Ray optics is the chapter where physics finally feels like something you can see. A mirror in the bathroom, the bend of a straw in a glass of nimbu paani, the fat lens in your grandmother’s reading glasses, the thin fibre carrying your Wi-Fi signal — all of it is in this chapter. And yet it is also the chapter where the largest number of students lose marks for the silliest reason: a sign. Not a concept, not a formula — a plus that should have been a minus.
So we are going to do something a little different here. Instead of racing through ray diagrams, we are going to build one habit and repeat it until it is boring: a four-step sign ritual that you perform identically on every single problem, whether it is a concave mirror, a prism or an astronomical telescope. Every worked example below uses the same four steps in the same order. By example twelve you will not be thinking about signs any more — your hand will do it for you.
This page is your full set of ray optics and optical instruments class 12 physics notes with solved examples — eighteen fully worked problems, a one-page formula list, a practice worksheet with answers, and a straight note on which topics have been trimmed from the 2026-27 syllabus so you do not waste a single evening on deleted material. Take it slowly. Nobody understands ray optics on the first read, and that is completely normal.
Meet Your Tutor
Ray Optics rewards a calm diagram and a disciplined sign convention more than fast formula recall. I will guide you through each mirror, lens, prism and instrument problem in the same repeatable order: draw, sign, substitute and interpret. Keep a pencil nearby and redraw the small ray sketches as you go.
What You’ll Learn
Jump straight to what you need
- The Four-Step Sign Discipline (Your Solving Ritual)
- The Cartesian Sign Convention, Set Up Once and For All
- Reflection at Spherical Mirrors: Mirror Formula and Magnification
- Reading the v–u Graph: What the Formula Actually Looks Like
- Refraction of Light, Snell’s Law and Apparent Depth
- Total Internal Reflection and Optical Fibres
- Refraction at a Spherical Surface
- Thin Lenses: Lens Formula, Lens Maker’s Formula and Power
- Combination of Thin Lenses in Contact
- Refraction of Light Through a Prism
- The Simple Microscope (Magnifying Glass)
- The Compound Microscope and Its Magnifying Power
- The Astronomical Telescope: Refracting and Reflecting
- Real or Virtual? The Decision Flow
- Ray Optics Class 12 Physics Formula List (One-Page Revision)
- Check the Current 2026-27 Scope
- Ray Optics Class 12 Physics Important Questions — How They Are Set
- Practice Worksheet with Answers
Your Game Plan
- Day 1 — Learn the ritual, not the formulas. Read the first two sections and copy the sign-convention card into your notebook by hand. Do not move on until you can state the four steps from memory.
- Day 2 — Mirrors. Work Examples 1 to 4 with the book closed, then check. Redo any you got wrong the same evening, not tomorrow.
- Day 3 — Refraction, total internal reflection, spherical surfaces. Examples 5 to 9.
- Day 4 — Lenses and prisms. Examples 10 to 15. This is the highest-yield day in the chapter.
- Day 5 — Optical instruments. Examples 16 to 18, then the formula list.
- Day 6 — The worksheet, timed. Forty minutes, no notes. Then mark honestly and list only the mistakes you made twice.
Study Notes
The Four-Step Sign Discipline (Your Solving Ritual)
Here is the honest truth about this chapter: the physics is not hard. The bookkeeping is. A mirror problem and a lens problem look almost identical on paper, and the only thing standing between you and full marks is whether you kept track of which side of the mirror or lens each distance was measured on. So we build a ritual. Four steps. Same four, every time.
2. LABEL with signs. Write every given quantity next to its position on the axis, with its sign already attached. Object 30 cm to the left becomes u = −30 cm, not “30 cm”. Do this before you touch a formula.
3. SUBSTITUTE, signs intact. Put the signed numbers into the formula exactly as written. Do not “fix” a minus because the answer looks odd. The formula is smarter than your intuition.
4. SANITY-CHECK the sign. Read the sign of your answer back into English: “v came out negative, so for a mirror the image is in front — it is real.” If that sentence contradicts your rough diagram, something in step 2 was wrong.
Step 4 is the one everybody skips and the one that saves the most marks. It costs you eight seconds and it catches almost every arithmetic slip, because a wrong sign in step 3 nearly always produces a physically silly answer — a real image behind a convex mirror, say, which simply cannot happen.
Mirror: 1/v + 1/u = 1/f | Lens: 1/v − 1/u = 1/f
“Mirrors carry the minus, lenses don’t.”
Mirror: m = −v/u | Lens: m = v/u
Notice the pattern: the mirror formula and the mirror magnification each carry exactly one extra minus sign compared with the lens versions. That is the whole difference. Say it out loud twice and it sticks.
Why it works: light reflects back off a mirror, so a real image forms on the same side as the object (negative side). Light passes through a lens, so a real image forms on the far side (positive side). The extra minus in the mirror formula is simply that reversal, written down once and for all.
The Cartesian Sign Convention, Set Up Once and For All
The convention itself is short enough to say in one breath: put the origin at the pole of the mirror or the optical centre of the lens, draw the incident light travelling from left to right, measure all distances from that origin, call rightward distances positive and leftward distances negative, and call heights above the principal axis positive. That is genuinely all of it.
Two consequences follow immediately and they are worth memorising as facts, because they turn up in almost every question. First, for a real object — an actual thing you can point at, sitting in front of the mirror or lens — the object distance u is always negative. Second, the focal length carries the sign of the surface: a concave mirror has a negative focal length, a convex mirror positive; a convex (converging) lens has a positive focal length, a concave (diverging) lens negative.
Why it works: the convention is not a law of nature, it is an agreement — the same agreement you already use in coordinate geometry, where points left of the origin have negative x. Because everyone agrees, one single formula can describe concave and convex mirrors at once, and one single formula can describe converging and diverging lenses at once. Without the convention you would need four formulas and four sets of rules. The signs are doing real work for you.
Reflection at Spherical Mirrors: Mirror Formula and Magnification
A spherical mirror is just a slice cut out of a shiny sphere. If the reflecting side is the hollow inside, you have a concave mirror; if it is the bulging outside, a convex mirror. Think of a steel katori: the inside curve is concave, the outside curve is convex. Everything else is vocabulary attached to that one picture.
Diagram: Ray construction for a concave mirror — object beyond C, at C, between C and F, and inside F, with pole, centre of curvature and principal focus labelled — to be added by illustrator
Four words to fix in your head, all measured along the principal axis. The pole P is the centre of the mirror’s surface — this is your origin. The centre of curvature C is the centre of the sphere the mirror was cut from. The distance PC is the radius of curvature R. The principal focus F is the point where rays travelling parallel to the axis converge after reflection (concave), or appear to diverge from (convex), and it sits exactly halfway: f = R/2.
If you studied Light: Reflection and Refraction in Class 10 Science, all of this will feel familiar — the difference in Class 12 is that we now insist on signs and treat the formula, not the ray diagram, as the primary tool.
Magnification: m = −v/u = h′/h
Here u is the object distance, v the image distance, h the object height and h′ the image height — all signed, all measured from P.
These are derived using the paraxial approximation: we assume every ray stays close to the principal axis and makes a small angle with it, so that the small arc of the mirror behaves like a section of a circle whose sagitta we can ignore. That assumption is why the formula is exact only near the axis, and why real mirrors show spherical aberration at the edges.
| Object position (concave mirror) | Image position | Nature and size |
|---|---|---|
| At infinity | At F | Real, inverted, a point |
| Beyond C | Between F and C | Real, inverted, diminished |
| At C | At C | Real, inverted, same size |
| Between C and F | Beyond C | Real, inverted, magnified |
| At F | At infinity | Real, inverted, hugely magnified |
| Between F and P | Behind the mirror | Virtual, erect, magnified |
| Anywhere (convex mirror) | Between P and F, behind | Virtual, erect, diminished |
1. Draw. Axis, pole P at the origin, light travelling left to right, object on the left.
2. Label. u = −40 cm (object is to the left). f = −15 cm (concave).
3. Substitute. 1/v = 1/f − 1/u = (−1/15) − (−1/40) = −8/120 + 3/120 = −5/120 = −1/24.
So v = −24 cm. Then m = −v/u = −(−24)/(−40) = −0.6.
4. Sanity-check. v is negative, so the image is 24 cm in front of the mirror — real. It lies between F (15 cm) and C (30 cm), exactly as the table predicts. m is negative and smaller than 1 in size, so the image is inverted and diminished to 0.6 of the object’s height. Everything agrees.
2. Label. u = −10 cm, f = −15 cm.
3. Substitute. 1/v = −1/15 + 1/10 = −2/30 + 3/30 = 1/30, so v = +30 cm.
m = −v/u = −(30)/(−10) = +3.
4. Sanity-check. v came out positive — the image is 30 cm behind the mirror, so it is virtual. m is positive and equal to 3, so the image is erect and three times as tall. This is exactly what happens when you bring your face close to a shaving mirror: a big, upright reflection. The formula told us that without a single ray drawn.
2. Label. Convex, so C is behind the mirror: R = +40 cm and f = R/2 = +20 cm. u = −30 cm.
3. Substitute. 1/v = 1/20 + 1/30 = 3/60 + 2/60 = 5/60 = 1/12, so v = +12 cm.
m = −12/(−30) = +0.4.
4. Sanity-check. Positive v → virtual, behind the mirror; positive m < 1 → erect and diminished. And notice v = 12 cm lies between P and F (20 cm), just as the last row of the table promises. That shrinking is precisely why convex mirrors are used as rear-view mirrors — a wide field of view squeezed into a small glass.
2. Label. f = −12 cm. A real image from a concave mirror is inverted, so m = −3 (not +3 — this is the step people rush).
3. Substitute. From m = −v/u = −3 we get v = 3u. Put that into the mirror formula:
1/(3u) + 1/u = 1/f ⇒ 4/(3u) = 1/f ⇒ u = 4f/3 = 4(−12)/3 = −16 cm, and v = 3(−16) = −48 cm.
Image height h′ = mh = (−3)(2) = −6 cm, i.e. 6 cm tall and inverted.
4. Sanity-check. Substitute back: 1/(−48) + 1/(−16) = −1/48 − 3/48 = −4/48 = −1/12 = 1/f. ✓ Both u and v are negative, so object and image are both in front — correct for a real image. The object at 16 cm sits between F (12) and C (24), and the image at 48 cm is beyond C: row four of the table again.
Reading the v–u Graph: What the Formula Actually Looks Like
Before we leave formulas behind, it is worth seeing one. If you rearrange the lens formula 1/v − 1/u = 1/f you get v = uf/(u + f). That is a hyperbola, and plotting it explains, in one picture, several things students otherwise memorise as unrelated facts. The graph below is for a converging lens of focal length +10 cm, and every plotted point was computed from that formula.
Three things jump out of that picture. One: as the object goes far away (u → −∞) the blue curve flattens towards v = +10 cm — the image collapses onto the focal point, which is exactly why distant objects image at F. Two: at u = −20 cm (that is, u = −2f) we get v = +20 cm, the famous same-size case. Three: the curve tears apart at u = −10 cm. Approach the focus from the left and v rockets to +∞; step just inside the focus and v reappears at −∞ on the virtual branch. That is not a glitch — it is the mathematical fingerprint of an object at the focus producing a beam of parallel rays and no image at all.
Refraction of Light, Snell’s Law and Apparent Depth
Drop a spoon into a glass of water and it looks broken at the surface. Nothing happened to the spoon; something happened to the light. Light travels at different speeds in different materials, and when it crosses a boundary at an angle it changes direction. That bending is refraction.
The refractive index of a medium is n = c/v, the speed of light in vacuum divided by its speed in that medium. Since nothing outruns light in vacuum, n is always at least 1: about 1.33 for water, 1.5 for ordinary glass, 2.42 for diamond. If you have already met light as an electromagnetic disturbance in Electromagnetic Waves, this is the same story from the geometry side.
Angles are measured from the normal, never from the surface. The incident ray, refracted ray and normal all lie in one plane.
Going into a denser medium (n2 > n1): the ray bends towards the normal. Coming out into a rarer medium: it bends away.
Substitute. 1 × sin 30° = 1.5 × sin r ⇒ sin r = 0.5/1.5 = 1/3 = 0.3333.
r = sin−1(0.3333) = 19.47° ≈ 19.5°.
Sanity-check. r < i, so the ray bent towards the normal — correct, because glass is optically denser than air. Had the answer come out larger than 30°, we would know a reciprocal had been flipped.
A neat consequence of refraction at a flat surface is apparent depth. Look straight down into water and the bottom seems closer than it is, because rays from the bottom bend away from the normal as they leave the water and your eye traces them back to a shallower point. For near-normal viewing, apparent depth = real depth / n, and the apparent upward shift is t(1 − 1/n) for a slab of thickness t.
Substitute. Apparent depth = 12 ÷ (4/3) = 12 × 3/4 = 9 cm.
Apparent shift = 12 − 9 = 3 cm, which matches t(1 − 1/n) = 12(1 − 0.75) = 3 cm. ✓
Sanity-check. The apparent depth must be less than the real depth (things look shallower, never deeper), and 9 < 12. This is exactly why a swimming pool always turns out deeper than it looked — and why you should never judge water depth by eye.
Why it works: refraction is not light “choosing” to bend. A wavefront entering a slower medium at an angle has one edge slowed before the other, so the whole front pivots — like a marching column swinging round when the soldiers on one side take shorter steps. The amount of pivot is fixed entirely by the speed ratio, which is what n measures.
Total Internal Reflection and Optical Fibres
Diagram: Ray paths at a denser-to-rarer boundary showing partial refraction below the critical angle, grazing emergence at the critical angle, and total internal reflection beyond it — plus a zig-zag ray inside an optical fibre core — to be added by illustrator
Send light the other way — from glass out into air — and the ray bends away from the normal. Increase the angle of incidence inside the glass and the refracted ray leans further and further over, until at one particular angle it grazes along the surface at 90°. Push past that angle and there is no refracted ray at all: every bit of light is reflected back into the glass. That angle is the critical angle C, and the effect is total internal reflection.
(i) light travels from the denser medium to the rarer medium, and
(ii) the angle of incidence exceeds the critical angle.
Putting r = 90° in Snell’s law: sin C = n2/n1, and for a medium of index n against air, sin C = 1/n.
Glass: sin C = 1/1.5 = 0.6667 ⇒ C = 41.81°
Water: sin C = 3/4 = 0.7500 ⇒ C = 48.59°
Diamond: sin C = 1/2.42 = 0.4132 ⇒ C = 24.41°
Sanity-check. The denser the medium, the smaller the critical angle — and 24.41° < 41.81° < 48.59° follows exactly that order. Diamond’s tiny critical angle means light that enters a cut stone bounces around inside many times before escaping. That is the whole secret of a diamond’s sparkle: geometry, not magic.
(a) sin C = ncladding/ncore = 1.45/1.50 = 0.96667 ⇒ C = 75.16°.
(b) A ray entering the end face at angle i refracts to r inside, and strikes the side wall at (90° − r). Requiring 90° − r ≥ C gives the numerical aperture
NA = sin imax = √(ncore2 − ncladding2) = √(2.2500 − 2.1025) = √0.1475 = 0.3841
so imax = sin−1(0.3841) = 22.59°.
Sanity-check. The critical angle is large (75°) because core and cladding are so close in index — which is deliberate: it keeps the trapped rays nearly parallel to the fibre axis so that pulses do not smear out over kilometres. Note also that C is measured from the normal to the side wall, which is why the useful quantity is 90° − r, not r. Mixing those two up is the single most common error in fibre problems.
Refraction at a Spherical Surface
Diagram: Refraction of a paraxial ray at a single convex spherical surface separating two media, with pole, centre of curvature, object point and image point labelled — to be added by illustrator
This is the bridge between refraction and lenses, and it is the section students most often skip — a mistake, because the lens maker’s formula is built out of it. The setup is one curved boundary between two media: light starts in medium 1 (index n1), meets a spherical surface of radius R, and continues in medium 2 (index n2).
Same sign convention as always. R is positive if the centre of curvature lies to the right of the surface (a surface bulging towards the incoming light has R positive), negative if to the left.
2. Label. n1 = 1 (air, where the light starts), n2 = 1.5 (glass), u = −100 cm, R = +20 cm (the centre of curvature is inside the glass, to the right).
3. Substitute. 1.5/v − 1/(−100) = (1.5 − 1)/20 = 0.5/20 = 0.025
1.5/v = 0.025 − 0.010 = 0.015 ⇒ v = 1.5/0.015 = +100 cm.
4. Sanity-check. v is positive, so the image lies 100 cm to the right of the surface — i.e. inside the glass, which is where light actually goes after refracting. Real image, correctly on the transmission side. If your answer had come out negative you would know you had put the wrong medium as n1: n1 is always the medium the light is coming from.
Thin Lenses: Lens Formula, Lens Maker’s Formula and Power
Diagram: Ray constructions for a convex lens (object beyond 2F, between F and 2F, and inside F) and for a concave lens, with optical centre, both focal points and 2F marked — to be added by illustrator
A lens is simply two refracting surfaces back to back. Apply the spherical-surface formula at the first surface, feed its image in as the object for the second surface, and let the glass thickness go to zero — out drops everything you know about lenses. That is the entire derivation in one sentence, and it is worth carrying in your head because it tells you why the lens maker’s formula has two radii in it.
Magnification: m = v/u = h′/h (no minus sign — that is the mirror’s)
Lens maker’s formula: 1/f = (n2/n1 − 1)(1/R1 − 1/R2), which for a lens in air becomes 1/f = (n − 1)(1/R1 − 1/R2)
Power: P = 1/f in dioptres when f is in metres, or P = 100/f when f is in centimetres.
2. Label. u = −30 cm, f = +20 cm (convex).
3. Substitute. 1/v = 1/f + 1/u = 1/20 − 1/30 = 3/60 − 2/60 = 1/60 ⇒ v = +60 cm.
m = v/u = 60/(−30) = −2. h′ = mh = (−2)(5) = −10 cm.
4. Sanity-check. Positive v → the image is 60 cm beyond the lens, on the far side — real. Negative m → inverted; |m| = 2 → twice as tall, so 10 cm and upside down. The object at 30 cm lies between f = 20 and 2f = 40, and the rule says the image should then be beyond 2f — and 60 > 40. ✓
2. Label. u = −30 cm, f = −20 cm (concave — this single minus sign is the whole question).
3. Substitute. 1/v = −1/20 − 1/30 = −3/60 − 2/60 = −5/60 = −1/12 ⇒ v = −12 cm.
m = (−12)/(−30) = +0.4.
4. Sanity-check. Negative v → the image is on the same side as the object, 12 cm from the lens — virtual. Positive m < 1 → erect and diminished. A diverging lens can only ever do this, whatever the object distance, which is a useful thing to state in one line if a question asks you to justify your answer.
2. Label. Light hits the first surface, whose centre of curvature is to the right: R1 = +20 cm. The second surface curves the other way, its centre to the left: R2 = −20 cm.
3(a). Substitute. 1/f = (1.5 − 1)(1/20 − 1/(−20)) = 0.5 × (0.05 + 0.05) = 0.5 × 0.10 = 0.05
⇒ f = +20 cm, and P = 100/20 = +5 D.
3(b). In water the ratio becomes n2/n1 = 1.5 ÷ (4/3) = 1.125, so
1/f′ = (1.125 − 1)(0.10) = 0.125 × 0.10 = 0.0125 ⇒ f′ = +80 cm, P′ = +1.25 D.
4. Sanity-check. The focal length grew four-fold, so the lens became much weaker in water. That is exactly right: the lens bends light because of the contrast between its index and the surroundings, and water shrinks that contrast. It is also why your vision is blurry underwater but sharp behind goggles — the goggles restore an air gap in front of your eye.
Combination of Thin Lenses in Contact
Put two thin lenses flat against each other and they behave as a single lens. The image made by the first becomes the object for the second, and when you write that out the algebra collapses to something beautifully simple: focal lengths combine as reciprocals, and powers simply add.
The total magnification is the product: m = m1 × m2 × …
Work in powers whenever you can — adding is far safer than adding fractions.
2. Label. f1 = +25 cm, f2 = −50 cm.
3. Substitute. P1 = 100/25 = +4 D, P2 = 100/(−50) = −2 D.
P = 4 + (−2) = +2 D ⇒ F = 100/2 = +50 cm.
4. Sanity-check. P is positive, so the pair acts as a converging lens — and it should, because the convex lens is the stronger of the two (4 D beats 2 D). Check the fraction route too: 1/F = 1/25 − 1/50 = 2/50 − 1/50 = 1/50, giving F = 50 cm. ✓ Same answer by both roads.
Why it works: power measures how sharply a lens bends light per unit distance, and bending is additive as long as the light has no room to travel between the two lenses. The moment you separate them by a distance d, the simple sum fails and you must use the two-lens sequence step by step — which is exactly what a compound microscope and a telescope are.
Refraction of Light Through a Prism
Diagram: A ray passing through a triangular prism, showing angles i₁, r₁, r₂, i₂, the refracting angle A and the angle of deviation D, plus the D-versus-i curve with its minimum — to be added by illustrator
A prism is a wedge of glass with two flat refracting faces meeting at the refracting angle A. A ray refracts on the way in, travels through the glass, and refracts again on the way out — and because both bends push it the same way, it emerges turned through an overall angle of deviation D.
D = i1 + i2 − A
At minimum deviation the path is symmetric: i1 = i2 = i and r1 = r2 = A/2, so i = (A + Dm)/2, giving the prism formula
n = sin[(A + Dm)/2] ÷ sin(A/2)
3. Substitute. n = sin[(60° + 30°)/2] / sin(60°/2) = sin 45° / sin 30° = 0.70711 / 0.5 = 1.4142, i.e. n = √2.
Angle of incidence: i = (A + Dm)/2 = 90°/2 = 45°, and inside the prism r1 = r2 = A/2 = 30°.
4. Sanity-check. Verify with Snell’s law at the first face: sin 45°/sin 30° = 0.70711/0.5 = 1.4142 = n. ✓ And r1 + r2 = 30 + 30 = 60 = A. ✓ Also D = i1 + i2 − A = 45 + 45 − 60 = 30°. ✓ Three independent checks all agree, which is how you should finish a prism question.
3. Substitute. sin[(60° + Dm)/2] = n sin(A/2) = 1.5 × sin 30° = 1.5 × 0.5 = 0.75
(60° + Dm)/2 = sin−1(0.75) = 48.59° ⇒ 60° + Dm = 97.18° ⇒ Dm = 37.18°.
4. Sanity-check. Glass with n = 1.5 is denser than the √2 ≈ 1.414 glass of Example 14, so it should deviate light more — and 37.18° > 30°. ✓ The incidence angle here would be (60 + 37.18)/2 = 48.59°.
The Simple Microscope (Magnifying Glass)
Every optical instrument in this chapter is answering the same question: how do I make something take up a bigger angle at my eye? That is what magnification really means — not making the object bigger, but making its image spread over a wider angle on your retina. The reference distance is the least distance of distinct vision, D = 25 cm, the closest a normal eye can focus comfortably.
A simple microscope is one converging lens of short focal length held close to the eye, with the object just inside its focus. It produces an enlarged, erect, virtual image.
Image at infinity (normal adjustment, eye fully relaxed): M = D/f
The near-point value is always exactly 1 larger. Every instrument in this chapter has these same two settings — learn the pair, not eight separate formulas.
(a) M = 1 + D/f = 1 + 25/5 = 6.
Object position: the virtual image is at the near point, so v = −25 cm and f = +5 cm.
1/u = 1/v − 1/f = −1/25 − 1/5 = −0.04 − 0.20 = −0.24 ⇒ u = −4.17 cm.
(b) M = D/f = 25/5 = 5, with the object exactly at the focus, u = −5 cm.
4. Sanity-check. In case (a) the object sits 4.17 cm away, just inside the 5 cm focus — which is the only place a single convex lens gives an erect virtual image. Moving the object from 4.17 cm out to 5 cm drops the magnification from 6 to 5 but lets your eye relax completely. That trade-off is the entire physics of “normal adjustment”.
The Compound Microscope and Its Magnifying Power
Diagram: Compound microscope ray layout — objective forming a real inverted intermediate image inside the focus of the eyepiece, which forms the final magnified virtual image; tube length and both focal lengths labelled — to be added by illustrator
One lens can only do so much. A compound microscope magnifies twice: a short-focus objective makes a real, inverted, enlarged image inside the tube, and a longer-focus eyepiece then acts as a simple microscope on that image. Two magnifications multiply, which is how you reach hundreds of times rather than five or six.
Image at near point: me = 1 + D/fe → M = (vo/uo)(1 + D/fe)
Image at infinity: me = D/fe → M = (vo/uo)(D/fe), and tube length L = vo + fe
Both fo and fe are small, with fo < fe. The final image is inverted with respect to the object.
(a) Objective. uo = −1.2 cm, fo = +1.0 cm.
1/vo = 1/1.0 − 1/1.2 = 1 − 0.83333 = 0.16667 ⇒ vo = +6.0 cm.
mo = vo/uo = 6.0/(−1.2) = −5.
(b) Near point. me = 1 + 25/5 = 6, so M = (−5)(6) = −30, i.e. 30× and inverted.
For the eyepiece with ve = −25 cm: 1/ue = −1/25 − 1/5 = −0.24 ⇒ ue = −4.17 cm.
Separation L = vo + |ue| = 6.0 + 4.17 = 10.17 cm.
(c) Normal adjustment. me = 25/5 = 5, so M = (−5)(5) = −25, and L = vo + fe = 6.0 + 5.0 = 11.0 cm.
4. Sanity-check. The specimen at 1.2 cm sits just outside the objective’s 1.0 cm focus — exactly where it must be to give a real, strongly enlarged intermediate image. Near-point viewing gives more magnification (30 vs 25) but a shorter tube (10.17 vs 11.0 cm), which is the same trade-off as the magnifying glass. And the near-point magnification is larger by the factor 6/5, precisely the ratio of the two eyepiece formulas. ✓
The Astronomical Telescope: Refracting and Reflecting
Diagram: Astronomical telescope in normal adjustment — parallel rays from a distant object, objective forming a real image at its focus which coincides with the eyepiece focus, final image at infinity; alongside a Cassegrain reflecting telescope with concave primary mirror and secondary mirror — to be added by illustrator
A telescope faces the opposite problem to a microscope. The object is not small — it is enormous — but it is so far away that it subtends a tiny angle. So the objective is given a long focal length and a large aperture: long focal length for angular magnification, large aperture to gather enough light.
Final image at the near point: M = (fo/fe)(1 + fe/D)
Here fo is large and fe small — the exact opposite of the microscope, where both are small.
(a) Magnitude |M| = fo/fe = 150/6 = 25; with sign, M = −25 because the final image is inverted. L = 150 + 6 = 156 cm.
(b) M = (150/6)(1 + 6/25) = 25 × 1.24 = 31.
Tube length: for the eyepiece, ve = −25 cm, fe = +6 cm, so
1/ue = −1/25 − 1/6 = −(6 + 25)/150 = −31/150 ⇒ ue = −150/31 = −4.84 cm.
L = fo + |ue| = 150 + 4.84 = 154.84 cm.
4. Sanity-check. Near-point viewing again gives more magnification (31 vs 25) and a shorter tube (154.84 vs 156 cm) — the same pattern as the microscope, for the same reason. Note 31/25 = 1.24 = 1 + fe/D, so the two answers are consistent by construction. ✓
A reflecting telescope replaces the objective lens with a large concave mirror. Two advantages follow immediately, and both are standard exam answers. First, a mirror shows no chromatic aberration at all, because reflection does not depend on wavelength the way refraction does. Second, a big mirror can be supported across its whole back surface, whereas a big lens can only be held at its rim and sags under its own weight — which is why every giant telescope built in the last century is a reflector. In the Cassegrain design the light bounces off a concave primary, then off a small convex secondary, and out through a hole in the primary to the eyepiece.
| Feature | Compound microscope | Astronomical telescope |
|---|---|---|
| Object distance | Just beyond fo, a few mm | Effectively infinite |
| Objective focal length | Very small | Very large |
| Objective aperture | Small | As large as possible |
| M (normal adjustment) | (vo/uo)(D/fe) | |M| = fo/fe; signed M = −fo/fe (inverted) |
| Tube length (normal) | vo + fe | fo + fe |
| Purpose | Enlarge a tiny nearby object | Widen the angle of a distant one |
Real or Virtual? The Decision Flow
Step 4 of the ritual asks you to read the sign of your answer back into English. Here is that step drawn out as a flow. Learn this one picture and you will never again write “real image behind the mirror” in an exam.
Look at the symmetry in that diagram. For a mirror, negative v means real; for a lens, positive v means real. They are opposite, and they are opposite for one physical reason: reflected light goes back the way it came, refracted light carries on forwards. If you can explain that sentence, you have understood the sign convention rather than memorised it.
Ray Optics Class 12 Physics Formula List (One-Page Revision)
Print this, stick it inside your cupboard door, and read it once every morning for the week before the exam. This is the complete ray optics and optical instruments class 12 physics formula list for the current syllabus — nothing extra, nothing missing.
| Situation | Formula | Watch out for |
|---|---|---|
| Spherical mirror | 1/v + 1/u = 1/f, f = R/2 | Concave f negative, convex positive |
| Mirror magnification | m = −v/u = h′/h | The minus sign is part of the formula |
| Refractive index | n = c/vmedium | Always ≥ 1 |
| Snell’s law | n1 sin i = n2 sin r | Angles from the normal, not the surface |
| Apparent depth | apparent = real/n; shift = t(1 − 1/n) | Near-normal viewing only |
| Critical angle | sin C = n2/n1 (= 1/n against air) | Denser → rarer only |
| Optical fibre | NA = sin imax = √(ncore2 − nclad2) | Wall angle is 90° − r |
| Spherical surface | n2/v − n1/u = (n2 − n1)/R | n1 is where light comes from |
| Thin lens | 1/v − 1/u = 1/f, m = v/u | Minus in the formula, none in m |
| Lens maker’s formula | 1/f = (n − 1)(1/R1 − 1/R2) | Use nlens/nmedium if not in air |
| Power of a lens | P = 1/f(m) = 100/f(cm) | Dioptres need metres |
| Lenses in contact | 1/F = 1/f1 + 1/f2; P = P1 + P2 | Only when they touch |
| Prism | A = r1 + r2; D = i1 + i2 − A | Both hold for every ray, not just at Dm |
| Prism formula | n = sin[(A + Dm)/2] / sin(A/2) | Minimum deviation only |
| Simple microscope | M = 1 + D/f (near pt); M = D/f (∞) | D = 25 cm |
| Compound microscope | M = (vo/uo)(1 + D/fe) or (vo/uo)(D/fe) | L = vo + fe only at ∞ |
| Telescope | M = fo/fe; L = fo + fe | Near point: multiply by (1 + fe/D) |
Check the Current 2026-27 Scope
The current official CBSE 2026-27 Physics outline lists mirrors, refraction, total internal reflection and optical fibres, spherical surfaces, lenses, prisms, microscopes and astronomical telescopes for this chapter. The topics below do not appear in that Class 12 Ray Optics outline. Treat them as enrichment only unless your school gives separate instructions.
• The human eye, image formation on the retina, and accommodation
• Correction of eye defects — myopia and hypermetropia, and the lens powers needed
• Resolving power of a microscope and of a telescope
Several of these ideas are useful enrichment, and the same careful sign-convention habit also matters in the Class 10 Electricity notes and practice guide. For Class 12 board preparation, prioritise the topics that the current CBSE outline explicitly lists.
This page prioritises the topics explicitly named in the official outline: mirrors and the mirror formula; refraction, total internal reflection and optical fibres; spherical surfaces; thin lenses, lens maker’s formula, magnification, power and lenses in contact; refraction through a prism; microscopes; and reflecting and refracting astronomical telescopes with magnifying power. For anything beyond that list, use your school’s current instructions as the deciding reference.
Ray Optics Class 12 Physics Important Questions — How They Are Set
Across recent papers, questions from this chapter fall into a small number of recognisable shapes. Knowing the shapes is worth more than knowing extra facts, because it tells you what the first line of your answer should be.
- Plug-and-read (1–2 marks). A mirror or lens with given u and f; find v, m, or the nature of the image. Pure ritual. Never lose these.
- Work backwards (2–3 marks). Magnification and focal length given, find the object distance — like Example 4. The trap is always the sign of m.
- Two-element chains (3 marks). A lens followed by a mirror, or a lens in contact with another. Image of the first becomes object of the second; carry the sign through, do not restart.
- Medium-change questions (2–3 marks). “What happens to the focal length if the lens is put in water?” Answer with the lens maker’s formula and the ratio nlens/nmedium, as in Example 12.
- Derivations (3–5 marks). Mirror formula, refraction at a spherical surface, lens maker’s formula, prism formula, magnifying power of a compound microscope or telescope. Learn the sequence of steps, and always state the paraxial assumption.
- Reasoning one-liners (1–2 marks). Why convex mirrors are used as rear-view mirrors; why a diamond sparkles; why big telescopes use mirrors; why a lens is weaker in water. Each needs one clean sentence, and you now have all four above.
- Graph and assertion-reason. The 1/v versus 1/u straight line, or the v–u hyperbola from the section above.
Practice Worksheet
Twelve questions, all original, all solvable with the ritual. Give yourself forty minutes with a calculator and no notes. Write the signed substitution line for every numerical — that habit is what you are really practising here.
Q1. An object is placed 15 cm in front of a concave mirror of focal length 10 cm. Find the image distance and magnification, and describe the image.
Show Answer
Q2. A convex mirror has a radius of curvature of 30 cm. An object stands 10 cm from it. Locate the image and give its magnification.
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Q3. A transparent block has refractive index 1.6. Find its critical angle with air, and state whether a ray striking the inside of a face at 35° will emerge.
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Q4. A tank holds water (n = 4/3) to a real depth of 2.0 m. Viewed from directly above, what is the apparent depth, and by how much does the bottom appear raised?
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Q5. An object is placed 10 cm from a converging lens of focal length 15 cm. Find v and m, and say what kind of image forms.
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Q6. A plano-convex lens is made of glass with n = 1.5. Its curved face has radius 15 cm and the flat face is turned away from the incoming light. Find its focal length and power in air.
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Q7. A lens of power +6 D is placed in contact with one of power −4 D. Find the power and focal length of the combination and state its nature.
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Q8. A prism of refracting angle 60° is made of glass of refractive index 1.6. Calculate the angle of minimum deviation and the angle of incidence at which it occurs.
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Q9. An astronomical telescope has an objective of focal length 100 cm and an eyepiece of focal length 4 cm. Find its magnifying power and tube length in normal adjustment.
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Q10. A compound microscope has fo = 2.0 cm and fe = 6.25 cm. The object is 2.5 cm from the objective and the final image is at the near point. Find vo, mo and the total magnifying power. Take D = 25 cm.
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Q11. A glass rod (n = 1.5) has a convex end of radius 10 cm. A point object in air lies on the axis, 30 cm from that end. Find the image position.
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Q12. A concave mirror of focal length 20 cm produces a virtual image four times the size of the object. Where is the object placed, and where is the image?
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Once you have marked yourself, do not simply note the score. Note the step that failed. Almost every wrong answer in this chapter traces back to step 2 — a sign written down without thinking — and once you see that pattern in your own work, it stops happening.
When you are comfortable here, the natural next stop in your Class 12 Physics revision is Current Electricity, which shares the same discipline of signs and directions in a completely different setting. When you are ready to see what happens where the ray model finally breaks down, Wave Optics picks up exactly where this chapter stops.
One more than yesterday. You do not need to master ray optics tonight. You need one more correct sign than you got right yesterday, and then one more the day after. Six days of that and this chapter will feel like arithmetic. Kaizen — small, steady, unglamorous improvement — is what actually moves a physics score, and it is completely within your reach.