Lesson 11 of 11 · 20 min

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Must-know facts

20 facts

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

Common traps

Where marks are lost

Using the lens formula for a mirror, or the mirror formula for a lens.

Mirror: 1/v + 1/u = 1/f with m = −v/u. Lens: 1/v − 1/u = 1/f with m = v/u. Plus sign for mirrors, minus sign for lenses.

Entering the sign of a quantity twice, once in the formula and once when substituting.

Keep the formula in its standard form and put the sign only in the numbers: a concave mirror of focal length 20 cm has f = −20 cm.

Taking the refractive index in the lens maker's formula relative to vacuum when the lens sits in a liquid.

Use n_lens/n_medium. If the medium is denser than the lens, (n₂₁ − 1) is negative and a convex lens becomes diverging.

Adding focal lengths of lenses in contact instead of powers.

Powers add: P = P₁ + P₂, which means 1/f = 1/f₁ + 1/f₂. Give the diverging lens a negative focal length before adding.

Expecting total internal reflection when light goes from air into glass at a large angle.

TIR happens only when light goes from the denser to the rarer medium and the incidence exceeds the critical angle.

Using the thin-prism formula D_m = (n − 1)A for a 60° prism.

For a prism of large angle use n = sin[(A + D_m)/2] / sin(A/2); the thin-prism form is only for small A.

Mixing up the near-point and infinity formulas for microscopes.

Near point: eyepiece or magnifier gives 1 + D/f. Infinity (relaxed eye): D/f. Read which adjustment the question asks for.

Taking the tube length of a compound microscope as the separation of objective and eyepiece in m = (L/f_o)(D/f_e).

In NCERT's formula, L is measured from the objective's second focal point to the eyepiece's first focal point. If only object distance is given, find m_o = v/u of the objective directly.

Adding thicknesses and using a single refractive index when a coin lies under two liquid layers.

Apparent depth is found layer by layer: Σ(thickness ÷ refractive index of that layer).

Thinking the magnifying power of a telescope is f_e/f_o or that a long eyepiece focal length helps.

In normal adjustment m = f_o/f_e: long-focus objective, short-focus eyepiece.

Formulas

19 to know

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.

Key terms

16 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.

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