Ray Optics and Optical Instruments — NEET Notes

Reflection by spherical mirrors

A spherical mirror is a small section of a sphere with one side polished. If the inner (concave) surface reflects, we call it a concave mirror; if the outer (convex) surface reflects, a convex mirror. The geometry is fixed by three points on the principal axis: the pole P, the centre of curvature C, and the focus F midway between them. Distance PF is the focal length f, PC is the radius of curvature R. For paraxial rays a simple argument shows that f = R/2. The usual laws of reflection apply — angle of incidence equals angle of reflection — except the normal now points along the radius. Concave mirrors converge a parallel beam to a real focus in front; convex mirrors make a parallel beam appear to diverge from a virtual focus behind. That single difference produces every behaviour distinction NEET tests.

Sign convention & the mirror formula

Every formula that follows depends on one agreement: how to measure distances. NCERT uses the Cartesian sign convention — all distances measured from the pole (or optical centre). Distances along the direction of incident light are positive; against it, negative. Heights upward are positive; downward, negative. With this convention a single mirror equation works for every case — concave, convex, real, virtual.

1/v + 1/u = 1/f

The mirror formula — valid for all spherical mirrors and image types

Linear magnification follows from similar triangles: m = h′/h = −v/u. Negative m ⇒ inverted, real (mirror). Positive m ⇒ erect, virtual. |m| > 1 magnified; |m| < 1 diminished. NEET 2016 column-matched m = −2 (real, inverted, concave) with m = +2 (virtual, erect, concave).

Refraction & Snell's law

When light meets a boundary between two transparent media, part reflects and part crosses over. The refracted ray bends because light speed differs in the two media. Snell's law ties the angle of incidence i to the angle of refraction r: n₁ sin θ₁ = n₂ sin θ₂. The absolute refractive index n = c/v compares vacuum speed c to the speed v inside the medium. An optically denser medium has higher n; light slows and bends toward the normal on entry.

n₁ sin θ₁ = n₂ sin θ₂

Snell's law — the heart of refraction

Two NEET-favourite consequences: a glass slab refracts a ray twice and produces a parallel, laterally-shifted emergent ray; and an object in water shows apparent depth h₁ = h₂/n — a coin in water sits closer to the surface than it really is.

Total internal reflection

Snell's law has a built-in cap. When light travels from a denser medium to a rarer one, the refracted ray bends away from the normal. As i grows, r grows faster; at some i = i_c the refracted ray grazes the surface, with r = 90°. Beyond that critical angle, refraction is impossible — every photon is reflected back. This is total internal reflection (TIR), and unlike ordinary reflection it loses no light to transmission.

Set r = 90° in Snell's law: sin i_c = 1/n, where n is the refractive index of the denser medium with respect to the rarer one. Water–air gives i_c ≈ 48.6°; diamond–air, i_c ≈ 24.4°. The small i_c of diamond traps light in internal reflections, releasing it only through the top facets as brilliance.

Three classical applications: optical fibres guide signals via TIR in a high-n core/low-n cladding, even around bends. Totally reflecting prisms exploit i_c < 45° in crown glass, turning a 45° right-angle prism into a perfect 90° or 180° turning mirror in periscopes and binoculars. Mirages are TIR at the warm-air–cool-air interface above a hot road, producing a wet-looking patch.

Refraction at a spherical surface

The bridge between the mirror equation and the lens maker's formula is refraction at a single spherical surface — applied twice it gives the lens formula. Light passes from medium n₁ into a sphere of n₂ across a surface of radius R (centre C). Snell's law in the small-angle limit, plus the Cartesian sign convention, yields:

n₂/v − n₁/u = (n₂ − n₁)/R

Refraction at a single spherical surface

NCERT example: air (n₁ = 1) to glass (n₂ = 1.5), R = +20 cm, u = −100 cm → v = +100 cm. R is positive when C lies on the transmission side, negative otherwise.

Lens maker's formula & thin lens equation

A thin lens is bounded by two refracting surfaces. Apply the single-surface relation at each, add, and the intermediate image distance cancels — giving the lens maker's formula:

1/f = (n − 1) (1/R₁ − 1/R₂)

Lens maker's formula — design a lens to a target f

Here n is the refractive index of the lens material relative to the surrounding medium; R₁ is the radius of the surface light meets first, R₂ that of the second. For a biconvex lens, R₁ > 0 and R₂ < 0 → f > 0 (converging). For a biconcave lens, R₁ < 0 and R₂ > 0 → f < 0 (diverging). A flat surface contributes nothing (R = ∞).

With f known, the relation between object and image distance is the thin lens equation:

1/v − 1/u = 1/f

Thin lens formula — valid for both convex and concave lenses

Linear magnification, derived from similar triangles in the lens ray diagram, is m = h′/h = v/u. Note the sign difference from the mirror case: there is no minus. As with mirrors, a positive m means erect and virtual, a negative m means inverted and real.

Power & combination of lenses

The power of a lens measures how strongly it converges or diverges light: P = 1/f, where f is in metres. Its SI unit is the dioptre (D = m⁻¹). A converging lens has positive power, a diverging lens has negative power. An optician's prescription of +2.5 D means a convex lens of focal length 1/2.5 = 0.4 m = 40 cm; a −4 D prescription means a concave lens of 25 cm focal length.

When two thin lenses of focal lengths f₁ and f₂ are placed in contact, the image formed by the first lens acts as the object for the second. Applying the lens formula at each lens and adding gives the combination formula:

1/f = 1/f₁ + 1/f₂   ⇒   P = P₁ + P₂

Thin lenses in contact — powers add algebraically

The sum is algebraic — convex lenses contribute positively, concave negatively. NEET 2023 Q.50 used this directly: a convex lens of focal length f combined with a concave lens of focal length −f gives 1/f_eq = 1/f − 1/f = 0, so f_eq → ∞. The combined system behaves like a plane slab and does not converge or diverge light at all. Total magnification of the combination is the product of individual magnifications: m = m₁ m₂ m₃ ….

Refraction through a prism

A prism is a wedge with two refracting surfaces meeting at angle A. A ray enters at incidence i, refracts to r₁, traverses the glass, hits the second face at r₂, and emerges at e. The geometry gives r₁ + r₂ = A and total deviation δ = i + e − A.

A plot of δ vs i dips to a minimum then rises. At the minimum, by i ↔ e symmetry, i = e and r₁ = r₂ = A/2. This minimum deviation condition makes the ray inside parallel to the base; Snell's law then yields:

n = sin[(A + D_m)/2] / sin(A/2)

Refractive index from prism minimum deviation

NEET tests this derivation every year. NEET 2016: i = 45°, A = 60°, min deviation ⇒ D_m = 30°, n = √2. NEET 2020: small-angle prism with r₂ = 0 gives i ≈ μA. NEET 2018: silvered-face retracing condition ⇒ sin i = √2 sin 30° → i = 45°.

Dispersion & the rainbow

Refractive index depends weakly on wavelength. Violet (≈ 400 nm) sees a higher n in glass than red (≈ 750 nm), and refracts more steeply. White sunlight through a prism spreads into a coloured spectrum — dispersion. Since deviation ∝ (n − 1), the violet–red gap (n_v − n_r) sets angular dispersion; dividing by mean deviation gives dispersive power.

The rainbow is dispersion's natural masterpiece. A spherical water droplet acts as a tiny prism — one refraction on entry, one TIR at the back, one refraction on exit. The primary rainbow (red outside, violet inside) emerges at about 42° from the antisolar point; the secondary rainbow (colours reversed) uses two internal reflections and forms at about 51°. NEET 2017 used the dispersion-without-deviation condition: A′/A = −(μ − 1)/(μ′ − 1).

Scattering of light

When light passes through a medium with particles much smaller than its wavelength — N₂, O₂ molecules in air — it is scattered: each molecule absorbs a photon and re-emits it in a random direction. Short wavelengths scatter far more than long. Rayleigh's 1871 result:

I_scattered ∝ 1/λ⁴

Rayleigh's law of scattering

The blue sky: blue (λ ≈ 470 nm) scatters ~10× more than red (≈ 700 nm), so the sky away from the Sun looks blue. The red sunset: long oblique air path strips blue out of the line of sight, leaving red. White clouds: cloud droplets are not small compared to λ, so all wavelengths scatter equally (Mie regime) → white.

The microscope — simple & compound

Optical instruments extend what the eye can do. The two examined heaviest are the microscope (small, near objects) and the telescope (distant, large). Both produce a magnified image so it subtends a larger angle than the object alone.

A simple microscope is a single converging lens of short focal length, held close to the object. With the object just inside the focal point, the lens produces a virtual, erect, magnified image at — or before — the near point D = 25 cm. Near point: M = 1 + D/f. Infinity (relaxed eye): M = D/f. They differ by 1.

A compound microscope chains two lenses. The objective (small f_o, a few mm) sits near the object and forms a real, inverted, magnified intermediate image inside the tube. The eyepiece (small f_e) then acts as a simple microscope, producing the final virtual image at infinity. Magnifications multiply:

M = (L/f_o) × (D/f_e)

Compound microscope — final image at infinity

L is the tube length — distance between the second focal point of the objective and the first focal point of the eyepiece. With f_o = 1 cm, f_e = 2 cm, L = 20 cm, D = 25 cm: M = 20 × 12.5 = 250.

The telescope — refracting & reflecting

An astronomical telescope gives angular magnification of distant objects. Its objective is a converging lens of long focal length and large diameter, forming a real, inverted, diminished image at its focal plane. The eyepiece, of short f_e, magnifies this image with final image at infinity.

M = f_o / f_e

Astronomical telescope (final image at infinity)

Two figures of merit: light-gathering power scales with objective area; resolving power scales with diameter via Rayleigh criterion 1.22λ/D. NEET 2018 and 2021 both test this: telescopes want long f_o AND large diameter. Tube length L = f_o + f_e. NEET 2016 took finite object distance (200 cm): apply the lens formula to the objective ⇒ v_o = 50 cm; for final image at infinity, |u_e| = f_e = 4 cm; so L = 54 cm.

For the largest telescopes, a refracting design fails: glass beyond a metre sags under its own weight. Modern observatories use reflecting telescopes, with a concave-mirror objective — supportable across its back, free of chromatic aberration, and capable of 10 m (Keck) or 39 m (ELT) diameters. Cassegrain and Newtonian designs differ in how the beam is redirected to the eyepiece.