Is it correct to say: Taking u as positive for a real object, e.g. u = +25 cm?

In the New Cartesian convention distances measured against the incident light are negative. A real object in front of a mirror or lens has u < 0.

Is it correct to say: Using the mirror magnification −v/u for a lens?

Lens: m = v/u. Mirror: m = −v/u.

Is it correct to say: Mixing up the mirror and lens equations?

Mirror: 1/v + 1/u = 1/f. Lens: 1/v − 1/u = 1/f.

Is it correct to say: Using f in centimetres in P = 1/f?

Convert f to metres first: f = 25 cm gives P = 1/0.25 = 4 D. Equivalently P = 100/f with f in cm.

Is it correct to say: Stating only one condition for total internal reflection?

Both are needed: light must go from the denser to the rarer medium, and the angle of incidence must exceed the critical angle.

Is it correct to say: Applying D = (n − 1)A to a 60° prism?

That is the thin-prism approximation. For a large prism angle use n = sin((A + D_m)/2)/sin(A/2).

📄 Formula sheet (PDF)

Ray Optics and Optical Instruments

Q: Is it correct to say: Using the mirror magnification −v/u for a lens?

Lens: m = v/u. Mirror: m = −v/u.

Q: Is it correct to say: Mixing up the mirror and lens equations?

Mirror: 1/v + 1/u = 1/f. Lens: 1/v − 1/u = 1/f.

Q: Is it correct to say: Using f in centimetres in P = 1/f?

Convert f to metres first: f = 25 cm gives P = 1/0.25 = 4 D. Equivalently P = 100/f with f in cm.

Q: Is it correct to say: Stating only one condition for total internal reflection?

Both are needed: light must go from the denser to the rarer medium, and the angle of incidence must exceed the critical angle.

Q: Is it correct to say: Applying D = (n − 1)A to a 60° prism?

That is the thin-prism approximation. For a large prism angle use n = sin((A + D_m)/2)/sin(A/2).

Mirror and lens formulas, refraction, total internal reflection, refraction at spherical surfaces, lens maker's formula, power, the prism, and the microscope and telescope — NCERT Class 12 Physics Ch 9

📐 18 formulas✏️ 4 examples🎯 6 practice⚖️ Units V–VI: 18 marks (with Ch 8 and 10)🏫 CBSE📚 Class 12✓ 2025–26 syllabus

💡

Board Exam Tips

→Write the New Cartesian sign convention at the start of every numerical: distances from the pole or optical centre, positive along the incident light, negative against it. Put signs in before substituting.
→The mirror equation 1/v + 1/u = 1/f and the lens formula 1/v − 1/u = 1/f differ by one sign. So do the magnifications: −v/u for a mirror, v/u for a lens.
→Derivations to prepare: mirror equation, refraction at a spherical surface, lens maker's formula, lenses in contact, and the prism formula at minimum deviation.
→For optical instruments, draw the labelled ray diagram (final image at infinity unless told otherwise) and then write the magnifying power.
→Power in dioptres needs f in metres: P = 1/f(m) = 100/f(cm).

📊 Diagram

A ray parallel to the principal axis refracted through the focus F of a convex lens

📐 Formulas(18)

Focal Length of a Spherical Mirror

f = \frac{R}{2}

Symbol	Meaning
$f$	Focal length (m or cm)
$R$	Radius of curvature (m or cm)

Mirror Equation★ Board fav

\frac{1}{v} + \frac{1}{u} = \frac{1}{f}

Symbol	Meaning
$u$	Object distance from the pole
$v$	Image distance from the pole
$f$	Focal length

Magnification by a Mirror

m = \frac{h ^{'}}{h} = - \frac{v}{u}

Symbol	Meaning
$h^{'}$	Height of the image
$h$	Height of the object

Snell's Law

n_{21} = \frac{s i n i}{s i n r} = \frac{n _{2}}{n _{1}}

Symbol	Meaning
$i$	Angle of incidence
$r$	Angle of refraction
$n_{21}$	Refractive index of medium 2 relative to medium 1

Real and Apparent Depth

h_{1} = \frac{h}{n}

Symbol	Meaning
$h$	Real depth
$h_{1}$	Apparent depth
$n$	Refractive index of the medium relative to air

Critical Angle★ Board fav

sin i_{c} = \frac{1}{n}

Symbol	Meaning
$i_{c}$	Critical angle
$n$	Refractive index of the denser medium relative to the rarer medium

Refraction at a Spherical Surface

\frac{n _{2}}{v} - \frac{n _{1}}{u} = \frac{n _{2} - n _{1}}{R}

Symbol	Meaning
$n_{1}$	Refractive index of the medium containing the object
$n_{2}$	Refractive index of the medium into which light refracts
$R$	Radius of curvature of the surface

Lens Maker's Formula★ Board fav

\frac{1}{f} = (n_{21} - 1) (\frac{1}{R _{1}} - \frac{1}{R _{2}})

Symbol	Meaning
$R_{1}, R_{2}$	Radii of curvature of the first and second surfaces (with sign)
$n_{21}$	Refractive index of lens material relative to the surrounding medium

Thin Lens Formula★ Board fav

\frac{1}{v} - \frac{1}{u} = \frac{1}{f}

Magnification by a Lens

m = \frac{h ^{'}}{h} = \frac{v}{u}

Power of a Lens

P = \frac{1}{f}

Symbol	Meaning
$P$	Power (dioptre, D = m⁻¹)
$f$	Focal length (m)

Thin Lenses in Contact

\frac{1}{f} = \frac{1}{f _{1}} + \frac{1}{f _{2}}, P = P_{1} + P_{2}, m = m_{1} m_{2}

Angles in a Prism

r_{1} + r_{2} = A, A + δ = i + e

Symbol	Meaning
$A$	Angle of the prism
$δ$	Angle of deviation
$i, e$	Angles of incidence and emergence

Refractive Index from Minimum Deviation★ Board fav

n_{21} = \frac{s i n ( \frac{A + D _{m}}{2} )}{s i n ( \frac{A}{2} )}

Symbol	Meaning
$D_{m}$	Angle of minimum deviation

Deviation by a Thin Prism

D_{m} = (n_{21} - 1) A

Simple Microscope

m = 1 + \frac{D}{f} (image at D), m = \frac{D}{f} (image at infinity)

Symbol	Meaning
$D$	Least distance of distinct vision (about 25 cm)
$f$	Focal length of the convex lens

Compound Microscope★ Board fav

m = m_{o} m_{e} \approx \frac{L}{f _{o}} \cdot \frac{D}{f _{e}}

Symbol	Meaning
$f_{o}, f_{e}$	Focal lengths of objective and eyepiece
$L$	Tube length

Astronomical (Refracting) Telescope

m = \frac{f _{o}}{f _{e}}, L = f_{o} + f_{e}

Symbol	Meaning
$f_{o}$	Focal length of the objective
$f_{e}$	Focal length of the eyepiece
$L$	Length of the telescope tube

✏️ Solved Examples

1Solved Exampleeasy4 steps

An object is placed 25 cm in front of a concave mirror of focal length 15 cm. Find the position, nature and magnification of the image.

New Cartesian signs: the object and focus are in front of the mirror

u = - 25 cm, f = - 15 cm

2Solved Exampleboard5 steps

A double-convex lens is made of glass of refractive index 1.5 and both its faces have a radius of curvature of 20 cm. (a) Find its focal length and power in air. (b) An object is placed 30 cm from the lens. Find the position and magnification of the image.

Signs for a double-convex lens

R_{1} = + 20 cm, R_{2} = - 20 cm

3Solved Exampleboard3 steps

A prism of angle 60° gives a minimum deviation of 30°. Find (a) the refractive index of the prism material, (b) the angle of incidence at minimum deviation and (c) the critical angle for this material in air.

Prism formula at minimum deviation

n = \frac{s i n ( \frac{6 0 ^{\circ} + 3 0 ^{\circ}}{2} )}{s i n ( \frac{6 0 ^{\circ}}{2} )} = \frac{s i n 4 5 ^{\circ}}{s i n 3 0 ^{\circ}} = \frac{0.7071}{0.5} = 2 \approx 1.41

4Solved ExampleHOTS6 steps

A compound microscope has an objective of focal length 2.0 cm and an eyepiece of focal length 5.0 cm. An object is placed 2.5 cm in front of the objective and the final image is formed at the least distance of distinct vision (25 cm). Find (a) the magnifying power and (b) the separation between the two lenses.

Objective: u_o = −2.5 cm, f_o = +2.0 cm

\frac{1}{v _{o}} = \frac{1}{2.0} - \frac{1}{2.5} = 0.5 - 0.4 = 0.1 \Rightarrow v_{o} = + 10 cm

⚠️ Traps & Common Mistakes

⚠️Common Mistakes6

1
Taking u as positive for a real object, e.g. u = +25 cm
✓In the New Cartesian convention distances measured against the incident light are negative. A real object in front of a mirror or lens has u < 0.
2
Using the mirror magnification −v/u for a lens
✓Lens: m = v/u. Mirror: m = −v/u.
3
Mixing up the mirror and lens equations
✓Mirror: 1/v + 1/u = 1/f. Lens: 1/v − 1/u = 1/f.
4
Using f in centimetres in P = 1/f
✓Convert f to metres first: f = 25 cm gives P = 1/0.25 = 4 D. Equivalently P = 100/f with f in cm.
5
Stating only one condition for total internal reflection
✓Both are needed: light must go from the denser to the rarer medium, and the angle of incidence must exceed the critical angle.
6
Applying D = (n − 1)A to a 60° prism
✓That is the thin-prism approximation. For a large prism angle use n = sin((A + D_m)/2)/sin(A/2).

🎯 Practice Yourself

🎯Practice Yourself6 questions

Q1
A concave mirror has a radius of curvature of 30 cm. What is its focal length, with sign?
Q2
Find the power of a concave lens of focal length 25 cm.
Q3
Lenses of power +10 D and −4 D are placed in contact. Find the power and focal length of the combination.
Q4
Find the critical angle for a glass–air interface if the refractive index of glass is 1.5.
Q5
An astronomical telescope has an objective of focal length 150 cm and an eyepiece of focal length 5.0 cm. Find its magnifying power and tube length in normal adjustment.
Q6
A thin prism of angle 5° is made of glass of refractive index 1.6. Find the deviation it produces.

📝 Notes

Ray Optics and Optical Instruments

Almost every numerical in this chapter is one formula plus careful signs. Get the sign convention right and the rest is arithmetic.

Sign convention first

In the New Cartesian convention, measure every distance from the pole (mirror) or optical centre (lens). Distances in the direction of the incident light are positive; distances against it are negative; heights above the axis are positive. So a real object always has $u < 0$ , a concave mirror has $f < 0$ and a convex lens has $f > 0$ .

Mirrors and lenses side by side

Mirror: $\dfrac{1}{v} + \dfrac{1}{u} = \dfrac{1}{f}$ and $m = -\dfrac{v}{u}$ .
Lens: $\dfrac{1}{v} - \dfrac{1}{u} = \dfrac{1}{f}$ and $m = \dfrac{v}{u}$ .

In both cases a negative $m$ means a real, inverted image.

Refraction, TIR and the prism

Snell's law leads to the critical angle $\sin i_c = 1/n$ and, applied to a curved surface, to $\dfrac{n_2}{v} - \dfrac{n_1}{u} = \dfrac{n_2 - n_1}{R}$ . Using that relation twice gives the lens maker's formula. For a prism, combine $r_1 + r_2 = A$ and $A + \delta = i + e$ ; at minimum deviation the ray passes symmetrically, which gives $n = \sin\frac{A+D_m}{2} / \sin\frac{A}{2}$ .

Optical instruments

Simple microscope: $m = 1 + D/f$ (image at near point), $D/f$ (image at infinity).
Compound microscope: $m \approx (L/f_o)(D/f_e)$ — both focal lengths small.
Telescope: $m = f_o/f_e$ , tube length $f_o + f_e$ — large $f_o$ , small $f_e$ .

Reflecting telescopes use a concave mirror as the objective: there is no chromatic aberration, a parabolic mirror removes spherical aberration, and a large mirror is easier to support than a large lens.