Ray optics for JEE Main

Ray optics is a sign-convention chapter. Nearly every formula here is one line long, and nearly every wrong answer comes from a sign put in the wrong place. Fix one convention, use it every time, and this becomes one of the fastest sections of the paper.

The official 2026 syllabus covers reflection and spherical mirrors, the mirror formula, refraction at plane and spherical surfaces, the thin lens formula and lens maker's formula, total internal reflection and its applications, magnification, power of a lens, thin lenses in contact, refraction through a prism, and the microscope and astronomical telescope (reflecting and refracting) with their magnifying powers. Two experiments from the practical list also sit here: focal length by the parallax method, and the deviation–incidence graph of a prism.

Sign convention

Use the Cartesian convention:

• Measure all distances from the pole of the mirror or the optical centre of the lens.
• Distances in the direction of the incident light are positive; against it, negative.
• Heights above the principal axis are positive; below, negative.

With light coming from the left, a real object always has negative u. A concave mirror has negative f, and a convex lens has positive f.

Mirrors

1/v + 1/u = 1/f, with f = R/2 and magnification m = −v/u.

A negative m means an inverted, real image; a positive m means erect and virtual. A convex mirror always forms a virtual, erect, diminished image, which is why it is used as a rear-view mirror.

If a plane mirror turns through an angle θ, the reflected ray turns through 2θ.

Refraction and total internal reflection

Snell's law: n₁ sin i = n₂ sin r. Light bends towards the normal on entering a denser medium. Its frequency stays the same, while its speed and wavelength fall by the factor n.

Apparent depth. Seen from directly above, an object at real depth d in a medium of refractive index n appears at d/n. It seems raised by d(1 − 1/n).

Total internal reflection happens when light goes from denser to rarer and the angle of incidence exceeds the critical angle C, where sin C = 1/n (for the denser medium against air). Applications include optical fibres, totally reflecting prisms in binoculars, mirages and the sparkle of a diamond, whose critical angle is small.

Refraction at a spherical surface

For a surface of radius R separating media n₁ (where the light starts) and n₂:

n₂/v − n₁/u = (n₂ − n₁)/R

This single equation gives apparent depth (put R = ∞) and, applied at two surfaces one after the other, the lens maker's formula.

Thin lenses

• Lens formula: 1/v − 1/u = 1/f
• Magnification: m = v/u
• Lens maker's formula: 1/f = (n − 1)(1/R₁ − 1/R₂)
• Power: P = 1/f, with f in metres; unit dioptre (D)
• Lenses in contact: P = P₁ + P₂, so 1/f = 1/f₁ + 1/f₂

A lens in a medium of index n_m uses the relative index: 1/f = (n/n_m − 1)(1/R₁ − 1/R₂). A glass lens in water becomes much weaker. If n_m exceeds n, a convex lens starts to diverge.

Prism

For a prism of angle A, with incidence i, emergence e and internal refraction angles r₁ and r₂:

• r₁ + r₂ = A
• deviation δ = i + e − A

As i increases, δ first falls and then rises. At minimum deviation the ray passes symmetrically: i = e and r₁ = r₂ = A/2, so

n = sin[(A + δ_m)/2] / sin(A/2)

For a thin prism, δ = (n − 1)A.

Microscope and telescope

Compound microscope. The objective forms a real, magnified image, and the eyepiece acts as a magnifier. With the final image at infinity:

M ≈ (L/f_o)(D/f_e), where L is the tube length and D = 25 cm, the least distance of distinct vision.

Both focal lengths are short, and the objective's is the shorter.

Astronomical telescope. In normal adjustment (final image at infinity):

• M = f_o/f_e
• length of the tube = f_o + f_e
• with the final image at the near point: M = (f_o/f_e)(1 + f_e/D)

A large objective focal length and a small eyepiece focal length give high magnification. A reflecting telescope uses a concave mirror as the objective. It has no chromatic aberration, and a large mirror is easier to make and support than a large lens.

Worked problems

Problem 1: concave mirror. An object is 10 cm in front of a concave mirror of focal length 15 cm. Find the image.

u = −10, f = −15. 1/v = 1/f − 1/u = −1/15 + 1/10 = 1/30, so v = +30 cm, behind the mirror.
m = −v/u = −30/(−10) = +3: virtual, erect and three times the size.

Problem 2: lens maker and a lens in water. A biconvex lens has n = 1.5 and R₁ = 20 cm, R₂ = −20 cm. Find its focal length in air and in water (n = 4/3).

Air: 1/f = 0.5 × (1/20 + 1/20) = 0.05, so f = 20 cm.
Water: relative index = 1.5/(4/3) = 1.125, so 1/f = 0.125 × 0.1 = 0.0125 and f = 80 cm.

Problem 3: prism. A prism of angle 60° gives a minimum deviation of 30°. Find its refractive index and critical angle.

n = sin 45°/sin 30° = (1/√2)/(1/2) = √2 ≈ 1.41.
sin C = 1/√2, so C = 45°.

Problem 4 (numerical answer): lenses in contact. A convex lens of focal length 20 cm touches a concave lens of focal length 40 cm. An object is 60 cm in front of the pair. Find the image distance in cm.

P = 5 − 2.5 = 2.5 D, so f = +40 cm.
1/v = 1/40 + 1/(−60) = (3 − 2)/120, so v = 120 cm, real, on the far side.

Practice set

1. A telescope has f_o = 100 cm and f_e = 5 cm. Find the magnification and tube length in normal adjustment.
2. Find the critical angle for a medium of refractive index 2.
3. A coin lies 12 cm under water (n = 4/3). How deep does it appear?
4. An object in air is 30 cm from a convex glass surface (n = 1.5, R = 10 cm). Find where the image forms inside the glass.
5. A thin prism of angle 5° has n = 1.5. Find its deviation.
6. An object is 10 cm from a convex lens of focal length 20 cm. Find the image and its magnification.
7. A microscope has f_o = 1 cm, f_e = 5 cm and a tube length of 15 cm. Estimate its magnifying power with the final image at infinity.
8. A plane mirror is rotated through 10°. Through what angle does the reflected ray turn?

Answers

1. M = 20; length 105 cm.
2. sin C = 1/2, so 30°.
3. 12/(4/3) = 9 cm.
4. 1.5/v − 1/(−30) = 0.5/10, so 1.5/v = 1/60 and v = 90 cm inside the glass.
5. δ = 0.5 × 5° = 2.5°.
6. 1/v = 1/20 − 1/10 = −1/20, so v = −20 cm (virtual, same side); m = −20/−10 = +2.
7. M ≈ (15/1)(25/5) = 75.
8. 20°.

What to do next

• Write the sign convention on the top of your notes page and solve 20 mirror and lens problems without breaking it once.
• Derive the lens maker's formula from two refractions at spherical surfaces.
• Practise prism questions, including the δ–i graph from the practical list.
• Move on to wave optics, then keep the physics formula sheet for last-week revision.