Ray Optics and Optical Instruments: NEET notes.

01 · NCERT § "Reflection of Light by Spherical Mirrors"

Reflection by spherical mirrors and the sign convention

• Laws of reflection: the angle of reflection equals the angle of incidence, and the incident ray, the reflected ray and the normal at the point of incidence are coplanar. For a curved mirror the normal is along the radius at that point.
• NCERT uses the Cartesian sign convention: every distance is measured along the principal axis, taking the pole of the mirror (or, for a lens, its optical centre) as the origin.
• Distances measured in the direction of the incident light are positive; those measured against it are negative. Heights above the principal axis are positive and below it negative.
• With light coming from the left, a real object in front of a mirror has negative u; a concave mirror has negative f and R, and a convex mirror has positive f and R.
• For a spherical mirror of small aperture using paraxial rays, the focal length is half the radius of curvature: f = R/2.
• Paraxial rays are rays close to the principal axis making small angles with it; the simple mirror and lens formulas are valid only for them.

02 · NCERT § "The Mirror Equation"

The mirror equation and magnification

• Object distance u, image distance v and focal length f are related by 1/v + 1/u = 1/f, with all values taken with their signs.
• Linear magnification of a mirror is m = h′/h = −v/u.
• A negative m means an inverted image, which for a single mirror is real; a positive m means an erect image, which is virtual.
• A concave mirror forms a real inverted image when the object is beyond the focus, and a virtual, erect, enlarged image when the object lies between the focus and the pole.
• A convex mirror always gives a virtual, erect, diminished image located between the pole and the focus, for any real object.
• For a concave mirror, an object placed at its centre of curvature gives a real inverted image of the same size, also at the centre of curvature.
• Substitute signed values only once, into the formula; do not also insert signs by hand into the equation.

03 · NCERT § "Refraction"

Refraction and apparent depth

• Snell's law: sin i / sin r = n₂₁, the refractive index of medium 2 relative to medium 1; the incident ray, refracted ray and normal lie in one plane.
• n₂₁ = n₂/n₁ = v₁/v₂: light slows down in the optically denser medium (larger n) and bends towards the normal on entering it.
• The refractive index of a medium relative to vacuum is n = c/v, and n₁₂ = 1/n₂₁.
• A ray passing through a parallel-sided slab emerges parallel to its original direction but shifted sideways (lateral shift); the shift grows with slab thickness and angle of incidence.
• An object at the bottom of a water tank viewed from directly above appears raised: real depth / apparent depth = n of water relative to air.
• For several layers stacked one on another, viewed nearly normally, the total apparent depth is the sum of each layer's thickness divided by its own refractive index.
• The frequency of light does not change on refraction; its speed and wavelength change.

04 · NCERT § "Total Internal Reflection"

Total internal reflection

• When light travels from a denser to a rarer medium, the refracted ray bends away from the normal; at the critical angle i_c the refracted ray grazes the surface (r = 90°).
• For incidence greater than i_c there is no refracted ray and all the light is reflected back into the denser medium: total internal reflection.
• Two conditions are needed: light must go from denser to rarer, and the angle of incidence must exceed the critical angle.
• For a denser medium of refractive index n relative to the rarer one, sin i_c = 1/n. For glass of n = 1.5 in air, i_c is about 42°.
• Since n = c/v, the critical angle can also be found from sin i_c = v_denser / v_rarer.
• Optical fibres use total internal reflection: a core is surrounded by cladding of lower refractive index, so light entering the core is reflected again and again along its length with very little loss.
• Bundles of optical fibres (light pipes) are used to transmit and receive signals in telecommunication and to look inside the body in medical examinations.

05 · NCERT § "Refraction at a Spherical Surface"

Refraction at a spherical surface

• For a single spherical surface separating media of refractive indices n₁ (object side) and n₂, with paraxial rays: n₂/v − n₁/u = (n₂ − n₁)/R.
• R is positive when the centre of curvature lies on the side the light goes into (the refracted side), and negative otherwise.
• The formula applies to both convex and concave surfaces, with the sign of R taking care of the shape.
• Setting R → ∞ gives the flat-surface result, v/u = n₂/n₁, which is the apparent-depth relation.
• This result is the building block for the lens maker's formula, obtained by applying it to the two surfaces of a lens in turn.

06 · NCERT § "Refraction by a Lens"

Thin lenses: lens maker's formula and thin lens formula

• Lens maker's formula: 1/f = (n₂₁ − 1)(1/R₁ − 1/R₂), where n₂₁ is the refractive index of the lens material relative to the surrounding medium.
• For a biconvex lens with light from the left, R₁ is positive and R₂ is negative; for a biconcave lens, R₁ is negative and R₂ is positive.
• A convex (converging) lens has positive f and a concave (diverging) lens has negative f in this convention.
• Thin lens formula: 1/v − 1/u = 1/f. Note the minus sign, unlike the mirror equation.
• Linear magnification of a lens is m = h′/h = v/u, again without the minus sign used for mirrors.
• When a lens is placed in a liquid, n₂₁ becomes n_lens/n_liquid, so the focal length changes. If the liquid's n equals the lens's n, the lens has no effect (f becomes infinite); if the liquid's n is larger, a convex lens behaves as a diverging lens.
• A convex lens gives a real inverted image for objects beyond F and a virtual erect enlarged image for objects between F and the lens; a concave lens always gives a virtual, erect, diminished image of a real object.

07 · NCERT § "Power of a Lens"

Power of a lens and combinations of thin lenses

• The power of a lens is the reciprocal of its focal length in metres, P = 1/f. Its SI unit is the dioptre (D), 1 D = 1 m⁻¹.
• A converging lens has positive power; a diverging lens has negative power.
• Two thin lenses in contact behave as one lens with 1/f = 1/f₁ + 1/f₂, so powers add: P = P₁ + P₂ + ….
• The total magnification of a combination is the product of the individual magnifications: m = m₁m₂m₃….
• Lenses are combined in instruments to raise magnification and to reduce certain aberrations; a combination also lets designers make lenses of the required power.
• For lenses separated by a distance, find the image formed by the first lens and use it as the object (real or virtual) for the next lens, measuring each distance from that lens.

08 · NCERT § "Refraction through a Prism"

Refraction through a prism

• For a prism of refracting angle A, the angles of refraction at the two faces satisfy r₁ + r₂ = A, and the deviation is δ = i + e − A.
• As the angle of incidence increases, the deviation first falls, reaches a minimum value D_m, and then rises again.
• At minimum deviation the ray passes symmetrically through the prism: i = e and r₁ = r₂ = A/2; for the usual isosceles or equilateral prism the ray inside then runs parallel to the base.
• Refractive index from minimum deviation: n₂₁ = sin[(A + D_m)/2] / sin(A/2).
• For a thin prism (small A), D_m = (n₂₁ − 1)A, so the deviation does not depend on the angle of incidence for small angles.
• Deviation depends on the refractive index of the prism relative to the surrounding medium; the same prism deviates light less when immersed in a liquid whose refractive index is lower than the prism's.

09 · NCERT § "The Microscope"

Simple and compound microscope

• A simple microscope (magnifier) is a converging lens of short focal length held close to the eye with the object within its focal length, giving a virtual, erect, enlarged image.
• Its angular magnification is m = 1 + D/f if the image forms at the near point (D = 25 cm), and m = D/f for an image at infinity (relaxed eye).
• A compound microscope has an objective of short focal length that forms a real, inverted, enlarged image, which the eyepiece then magnifies further as a simple microscope.
• Total magnification m = m_o × m_e. For the objective, the linear magnification is m_o = L/f_o, where L, the tube length, is measured from the objective's second focal point to the eyepiece's first focal point.
• With the final image at infinity, m = (L/f_o)(D/f_e); with the final image at the near point, the eyepiece contributes (1 + D/f_e).
• The final image in a compound microscope is inverted with respect to the object.
• High magnification needs short focal lengths for both objective and eyepiece.

10 · NCERT § "Telescope"

Telescope

• A refracting (astronomical) telescope has an objective of large focal length and large aperture and an eyepiece of short focal length; it gives angular magnification of distant objects.
• In normal adjustment the final image is at infinity, the magnifying power is m = f_o/f_e, and the tube length (separation of the lenses) is f_o + f_e.
• The final image of an astronomical telescope is inverted.
• A large objective aperture collects more light and improves resolving power, but large lenses are heavy, hard to make free of defects, and can be supported only at their edges.
• Reflecting telescopes use a concave mirror as the objective; this avoids chromatic aberration, and a mirror weighs less and can be supported over its whole back surface.
• In the Cassegrain design the light from the concave primary mirror is reflected by a small convex secondary mirror through a hole in the primary to the eyepiece or detector.
• For a telescope, magnification is about angles, not sizes: the image subtends a larger angle at the eye than the distant object does.

Revise

Must-know facts

1. 01Cartesian sign convention: distances from the pole or optical centre; along incident light positive, against it negative; heights up positive.
2. 02Concave mirror: f and R negative. Convex mirror: f and R positive. Convex lens: f positive. Concave lens: f negative.
3. 03Mirror: 1/v + 1/u = 1/f, m = −v/u, f = R/2.
4. 04Lens: 1/v − 1/u = 1/f, m = v/u.
5. 05Convex mirror and concave lens always give virtual, erect, diminished images of real objects.
6. 06Snell's law: sin i / sin r = n₂₁ = n₂/n₁ = v₁/v₂.
7. 07Frequency is unchanged on refraction; speed and wavelength change.
8. 08Real depth / apparent depth = n (normal viewing).
9. 09Critical angle: sin i_c = 1/n (denser relative to rarer); about 42° for glass of n = 1.5.
10. 10Total internal reflection needs denser-to-rarer travel and i > i_c.
11. 11Optical fibre: core of higher refractive index than the cladding.
12. 12Refraction at a spherical surface: n₂/v − n₁/u = (n₂ − n₁)/R.
13. 13Lens maker's formula: 1/f = (n₂₁ − 1)(1/R₁ − 1/R₂); n₂₁ is relative to the surrounding medium.
14. 14Power P = 1/f (f in metres), unit dioptre; lenses in contact: P = P₁ + P₂.
15. 15Prism: r₁ + r₂ = A; δ = i + e − A; at minimum deviation i = e, r = A/2.
16. 16n = sin[(A + D_m)/2] / sin(A/2); thin prism D_m = (n − 1)A.
17. 17Simple microscope: m = 1 + D/f (near point), D/f (infinity), D = 25 cm.
18. 18Compound microscope (image at infinity): m = (L/f_o)(D/f_e).
19. 19Astronomical telescope (normal adjustment): m = f_o/f_e, length f_o + f_e, final image inverted.
20. 20Reflecting telescopes avoid chromatic aberration; Cassegrain uses a convex secondary mirror.

Where marks go

Common traps

Reference

Formulas

Focal length of a spherical mirror

f = R/2

Paraxial rays; concave: f, R negative; convex: f, R positive.

Mirror equation

1/v + 1/u = 1/f

Cartesian sign convention; distances from the pole.

Magnification by a mirror

m = h′/h = −v/u

m negative: inverted real image; m positive: erect virtual image.

Snell's law

sin i / sin r = n₂₁ = n₂/n₁ = v₁/v₂

n₂₁ is the index of medium 2 relative to medium 1.

Refractive index relative to vacuum

n = c/v

c = 3 × 10⁸ m s⁻¹.

Apparent depth

real depth / apparent depth = n

Viewing near the normal from the rarer medium; for layers, apparent depth = Σ(tᵢ/nᵢ).

Critical angle

sin i_c = 1/n

n is the index of the denser medium relative to the rarer one.

Refraction at a spherical surface

n₂/v − n₁/u = (n₂ − n₁)/R

n₁ on the object side; R positive if the centre of curvature is on the refracted-light side.

Lens maker's formula

1/f = (n₂₁ − 1)(1/R₁ − 1/R₂)

n₂₁ = n_lens/n_medium; biconvex: R₁ > 0, R₂ < 0.

Thin lens formula

1/v − 1/u = 1/f

Convex lens f > 0; concave lens f < 0.

Magnification by a lens

m = h′/h = v/u

No minus sign, unlike mirrors.

Power of a lens

P = 1/f

f in metres; unit dioptre (D = m⁻¹).

Thin lenses in contact

1/f = 1/f₁ + 1/f₂ + …; P = P₁ + P₂ + …; m = m₁m₂…

Use signed focal lengths.

Prism geometry and deviation

r₁ + r₂ = A; δ = i + e − A

A is the refracting angle.

Prism at minimum deviation

n₂₁ = sin[(A + D_m)/2] / sin(A/2)

At D_m: i = e, r₁ = r₂ = A/2.

Thin prism

D_m = (n₂₁ − 1)A

Small refracting angle only.

Simple microscope

m = 1 + D/f (image at near point); m = D/f (image at infinity)

D = 25 cm, least distance of distinct vision.

Compound microscope

m = m_o m_e; m_o = L/f_o; m = (L/f_o)(D/f_e) for final image at infinity

L is between the objective's second focal point and the eyepiece's first focal point; near-point: m_e = 1 + D/f_e.

Astronomical telescope (normal adjustment)

m = f_o/f_e; tube length = f_o + f_e

Final image at infinity and inverted.

Glossary

Key terms

Pole

The centre point of the reflecting surface of a spherical mirror.

Principal axis

The line through the pole (or optical centre) and the centre of curvature.

Paraxial rays

Rays close to and nearly parallel to the principal axis.

Cartesian sign convention

The rule fixing signs of distances by comparing them with the direction of incident light.

Focal length

The distance from the pole or optical centre to the principal focus.

Refractive index

For medium 2 relative to medium 1, the speed of light in medium 1 divided by that in medium 2; relative to vacuum, c/v.

Lateral shift

The sideways displacement of a ray passing through a parallel-sided slab.

Critical angle

The angle of incidence in the denser medium for which the refracted ray grazes the boundary.

Total internal reflection

Complete reflection of light back into a denser medium when incidence exceeds the critical angle.

Optical fibre

A thin strand with a high-index core and lower-index cladding that guides light by repeated total internal reflection.

Power of a lens

The reciprocal of focal length in metres, measuring how strongly a lens converges or diverges light.

Dioptre

The unit of lens power: a lens whose focal length is 1 m has a power of one dioptre.

Angle of minimum deviation

The smallest deviation a prism produces, reached when the ray passes through it symmetrically.

Angular magnification

The ratio of the angle an image subtends at the eye to the angle the object would subtend.

Normal adjustment

Setting of a telescope in which the final image is at infinity.

Cassegrain telescope

A reflecting telescope that uses a concave primary and a convex secondary mirror.