Optics

Optics is one of the highest-yielding chapters in the NEET physics paper: mirror and lens numericals, total internal reflection, prism deviation, Young's double slit, and polarisation together contribute a reliable cluster of questions every year, and almost all of them reduce to a handful of formulas applied with the correct sign convention.

Light as Rays: Reflection and Spherical Mirrors

When the obstacles and apertures light meets are much larger than its wavelength (~500 nm), light travels in effectively straight lines and we can treat it as a bundle of rays. This is the domain of ray (geometrical) optics.

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. For a spherical mirror of radius of curvature R, the focal length is f = R/2, measured from the pole.

The universal sign convention (Cartesian):

• All distances are measured from the pole, with the incident light travelling in the +x direction.
• Distances measured against the incoming light are negative; along it, positive.
• Heights above the principal axis are positive, below negative.

Consequences: object distance u is always negative for a real object; f is negative for a concave mirror, positive for convex.

The mirror equation and magnification:

$$\frac{1}{v}+\frac{1}{u}=\frac{1}{f}, \qquad m=\frac{h'}{h}=-\frac{v}{u}$$

A positive m means an erect (hence virtual) image for mirrors; negative m means inverted and real. Useful qualitative facts to memorise:

1. A plane mirror always gives a virtual, erect, same-size image at equal distance behind it; lateral inversion occurs.
2. A convex mirror always gives a virtual, erect, diminished image between P and F — hence its use as a rear-view mirror (wide field of view).
3. A concave mirror gives a real inverted image when the object is beyond F, and a virtual magnified erect image when the object lies between P and F (shaving/make-up mirror, dentist's mirror).
4. Object at C of a concave mirror → image at C, same size, inverted (m = −1).

Refraction, Total Internal Reflection and the Prism

At a boundary between two transparent media, light bends: Snell's law gives n₁ sin i = n₂ sin r, where the refractive index of a medium is n = c/v. A denser medium has larger n and smaller wave speed; frequency never changes on refraction, so the wavelength shrinks as λ_medium = λ_vacuum/n.

Everyday consequences worth remembering: a coin in water appears raised (apparent depth = real depth/n, so normal shift = t(1 − 1/n)), the sun is visible slightly before actual sunrise and after actual sunset due to atmospheric refraction, and a stick partly in water looks bent.

Total internal reflection (TIR) happens only when light goes from denser to rarer medium and the angle of incidence exceeds the critical angle C, where

$$\sin C = \frac{n_{rarer}}{n_{denser}} \quad \left(\sin C = \frac{1}{n}\ \text{for a medium–air boundary}\right)$$

Applications: sparkle of diamonds (very small C ≈ 24°), mirage, optical fibres (light guided by repeated TIR in the core), totally reflecting prisms in periscopes and binoculars.

For a thin prism of refracting angle A, the deviation δ = i + e − A and A = r₁ + r₂. Deviation is minimum when the ray passes symmetrically (i = e, r₁ = r₂), giving

$$n=\frac{\sin!\left(\frac{A+\delta_m}{2}\right)}{\sin (A/2)}, \qquad \delta_m \approx (n-1)A \ \text{for small } A$$

Since n depends on wavelength (larger for violet, smaller for red), white light splits into a spectrum — dispersion. Angular dispersion = (n_v − n_r)A and dispersive power ω = (n_v − n_r)/(n_y − 1). Rainbows arise from refraction, one or two internal reflections, and dispersion inside water droplets; the blue sky and red sunset come from Rayleigh scattering, whose intensity varies as 1/λ⁴.

Thin Lenses, Combinations and Optical Instruments

For refraction at a single spherical surface, n₂/v − n₁/u = (n₂ − n₁)/R. Applying this at both surfaces of a thin lens gives the lens maker's formula:

$$\frac{1}{f}=(n-1)\left(\frac{1}{R_1}-\frac{1}{R_2}\right)$$

with the thin lens equation 1/v − 1/u = 1/f and magnification m = v/u. Here f is positive for a converging (convex) lens and negative for a diverging (concave) lens, and a positive m means erect and virtual.

Power P = 1/f (in dioptres, with f in metres). For thin lenses in contact, powers add: P = P₁ + P₂ + …, i.e. 1/F = 1/f₁ + 1/f₂. For two lenses separated by distance d, 1/F = 1/f₁ + 1/f₂ − d/(f₁f₂). Total magnification of a combination is the product of individual magnifications.

Two chromatic/monochromatic defects of images:

• Spherical aberration: marginal and paraxial rays focus at different points (reduced by stops, parabolic mirrors).
• Chromatic aberration: different colours focus differently (removed by an achromatic doublet).

Instrument formulas (image at near point D = 25 cm, or at infinity):

1. Simple microscope: m = 1 + D/f (near point), m = D/f (relaxed eye).
2. Compound microscope: m = m_o × m_e ≈ (L/f_o)(D/f_e); needs both f_o and f_e small, with f_o < f_e.
3. Astronomical telescope: m = −f_o/f_e for image at infinity, tube length L = f_o + f_e; large f_o and large aperture give high magnification and brightness. Reflecting telescopes use a concave mirror objective — no chromatic aberration, mechanically easier to support, larger apertures possible.

Wave Optics: Huygens' Principle and Interference

To explain phenomena at the scale of the wavelength we need the wave picture. Huygens' principle states that every point on a wavefront acts as a source of secondary spherical wavelets; the envelope of these wavelets at a later instant is the new wavefront. This construction reproduces the laws of reflection and refraction and shows why the wave slows down (not speeds up) in a denser medium.

Interference requires two coherent sources (constant phase difference), which is why we split a single wavefront rather than use two independent lamps. Superposition of two waves of amplitudes a₁, a₂ with phase difference φ gives resultant intensity

$$I = I_1 + I_2 + 2\sqrt{I_1 I_2}\cos\phi$$

so I_max = (√I₁ + √I₂)², I_min = (√I₁ − √I₂)², and for equal sources I_max = 4I₀, I_min = 0. Energy is conserved — it is only redistributed.

In Young's double slit experiment with slit separation d and screen distance D:

• Path difference at a point y from the centre: Δ = yd/D.
• Bright fringes: Δ = nλ, so y_n = nλD/d.
• Dark fringes: Δ = (2n − 1)λ/2.
• Fringe width β = λD/d, the same for all fringes, giving equally spaced fringes of equal intensity.

Note the behaviour of β: it increases if λ increases (red fringes wider than violet) or D increases, and decreases if d increases. If the whole apparatus is immersed in a liquid of index n, λ and hence β shrink by a factor n. Introducing a transparent slab of thickness t and index n in one path adds an extra path (n − 1)t, shifting the pattern towards that slit. White light gives a white central fringe with a few coloured fringes on either side.

Diffraction, Resolution and Polarisation

Diffraction is the bending/spreading of light at edges and small apertures. For a single slit of width a, the central maximum is flanked by minima at a sinθ = nλ (n = 1, 2, 3…) and secondary maxima at a sinθ = (2n + 1)λ/2. The angular half-width of the central maximum is θ ≈ λ/a, and its linear width on a screen at distance D is 2λD/a. Contrast this with YDSE: in single-slit diffraction the central maximum is much brighter and wider than the rest, and the fringes are not of equal intensity. Ray optics is valid as long as the aperture is much larger than λ (Fresnel distance z_F = a²/λ marks the limit).

Resolution: the smallest resolvable detail is limited by diffraction.

• Telescope: limit of resolution Δθ = 1.22λ/D (D = objective aperture); resolving power ∝ D/λ.
• Microscope: minimum resolvable separation d_min = 1.22λ/(2n sinβ); resolving power improves with shorter wavelength (why electron microscopes resolve far better).

Polarisation proves light is a transverse wave. Unpolarised light has field vibrations in all directions perpendicular to propagation; a Polaroid transmits only the component along its pass axis, cutting intensity to half. For an analyser at angle θ to the polariser, Malus' law gives I = I₀cos²θ, so crossed Polaroids (θ = 90°) transmit nothing.

Polarisation also arises on reflection. At Brewster's angle i_B, where tan i_B = n, the reflected light is completely linearly polarised perpendicular to the plane of incidence, and the reflected and refracted rays are mutually perpendicular. Uses of Polaroids: sunglasses and camera filters to cut glare, windowpanes, and 3-D viewing.

Common Mistakes and Exam Traps

1. Sign convention slips. Students plug u as positive, or use f positive for a concave mirror. Always write u negative for real objects, f = −|f| for concave mirrors and concave lenses, and read the sign of the answer as physical information (negative v for a mirror = real image in front; positive v for a lens = real image beyond the lens).
2. Confusing the two magnification formulas. For mirrors m = −v/u; for lenses m = +v/u. Also remember that for lenses/mirrors, m > 0 means erect and virtual — never "real and erect" for a single mirror or lens.
3. Misapplying TIR. TIR is impossible from rarer to denser medium, and the critical-angle formula sin C = 1/n uses n of the denser medium relative to the rarer one. Also note that at exactly i = C the refracted ray grazes along the surface.
4. Wave-optics mix-ups. Frequency is unchanged in refraction (wavelength and speed change) — many candidates change frequency instead. In YDSE, remember β = λD/d changes when the setup is immersed in liquid, and do not use the diffraction condition a sinθ = nλ (minima) as if it were the interference condition for maxima.