Optics

Cartesian sign convention (the only one used in JEE/NEET):

• All distances measured from the optical centre.
• Distances in the direction of incident light: +ve. Opposite: −ve.
• Heights above principal axis: +ve. Below: −ve.
• Light always travels left to right (by convention).

Result: for a real object, u is negative.

Lens formula:

1/v − 1/u = 1/f

(Note the minus sign — opposite to the mirror formula 1/v + 1/u = 1/f.)

Magnification:

m = v / u = h_image / h_object

m is positive for upright (virtual) images, negative for inverted (real) images.

Lens-maker's equation: for a lens of refractive index n in air:

1/f = (n − 1) · (1/R₁ − 1/R₂)

where R₁ is the radius of the first surface (where light enters), R₂ the second.

Power of a lens: P = 1/f (with f in metres). Unit: dioptre (D). Lenses in contact: P_total = P₁ + P₂.

Worked example. Object 20 cm in front of a convex lens of focal length 15 cm. Find image position and size.

Using sign convention: u = −20 cm, f = +15 cm.
1/v − 1/(−20) = 1/15 → 1/v = 1/15 − 1/20 = (4 − 3)/60 = 1/60.
v = +60 cm (positive → image on opposite side, real).

Magnification m = v/u = 60/(−20) = −3 (inverted, 3× larger).

Young's double slit experiment (1801) proved light's wave nature by demonstrating interference.

Setup: monochromatic light → narrow slits S₁ and S₂ (separation d) → screen at distance D. Slits act as coherent sources (Huygens).

Path difference at point on screen at distance y from central axis:
Δ = (yd)/D (for small angles, y ≪ D)

Bright fringes (constructive interference): Δ = nλ → y_n = nλD/d.
Dark fringes (destructive): Δ = (n + ½)λ → y = (n + ½)λD/d.

Fringe width (distance between consecutive bright or consecutive dark):

β = λD/d

Fringe pattern is symmetric about the central bright fringe (n=0).

Conditions for visible interference:

1. Sources must be coherent (constant phase relationship). Achieved here by splitting a single wavefront via two slits.
2. Equal (or nearly equal) amplitudes for maximum contrast.
3. Path difference Δ < coherence length of source.
4. Monochromatic (or very narrow spectrum).

Effect of changes:

• Increase d → fringe width decreases (fringes closer).
• Increase D → fringe width increases.
• Decrease λ (e.g., blue vs red) → fringe width decreases.

Worked example. Slits 0.5 mm apart, screen 1 m away, light λ = 600 nm. Find fringe width.

β = λD/d = (600 × 10⁻⁹ × 1) / (0.5 × 10⁻³) = 1.2 × 10⁻³ m = 1.2 mm.

Intensity formula:

I = I₁ + I₂ + 2√(I₁I₂) cos δ, where δ = 2π Δ/λ.

For equal-intensity slits (I₁ = I₂ = I₀): I = 4I₀ cos²(δ/2). Maximum 4I₀ at bright fringes, minimum 0 at dark.

Diffraction (single slit): light bends around obstacles. Bright central maximum width = 2λD/a (a = slit width).

Diffraction occurs when slit width ~ wavelength. Visible-light diffraction is small but observable (~mm scale features).

Resolving power of optical instruments (Rayleigh's criterion): θ_min ≈ 1.22 λ/D for circular apertures. Why higher-resolution telescopes need bigger mirrors.

Polarization: transverse waves can be polarized; longitudinal cannot. Malus's law for intensity through a polarizer:

I = I₀ cos²θ

where θ is the angle between polarizer axes.

Brewster's angle (light reflected off a surface is completely plane-polarized): tan θ_B = n (refractive index of the medium).