Reflection at Spherical Mirrors and Refraction at Plane Surfaces notes: ISC Class 12 Physics

1. Spherical mirrors, R = 2f and the sign convention

Concave (converging) and convex (diverging) mirrors are parts of a sphere. Pole P, centre of curvature C, radius of curvature R = PC, principal axis through P and C, principal focus F, focal length f = PF. Results hold for paraxial rays (close to the axis, small aperture).
R = 2f: a ray parallel to the axis meets the mirror at M; the normal is CM. The angle of incidence θ equals the angle of reflection θ, and ∠MCF = θ (alternate angles), so triangle FCM is isosceles: FC = FM. For a paraxial ray FM ≈ FP, so F is the midpoint of PC and $f = R/2$ . The focal length does not depend on the surrounding medium.
New Cartesian sign convention (light drawn from left to right): distances measured from P; along the incident light positive, against it negative; heights above the axis positive. A real object has u < 0; concave mirror f < 0; convex mirror f > 0; a real image (in front) has v < 0, a virtual image (behind) v > 0.

Formulas

Focal length and radius of curvature
$f = \dfrac{R}{2}$

Key terms

Paraxial rays: Rays close to the principal axis and making small angles with it; for them all rays from a point meet at one image point.
Principal focus: The point where rays parallel to the axis meet after reflection (concave mirror) or from which they appear to diverge (convex mirror).

2. Mirror formula and magnification

Derivation (concave, object beyond C, real image): ray AM parallel to the axis reflects through F to A′; ray AP reflects at P to A′. MD is perpendicular to the axis. Triangles A′B′F ~ MDF and A′B′P ~ ABP; with MD = AB and D ≈ P (paraxial): A′B′/AB = B′F/FP = B′P/BP. Putting B′P = −v, BP = −u, FP = −f, B′F = f − v gives $(f - v)/(-f) = v/u$ , which rearranges to $\dfrac{1}{v} + \dfrac{1}{u} = \dfrac{1}{f}$ .
Magnification: $m = \dfrac{h'}{h} = -\dfrac{v}{u}$ . m negative: real, inverted; m positive: virtual, erect; |m| > 1 enlarged.
Concave mirror: object beyond C → image between F and C, smaller; at C → at C, same size; between C and F → beyond C, larger; at F → at infinity; inside F → virtual, erect, enlarged, behind the mirror.
Convex mirror: for every real object the image is virtual, erect, diminished and between P and F (1/v = 1/f − 1/u is a sum of positive terms). Wide field of view: rear-view and security mirrors.
Method: write u and f with signs, solve for v, then m = −v/u and h′ = mh; interpret the signs.

concave mirror, f = 12 cm (real images)
|v| = |u|

Object and image positions can be swapped: 18 cm and 36 cm are a conjugate pair.

Formulas

Mirror formula
$\dfrac{1}{v} + \dfrac{1}{u} = \dfrac{1}{f}$
Magnification
$m = \dfrac{h'}{h} = -\dfrac{v}{u}$

Key terms

Linear magnification: Ratio of image height to object height, m = h′/h = −v/u.
Virtual image: An image from which reflected rays only appear to come; it cannot be formed on a screen.

3. Snell’s law, relative refractive index and slabs

Laws of refraction: the incident ray, refracted ray and normal are in one plane; $n_1\sin i = n_2\sin r$ (Snell’s law). Absolute index $n = c/v$ . Frequency unchanged; speed and wavelength divided by n.
Relative index: $_1n_2 = \sin i/\sin r = n_2/n_1 = v_1/v_2$ ; $_2n_1 = 1/{}_1n_2$ .
Several parallel media: $n\sin\theta$ is the same in every layer. For a ray going 1 → 2 → 3 → 1: $_1n_2 \times {}_2n_3 \times {}_3n_1 = 1$ , so $_2n_3 = {}_1n_3/{}_1n_2$ . The emergent ray is parallel to the incident ray when the first and last media are the same.
Parallel-sided slab: e = i; lateral shift $d = t\sin(i - r)/\cos r$ (zero for normal incidence; increases with t, i and n).

slab 6.0 cm thick, n = 1.5

d = t sin(i − r)/cos r grows with i and is zero for normal incidence.

Formulas

Snell’s law
$n_1\sin i = n_2\sin r$
Three media
${}_1n_2\,{}_2n_3\,{}_3n_1 = 1$
Lateral shift
$d = \dfrac{t\sin(i - r)}{\cos r}$

Key terms

Absolute refractive index: Speed of light in vacuum divided by its speed in the medium, n = c/v.
Lateral shift: Perpendicular distance between the incident ray produced and the emergent ray of a parallel-sided slab.

4. Real and apparent depth

An object under a liquid seen from above (near-normal rays) appears raised: $n = \dfrac{\text{real depth}}{\text{apparent depth}}$ (from $n = \sin r/\sin i \approx \tan r/\tan i$ ).
Apparent shift through a slab or liquid layer of thickness t: $s = t(1 - 1/n)$ , independent of the object’s distance below the slab.
Several layers: apparent depth = $t_1/n_1 + t_2/n_2 + \dots$
Seen obliquely, the bottom looks even shallower. An object in air seen from under water appears higher (apparent height = n × real height).
Travelling microscope: n = real thickness ÷ apparent thickness of a slab or liquid.

Formulas

Apparent depth
$n = \dfrac{\text{real depth}}{\text{apparent depth}}$
Apparent shift
$s = t\left(1 - \dfrac{1}{n}\right)$

Key terms

Apparent depth: The depth at which an object under a refracting medium appears to lie when viewed from above.
Apparent shift: The distance an object appears to be raised by a slab or layer: t(1 − 1/n).

5. Total internal reflection, prisms and fibres

TIR happens when light travels from a denser to a rarer medium and meets the boundary at more than the critical angle C. At i = C the refracted ray grazes the surface (r = 90°).
$n\sin C = 1$ , so $\sin C = 1/n$ ; between two media $\sin C = n_2/n_1$ ( $n_1 > n_2$ ). Water 48.8°, crown glass about 41°, diamond 24.4°.
Total reflecting prisms: a 45°–90°–45° glass prism turns a ray through 90° (one TIR at 45°) or 180° (two TIRs, ray entering the hypotenuse); in a 30°–60°–90° prism an incidence of 60° is total for n > 1.155, while 30° would need n > 2. Prisms reflect almost 100%, do not tarnish and give no multiple images. In water the critical angle rises and a prism may stop working.
Optical fibre: glass core (n₁) inside cladding (n₂ < n₁); light meeting the wall at more than C (sin C = n₂/n₁) is guided by repeated TIR. Uses: communication, endoscopy, decorative lamps; low loss, high capacity, no electrical interference.

glass (n = 1.5) to air

Beyond the critical angle the light is totally reflected.

Formulas

Critical angle
$\sin C = \dfrac{1}{n} = \dfrac{n_2}{n_1}$

Key terms

Critical angle: Angle of incidence in the denser medium for which the angle of refraction is 90°; sin C = n₂/n₁.
Total internal reflection: Complete reflection of light at a boundary with a rarer medium when the angle of incidence exceeds the critical angle.

Exam tips

Mirrors: signs first (u < 0 for a real object; concave f < 0), then 1/v + 1/u = 1/f and m = −v/u; read the nature of the image from the signs.
Layers: n sin θ is constant; product of relative indices round a closed set of media is 1.
Apparent depth needs near-normal viewing; shift t(1 − 1/n) is independent of the object’s position.
TIR: denser to rarer and i > C; compare angles inside the glass with C for the actual surroundings.

1. Spherical mirrors, R = 2f and the sign convention

Formulas

Key terms

2. Mirror formula and magnification

Formulas

Key terms

3. Snell’s law, relative refractive index and slabs

Formulas

Key terms

4. Real and apparent depth

Formulas

Key terms

5. Total internal reflection, prisms and fibres

Formulas

Key terms

Exam tips

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