Ray Optics: theory | Physics Circle

CONCEPT

Reflection & Mirrors

In one line
Light falling on a polished surface comes back into the same medium. This is reflection.

LAWS

9.1.1 Laws of reflection

∠i = ∠r

Angle of incidence = angle of reflection.
Both angles are measured from the normal.
Incident ray, normal and reflected ray lie in one plane.
Speed, wavelength and frequency do not change.

i and r are measured from the normal

PLANE MIRROR

9.1.2 Image in a plane mirror

Image is virtual and erect.
Image is of the same size: m = +1.
Image is laterally inverted (left ↔ right).
Image is as far behind the mirror as the object is in front.

Full image of a person needs a mirror of half his height.
Mirror turned through θ ⇒ reflected ray turns through 2θ.

Image distance = object distance

INCLINED MIRRORS

9.1.3 Number of images

n = 360°θ − 1

360°θ	Object	Images n
even	anywhere	360°θ − 1
odd	on the bisector	360°θ − 1
odd	not on the bisector	360°θ

Mirrors at 90° ⇒ 3 images.
Mirrors at 60° ⇒ 5 images.

Mirrors at 90°: three images

SPHERICAL MIRRORS

9.1.4 Concave and convex

Parallel rays meet at F (concave) or seem to come from F (convex)

f = R2

Concave mirror converges light.
Uses of concave: shaving mirror, headlight reflector, telescope.
Convex mirror diverges light.
Convex always gives an erect, diminished image.
Use of convex: rear-view mirror (wide field of view).

SIGN CONVENTION

9.1.5 New Cartesian

Measure all distances from the pole P.
Along the incident light: +. Against it: −.
Heights above the axis: +. Below: −.
Real object: u is always negative.
Concave mirror: f negative. Convex mirror: f positive.

MIRROR FORMULA

9.1.6 Linking u, v and f

1v + 1u = 1f

m = h_ih_o = −vu

m negative ⇒ real, inverted image.
m positive ⇒ virtual, erect image.
Concave mirror, real image n times the object:

|u| = f(n + 1n) · |v| = f(n + 1)

IMAGE FORMATION

9.1.7 Where is the image?

Object at	Image at	Nature and size
infinity	F	real, inverted, point-sized
beyond C	between F and C	real, inverted, diminished
C	C	real, inverted, same size
between C and F	beyond C	real, inverted, enlarged
F	infinity	real, inverted, very large
between F and P	behind the mirror	virtual, erect, enlarged
convex: anywhere	between P and F	virtual, erect, diminished

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Real and virtual objects

Real and virtual objects and images

Vector form of reflection

r̂ = î − 2(î · n̂) n̂

Component along the normal reverses; component along the mirror stays the same.

Moving object, moving mirror

Image speed: u for a moving object, 2u for a moving mirror

v_image = 2v_mirror − v_object

All velocities measured along the normal.
Component along the mirror is the same for object and image.

Mirror to see the whole wall behind

Person in the middle of the room

L_min = H3

Speed of image (spherical mirror)

v_image = −(vu)² v_object

Along the axis the image moves opposite to the object.
Perpendicular to the axis: v_image = m v_object.

Axial and areal magnification

m_L = −v₂ − v₁u₂ − u₁ · small object: m_L = −m²

areal magnification m_A = m²

Newton’s formula and graphs

x₁x₂ = f²

x₁, x₂ are measured from F.
1v against 1u: straight line of slope −1.
u against v (real images): rectangular hyperbola.
A mirror dipped in a liquid keeps the same f.

Practise PYQs on this topicReflection & Mirrors · 54 questions with solutions

CONCEPT

Refraction, Glass Slab & Apparent Depth

In one line
Light bends when it goes from one medium into another, because its speed changes. This is refraction.

REFRACTION

9.2.1 Rarer and denser media

Light is faster in an optically rarer medium.
Light is slower in an optically denser medium.
Bigger difference in speeds ⇒ more bending.
Rarer → denser: bends towards the normal.
Denser → rarer: bends away from the normal.

Deviation on refraction at a plane surface

SNELL’S LAW

9.2.2 The law of refraction

sin isin r = ₁μ₂ = μ₂μ₁

μ₁ sin i = μ₂ sin r

Incident ray, refracted ray and normal lie in one plane.
μ sin θ stays the same across any number of parallel boundaries.

REFRACTIVE INDEX

9.2.3 How much a medium slows light

μ = cv

μ is a scalar: no unit, no dimension.
Smallest value is 1 (vacuum); air ≈ 1.

Medium	μ
Glass	1.5 (32)
Water	1.33 (43)
Diamond	2.4 (125)
Air	≈ 1.0003

₁μ₂ = μ₂μ₁ = v₁v₂ = λ₁λ₂

Frequency does not change on refraction.
Speed and wavelength both fall by the factor μ.
So λ_medium = λμ.

DEPENDENCE OF μ

9.2.4 What μ depends on

μ = A + Bλ² (Cauchy)

Nature of the two media.
Colour of light: λ_red > λ_violet ⇒ μ_red < μ_violet.
Temperature: μ decreases as temperature rises.

REVERSIBILITY

9.2.5 Light can retrace its path

₁μ₂ = 1₂μ₁

₁μ₂ × ₂μ₃ × ₃μ₁ = 1

Light through several layers

APPARENT DEPTH

9.2.6 Object in the denser medium, observer in the rarer

The bottom seems raised

μ = real depthapparent depth = hh′

shift d = h − h′ = h(1 − 1μ)

Real depth > apparent depth.
Water (μ = 43): shift = h4.
Glass (μ = 32): shift = h3.

SEVERAL LAYERS

9.2.7 Immiscible liquids in a vessel

apparent depth = d₁μ₁ + d₂μ₂ + d₃μ₃ + …

μ_combination = d₁ + d₂ + …d₁μ₁ + d₂μ₂ + …

Two liquids of equal depth: μ = 2μ₁μ₂μ₁ + μ₂.

Each layer adds dμ

OBJECT IN THE RARER MEDIUM

9.2.8 Observer in the denser medium

μ = h′h

Object seems farther: apparent > real.
Shift d = (μ − 1)h.
Water: h3. Glass: h2.
Example: a fish sees a bird higher than it is.

Object in air seen from water

GLASS SLAB

9.2.9 Lateral shift and normal shift

Emergent ray ∥ incident ray

lateral shift d = t sin(i − r)cos r

The two faces are parallel.
Emergent ray ∥ incident ray (e = i).
Net deviation is zero; only a sideways shift.

A slab moves the point of convergence away

normal shift x = t(1 − 1μ)

An ink dot under a slab appears raised by x.
A converging beam meets x farther away.

OPTICAL PATH

9.2.10 Equivalent distance in vacuum

optical path = μx

t = μxc

Distance light would cover in vacuum in the same time.
Number of waves in thickness t: N = μtλ

Path x inside a medium μ

BREWSTER’S ANGLE

9.2.11 Reflected and refracted rays at 90°

tan i_p = μ

i_p + r = 90°

At i_p the reflected light is completely plane polarised.
Reflected ray ⊥ refracted ray.

At i_p, reflected ⊥ refracted

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Lateral magnification inside a medium

Object of height x inside medium μ₁, seen from medium μ₂ along the normal.

image length = μ₂μ₁ x

Apparent speed

An object moving at v inside water (seen from air) appears to move at vμ.
A bird diving at v (seen by a fish) appears to move at μv.

Practise PYQs on this topicRefraction, Glass Slab & Apparent Depth · 87 questions with solutions

CONCEPT

Total Internal Reflection

In one line
Light going from a denser to a rarer medium at an angle greater than the critical angle is reflected back completely.

CRITICAL ANGLE

9.3.1 When the refracted ray grazes the surface

Three rays from a source under water

As i increases (denser → rarer), r increases.
At i = C, the refracted ray grazes the surface (r = 90°).
For i > C, no light is refracted: all of it is reflected.

sin C = 1μ = μ_rarerμ_denser

sin C = v_denserv_rarer = λ_denserλ_rarer

CONDITIONS

9.3.2 For total internal reflection

Light must travel from a denser to a rarer medium.
Angle of incidence must be greater than C.

DEPENDENCE OF C

9.3.3 What the critical angle depends on

Colour: μ_violet > μ_red ⇒ C_violet < C_red.
Pair of media: larger ratio of indices ⇒ smaller C.
Temperature: μ falls as temperature rises ⇒ C increases.

Pair of media	C
glass → air	42°
water → air	49°
diamond → air	24°

APPLICATIONS

9.3.4 Where TIR is used

Brilliance of diamond: small C ⇒ light is trapped by repeated TIR.
Mirage (deserts) and looming (cold countries): TIR in air layers of different density.
Totally reflecting prism (45°–90°–45°, μ ≥ √2): turns light through 90° or 180° — periscope, binoculars.

Totally reflecting prisms

OPTICAL FIBRE

9.3.5 Light pipe

Light guided by repeated TIR

Each fibre has a core and a cladding.
μ_core > μ_cladding.
Light inside meets the wall at more than C ⇒ repeated TIR.
Works even when the fibre is bent.
Used in medical endoscopy and in communication.

CIRCLE OF LIGHT

9.3.6 A source or an eye under water

Light leaves only through a circle of radius r

r = h tan C = h√(μ² − 1)

A source at depth h lights only a circle of radius r on the surface.
A disc of radius r on the surface cuts off all the light.
A fish sees the whole outside world inside a cone of half-angle C.

Practise PYQs on this topicTotal Internal Reflection · 64 questions with solutions

CONCEPT

Spherical Surfaces & Lens Maker’s Formula

In one line
A lens is a transparent medium bounded by two refracting surfaces, at least one of them curved. Each surface refracts light on its own.

REFRACTION AT A CURVED SURFACE

9.4.1 One spherical surface

Refraction at a convex surface

μ₂v − μ₁u = μ₂ − μ₁R

Light goes from medium μ₁ into medium μ₂.
u, v and R are measured from the pole P.
R is + when the centre of curvature is on the side light goes to.

TYPES OF LENSES

9.4.2 Convex and concave

1 biconvex · 2 plano-convex · 3 concavo-convex · 4 biconcave · 5 plano-concave · 6 convexo-concave

Convex lenses are thicker at the middle; they converge light (in air).
Concave lenses are thinner at the middle; they diverge light (in air).
A ray through the optical centre goes on undeviated.

Convex: rays meet at F. Concave: rays seem to come from F

Focus F: where rays parallel to the axis meet (convex) or seem to come from (concave).
f is + for a convex lens and − for a concave lens.

LENS MAKER’S FORMULA

9.4.3 f from shape and material

R₁ and R₂ of the two surfaces

1f = (μ − 1)(1R₁ − 1R₂)

Biconvex lens: R₁ = +R, R₂ = −R.
f depends on μ, so it is different for each colour.

Lens	f	f for μ = 1.5
equiconvex (R, R)	R2(μ − 1)	R
plano-convex	Rμ − 1	2R
equiconcave	−R2(μ − 1)	−R
plano-concave	−Rμ − 1	−2R

Grinding one face of an equiconvex lens flat doubles f and halves P.

LENS IN A LIQUID

9.4.4 Only the relative index matters

1f_l = (μμ_l − 1)(1R₁ − 1R₂)

f_lf_a = μ − 1μμ_l − 1

A convex lens in three liquids

μ_l < μ: still converging, but f increases (power falls).
μ_l = μ: f = ∞; the lens becomes invisible.
μ_l > μ: nature reverses; convex lens diverges.
An air bubble (convex shape) in water acts as a diverging lens.

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Magnification at a curved surface

m = μ₁vμ₂u

Practise PYQs on this topicSpherical Surfaces & Thin Lenses · 112 questions with solutions

CONCEPT

Thin Lens Formula, Magnification & Power

In one line
For a thin lens, the object distance u, image distance v and focal length f are linked by the lens formula.

LENS FORMULA

9.5.1 Linking u, v and f

1v − 1u = 1f

m = h_ih_o = vu = ff + u = f − vf

Use the sign convention: u negative for a real object.
m negative ⇒ real, inverted image.
m positive ⇒ virtual, erect image.

POWER

9.5.2 Converging or diverging ability

P = 1f (m) = 100f (cm) dioptre

Convex lens: P positive. Concave lens: P negative.
Small change: Δff = −ΔPP.

CONVEX LENS

9.5.3 Where is the image?

Object at	Image at	Nature and size
infinity	F	real, inverted, point-sized
beyond 2F	between F and 2F	real, inverted, diminished
2F	2F	real, inverted, same size
between F and 2F	beyond 2F	real, inverted, enlarged
F	infinity	real, inverted, very large
between F and O	same side as object	virtual, erect, enlarged

CONCAVE LENS

9.5.4 Always virtual

Image is always virtual, erect, diminished.
Image lies between the optical centre and F.
Object at F ⇒ image at f2, m = 12.

USEFUL RESULTS

9.5.5 Asked often

real image n times: |u| = f(n + 1n), v = f(n + 1)

Minimum distance between object and its real image = 4f (u = v = 2f).
Equal-size image on a screen: lens midway, u = v = 2f.
Covering part of the lens keeps the whole image; only brightness falls.

DISPLACEMENT METHOD

9.5.6 Two positions, two sharp images

f = D² − x²4D · O = √(I₁I₂)

Object and screen are D apart (D > 4f).
Two lens positions x apart give sharp images I₁ and I₂.

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Axial and areal magnification (lens)

m_L = (vu)² = (ff + u)² · m_A = m²

Speed of image (lens)

v_image = (vu)² v_object = (ff + u)² v_object

An object moving from ∞ to F towards a convex lens: image moves slowly at first, then very fast.

Newton’s formula (lens)

f² = x₁x₂

x₁, x₂ are measured from the first and second principal foci.

Practise PYQs on this topicSpherical Surfaces & Thin Lenses · 112 questions with solutions

CONCEPT

Combination of Lenses

In one line
Thin lenses in contact simply add their powers.

LENSES IN CONTACT

9.6.1 Add the powers

1F = 1f₁ + 1f₂ + … · P = P₁ + P₂ + …

F = f₁f₂f₁ + f₂ · m = m₁ × m₂ × …

Equal convex and concave lenses in contact: P = 0, F = ∞ (acts like a glass plate).
The combination is diverging if the concave lens is stronger.
n identical lenses in contact: P = nP₁.

SEPARATED LENSES

9.6.2 Distance d apart

Convex and concave lenses d apart

1F = 1f₁ + 1f₂ − df₁f₂

P = P₁ + P₂ − dP₁P₂

Use d in metre when P is in dioptre.
Parallel beam in, parallel beam out: d = f₁ + f₂ (f₂ negative for concave).

CUTTING A LENS

9.6.3 What happens to f

Two ways of cutting a biconvex lens

Cut along the axis: each half has the same f; image intensity falls.
Cut perpendicular to the axis: each part is plano-convex, f doubles, P halves.
The two plano-convex parts put back together give f again.

JOINING LENSES

9.6.4 Lenses fitted together

plano-convex (n₁) in plano-concave (n₂): F = Rn₁ − n₂

Two plano-convex lenses of μ₁ and μ₂ (radius R) make a biconvex lens: 1F = μ₁ − 1R + μ₂ − 1R.
Liquid filling the gap between two lenses acts as a third lens (concave if the gap is biconcave).

SILVERED LENS

9.6.5 In short

1F = 2f_l + 1f_m

Plano-convex silvered on the plane face: F = f2.
Silvered on the curved face: F = R2μ.

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Silvered lens

A silvered lens behaves as a mirror

1F = 2f_l + 1f_m (P = 2P_l + P_m)

Plano-convex, plane face silvered: F = R2(μ − 1).
Plano-convex, curved face silvered: F = R2μ.
Biconvex (equal R), one face silvered: F = R2(2μ − 1).

Lens–mirror systems

Find the lens image first; it is the object for the mirror.
Light returns through the lens: apply the lens formula again.
Image coincides with the object when rays hit the mirror normally: at its centre of curvature (curved mirror) or parallel to the axis (plane mirror at the lens focus).

Practise PYQs on this topicCombination of Lenses · 82 questions with solutions

CONCEPT

Prism

In one line
A prism bends light towards its base at both faces. The deviation is least when the ray passes symmetrically.

REFRACTION THROUGH A PRISM

9.7.1 Angles in a prism

Minimum deviation: ray inside parallel to the base

r₁ + r₂ = A

δ = i + e − A

A = angle of prism, i = angle of incidence, e = angle of emergence.
r₁, r₂ = angles of refraction at the two faces.

MINIMUM DEVIATION

9.7.2 Symmetric path

μ = sin(A + δ_m2)sin(A2)

At minimum deviation i = e and r₁ = r₂ = A2.
In an equilateral prism the ray inside is parallel to the base.
δ_m = 2i − A.
The same δ occurs for two angles: i and e can be swapped.

δ–i graph: one minimum at i = e

Condition	Result
δ_m = A	μ = 2 cos(A2), so √2 < μ < 2
μ = cot(A2)	δ_m = 180° − 2A

THIN PRISM

9.7.3 Small angle A

δ = (μ − 1)A

in a liquid: δ_l = (μμ_l − 1)A

Glass prism (μ = 1.5) in water: δ falls to one-fourth of its value in air.
Hollow prism filled with liquid: μ_l = 1 + δA.

NORMAL INCIDENCE AND EMERGENCE

9.7.4 One face at 0°

Case	Condition	Result
normal incidence	i = 0, r₁ = 0	r₂ = A, sin e = μ sin A
normal emergence	e = 0, r₂ = 0	r₁ = A, sin i = μ sin A
thin prism, normal emergence	—	i = μA

GRAZING AND NO EMERGENCE

9.7.5 TIR at the second face

Grazing emergence

Grazing emergence: r₂ = C, so r₁ = A − C and sin i = μ sin(A − C).
No emergence for any i when A > 2C.
With normal incidence, no emergence when μ > cosec A.

SILVERED PRISM

9.7.6 Ray retraces its path

Ray returns along the same path

The ray must strike the silvered face normally, so r₁ = A.
μ = sin isin A.
If i = 2A, then μ = 2 cos A.

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Maximum deviation

Grazing incidence

i = 90° ⇒ r₁ = C, r₂ = A − C.

e = sin⁻¹[sin(A − C)sin C]

δ_max = π2 + e − A

Practise PYQs on this topicPrism, Dispersion & Scattering · 96 questions with solutions

CONCEPT

Dispersion, Aberration & Scattering

In one line
White light splits into colours because μ depends on wavelength: violet bends most, red least.

DISPERSION

9.8.1 Splitting of white light

Violet deviates more than red

μ_violet > μ_red ⇒ δ_violet > δ_red.
Speed in glass: v_violet < v_red.
Same angle of incidence ⇒ different angles of refraction for each colour.

angular dispersion θ = δ_V − δ_R = (μ_V − μ_R)A

dispersive power ω = δ_V − δ_Rδ = μ_V − μ_Rμ − 1

Mean μ = μ_V + μ_R2.
ω depends only on the material, not on A.

COMBINATION OF PRISMS

9.8.2 Crown and flint prisms

Two prisms placed opposite to each other

Aim	Condition
dispersion without deviation	(μ₁ − 1)A₁ = (μ₂ − 1)A₂
deviation without dispersion (achromatic)	ω₁δ₁ + ω₂δ₂ = 0

CHROMATIC ABERRATION

9.8.3 Colours focus at different points

f_V < f_R

μ_V > μ_R ⇒ f_V < f_R (convex lens).
Image of a white object is coloured and blurred.
Removed by an achromatic doublet: convex crown + concave flint.

achromatic doublet: ω₁f₁ + ω₂f₂ = 0 ⇒ ω₁P₁ = −ω₂P₂

SPHERICAL ABERRATION

9.8.4 Point object, blurred image

Marginal and paraxial rays

Marginal rays focus nearer the lens than paraxial rays.
The smallest blur patch is the circle of least confusion.
Reduced by stops, or by a plano-convex lens.

RAINBOW

9.8.5 Dispersion in rain drops

	Primary	Secondary
inside a drop	2 refractions + 1 TIR	2 refractions + 2 TIR
angle at the eye	≈ 42°	≈ 51° (above primary)
colours	red outside, violet inside	reversed, fainter

A rainbow is seen with the sun behind the observer.
Interference plays no part.

SCATTERING

9.8.6 Why the sky is blue

intensity of scattered light ∝ 1λ⁴

Blue is scattered most ⇒ the sky looks blue.
At sunrise and sunset the sun looks red.
Clouds (large drops) scatter all colours equally ⇒ white.
Danger signals are red: least scattered.

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Other defects of images

Coma: a point object off the axis gives a comet-shaped image.
Curvature: best image lies on a curved surface, not a plane.
Distortion: magnification differs across the object; straight lines look curved.
Astigmatism: image of an off-axis point spreads along the axis.
Chromatic aberration: f_R − f_V = ωf, where f = √(f_Rf_V).

Human Eye & Microscopes

In one line
A microscope makes a small near object look bigger by increasing the angle it subtends at the eye.

HUMAN EYE

9.9.1 In short

Eye lens forms a real, inverted image on the retina.
Ciliary muscles change its focal length (accommodation).
Normal eye: near point D = 25 cm, far point ∞.
Persistence of vision ≈ 110 s.

Defect	Correction
myopia (far point closer than ∞)	concave lens, f = −(far point)
hypermetropia (near point beyond 25 cm)	convex lens
presbyopia (old age)	bifocal lens
astigmatism	cylindrical lens

SIMPLE MICROSCOPE

9.9.2 One convex lens of short focal length

Object inside F: virtual, erect, enlarged image

image at D: M = 1 + Df

image at ∞ (relaxed eye): M = Df

Also called a magnifying glass or reading lens.
Smaller f ⇒ larger M.
f_red > f_blue ⇒ M is smaller in red light.

COMPOUND MICROSCOPE

9.9.3 Objective + eyepiece

Final image A″B″: virtual, inverted

Objective: small f and small aperture.
Eyepiece: larger f and larger aperture.
Objective gives a real, inverted, enlarged image.
Final image is virtual, inverted, enlarged.

M = m_o × m_e

image at D: M = −v_ou_o(1 + Df_e), L = v_o + u_e

image at ∞: M = −v_ou_o · Df_e, L = v_o + f_e

M ≈ Lf_o · Df_e

Larger M needs small f_o and small f_e.
Longer tube ⇒ larger M.

RESOLVING POWER

9.9.4 Of a microscope

RP = 2μ sin θλ

Use shorter λ or oil between object and objective (larger μ) to raise RP.

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Parts of the eye

Human eye

Eye lens behaves as a convex lens (μ ≈ 1.437).
Yellow spot: most sensitive; blind spot: no sensitivity (optic nerve).
Resolving limit of the eye ≈ 1′ = (160)°.
Binocular vision: seeing with two eyes gives depth.

Power of correcting lenses

Myopia, far point d: concave lens, f = −d.
Can see up to x, wants to see up to y: f = xyx − y.
Hypermetropia, near point d: convex lens, f = dDd − D, P = d − DdD.

Camera

f-number = focal lengthaperture

Time of exposure t ∝ (f-number)²; for fixed aperture, I t = constant.
Larger f-number ⇒ smaller aperture ⇒ longer exposure, larger depth of focus.

Simple microscope held at distance a from the eye

m_D = 1 + D − af · m_∞ = D − af

Practise PYQs on this topicOptical Instruments & the Eye · 55 questions with solutions

CONCEPT

Telescopes & Resolving Power

In one line
A telescope makes a distant object subtend a larger angle at the eye: it looks nearer, not bigger.

ASTRONOMICAL TELESCOPE

9.10.1 Refracting type

Normal adjustment: final image at ∞

Objective: large f and large aperture.
Eyepiece: small f and small aperture.
Final image is virtual, inverted and small.

normal adjustment: M = −f_of_e, L = f_o + f_e

image at D: M = −f_of_e(1 + f_eD), L = f_o + u_e

High M ⇒ large f_o, small f_e.
Doubling f_e halves M.

OTHER TELESCOPES

9.10.2 Erect images

Telescope	Length	Image
astronomical	f_o + f_e	inverted
terrestrial (erecting lens of focal length f)	f_o + 4f + f_e	erect
Galilean (concave eyepiece)	f_o − f_e	erect

REFLECTING TELESCOPE

9.10.3 Concave mirror as objective

Light reflected to an eyepiece by a secondary mirror

M = f_of_e

No chromatic aberration (mirrors reflect all colours alike).
Parabolic mirror ⇒ no spherical aberration.
A large mirror is easier to make and support than a large lens.
The secondary mirror sends light out of the tube to the eyepiece.

RESOLVING POWER

9.10.4 Seeing two close stars as two

dθ = 1.22λa · RP = 1dθ = a1.22λ

a = diameter (aperture) of the objective.
Large aperture ⇒ higher resolving power and brighter image.
RP does not depend on the focal length.
RP ∝ 1λ.

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Terrestrial and Galilean telescopes

Telescope	M (normal)	M (image at D)	Length
terrestrial	f_of_e	f_of_e(1 + f_eD)	f_o + 4f + f_e
Galilean	f_of_e	f_of_e(1 − f_eD)	f_o − f_e

Brightness and binoculars

brightness ratio β = D²d²

D = diameter of the objective, d = diameter of the eye pupil.
Binocular: two telescopes side by side with totally reflecting prisms; erect image, shorter tube, 3D view.

Photometry

Radiant flux R: total energy given out per second (W).
Luminous flux φ: visible light given out per second (lumen).
Luminous efficiency η = φR.

luminous intensity L = φω (candela) · φ = 4πL

illuminance I = φA (lux)

point source: I = Lr² · line source: I ∝ 1r

Lambert’s cosine law: I = L cos θr²

photometer: L₁L₂ = (r₁r₂)²

Practise PYQs on this topicOptical Instruments & the Eye · 55 questions with solutions

EXAM-DAY CARD

Just see this before the exam

1v + 1u = 1f

mirror

μ = cv = sin isin r

Snell

sin C = 1μ

TIR

1v − 1u = 1f

thin lens

δ = (μ − 1)A

thin prism

M = f_of_e

telescope

FORMULA SHEET

Whole chapter on one page

Mirrors

i = r, δ = 180° − 2i

n = 360°θ − 1

f = R2, 1v + 1u = 1f

m = −vu

Refraction

μ = cv, μ₁ sin i = μ₂ sin r

d = t sin(i − r)cos r

h_app = Σ tμ

x = t(1 − 1μ)

TIR

sin C = 1μ

denser → rarer, i > C

r = h√(μ² − 1)

tan i_p = μ

Lenses

μ₂v − μ₁u = μ₂ − μ₁R

1f = (μ − 1)(1R₁ − 1R₂)

1v − 1u = 1f, m = vu

P = P₁ + P₂ − dP₁P₂

Prism & colours

A = r₁ + r₂, δ = i + e − A

μ = sin(A + δ_m2)sin(A2)

δ = (μ − 1)A

ω = μ_v − μ_rμ − 1

Instruments

simple: M = 1 + Df

compound: M ≈ (Lf_o)(Df_e)

telescope: M = f_of_e

dθ = 1.22λD

TOP MISTAKES

Avoid these = easy marks

MISTAKE ✗

Dropping the signs of u, v, f

Lens in liquid: using μ instead of μμ_l

TIR from rarer to denser

Cut lens: f always halves

CORRECT ✓

Measure from P; along light: +

1f_l = (μμ_l − 1)(1R₁ − 1R₂)

TIR only denser → rarer, i > C

Along axis: same f; across: 2f