 = f^2$ where $x_1, x_2$ are distances of object and image from respective foci.
• Refraction at spherical surface: $n_2/v - n_1/u = (n_2 - n_1)/R$.

Practice MCQs

Q1. A convex lens of focal length 20 cm forms a real image at 30 cm from the lens. The object distance is A) −60 cm B) −30 cm C) −40 cm D) −15 cm

Q2. Critical angle of a medium with refractive index √2 (light going to air) is A) 30° B) 45° C) 60° D) 90°

Q3. Fringe width in YDSE doubles when A) λ is halved B) D is doubled C) d is doubled D) d and D are both doubled

Q4. A prism of angle 6° and n=1.5 gives minimum deviation of A) 1° B) 2° C) 3° D) 6°

Q5. Two coherent sources of intensity ratio 4:1 produce interference. Ratio Iₘₐₓ/Iₘᵢₙ equals A) 3 B) 9 C) 5 D) 25

Q6. A ray strikes glass (n=1.5) at Brewster angle. That angle is approximately A) 33.7° B) 45° C) 56.3° D) 60°

Q7. An object is placed 10 cm in front of a concave mirror of focal length 15 cm. The image is A) Real, inverted B) Virtual, erect, magnified C) Virtual, erect, diminished D) Real, magnified

Q8. Angular width of central maximum in single-slit diffraction with a=0.1 mm, λ=600 nm is A) 1.2×10⁻² B) 1.2×10⁻³ C) 6×10⁻³ D) 6×10⁻⁴

Q9. A slab of thickness t and index n introduces an extra path of A) nt B) (n−1)t C) t/n D) (n+1)t

Q10. Two thin lenses of powers +5 D and −2 D are placed in contact. Combined power is A) +3 D B) +7 D C) −3 D D) +10 D

Answers: 1-A (1/30 − 1/u = 1/20 → u = −60 cm.), 2-B (sinθc = 1/√2 → 45°.), 3-B (β ∝ λD/d.), 4-C (δ = (n−1)A = 0.5×6 = 3°.), 5-B ((2+1)²/(2−1)² = 9.), 6-C (tan⁻¹(1.5) ≈ 56.3°.), 7-B (Object inside f gives virtual erect enlarged image.), 8-A (2λ/a = 2×6×10⁻⁷/10⁻⁴ = 1.2×10⁻².), 9-B (Optical path − geometrical = (n−1)t.), 10-A (Simple sum: 5 − 2 = 3 D.)

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