The Optics Formula is a set of mathematical expressions that describe how light behaves when it reflects, refracts, and diffracts through various media and surfaces. These formulas are central to NCERT Physics for Class 10 (Chapter 10) and Class 12 (Chapters 9 and 10). They are equally critical for JEE Main, JEE Advanced, and NEET, where optics questions appear every year. This article covers every important optics formula — mirror formula, lens formula, Snell's law, magnification, wave optics, and more — with derivations, a complete formula sheet, solved examples, and CBSE exam tips.
Key Optics Formulas at a Glance
Quick reference for the most important optics formulas used in CBSE and competitive exams.
- Mirror Formula: \( \frac{1}{f} = \frac{1}{v} + \frac{1}{u} \)
- Lens Formula: \( \frac{1}{f} = \frac{1}{v} – \frac{1}{u} \)
- Snell's Law: \( n_1 \sin\theta_1 = n_2 \sin\theta_2 \)
- Magnification (Mirror): \( m = -\frac{v}{u} \)
- Lensmaker's Equation: \( \frac{1}{f} = (n-1)\left(\frac{1}{R_1} – \frac{1}{R_2}\right) \)
- Power of a Lens: \( P = \frac{1}{f} \) (in dioptres)
- Refractive Index: \( n = \frac{c}{v} \)
What is the Optics Formula?
The Optics Formula refers to the collection of equations that govern the behaviour of light as it interacts with mirrors, lenses, prisms, and other optical elements. Optics is broadly divided into two branches: Geometrical (Ray) Optics and Wave Optics. Geometrical optics deals with reflection and refraction using ray diagrams and algebraic formulas. Wave optics deals with interference, diffraction, and polarisation using wave theory.
In the NCERT curriculum, ray optics is introduced in Class 10 (Chapter 10: Light — Reflection and Refraction) and expanded significantly in Class 12 (Chapter 9: Ray Optics and Optical Instruments; Chapter 10: Wave Optics). Students must master every optics formula to score well in CBSE board exams and to crack JEE and NEET.
The sign convention used throughout optics is the New Cartesian Sign Convention. All distances are measured from the pole of the mirror or the optical centre of the lens. Distances in the direction of incident light are positive. Distances opposite to incident light are negative.
Optics Formula — Expression and Variables
Mirror Formula
\[ \frac{1}{f} = \frac{1}{v} + \frac{1}{u} \]
| Symbol | Quantity | SI Unit |
|---|---|---|
| f | Focal length of the mirror | metre (m) |
| v | Image distance from the pole | metre (m) |
| u | Object distance from the pole | metre (m) |
Lens Formula
\[ \frac{1}{f} = \frac{1}{v} – \frac{1}{u} \]
| Symbol | Quantity | SI Unit |
|---|---|---|
| f | Focal length of the lens | metre (m) |
| v | Image distance from optical centre | metre (m) |
| u | Object distance from optical centre | metre (m) |
Snell's Law of Refraction
\[ n_1 \sin\theta_1 = n_2 \sin\theta_2 \]
| Symbol | Quantity | SI Unit |
|---|---|---|
| \( n_1 \) | Refractive index of medium 1 | Dimensionless |
| \( n_2 \) | Refractive index of medium 2 | Dimensionless |
| \( \theta_1 \) | Angle of incidence | degree or radian |
| \( \theta_2 \) | Angle of refraction | degree or radian |
Derivation of the Mirror Formula
Consider an object placed in front of a concave mirror. Using the geometry of similar triangles formed by the incident ray, reflected ray, and the principal axis, we can show that:
Step 1: Consider a ray from the object hitting the mirror at point P on the principal axis. Two triangles are formed — one involving the object and one involving the image.
Step 2: By the property of similar triangles and the law of reflection (angle of incidence = angle of reflection), we get the relation \( \frac{AB}{A’B’} = \frac{u}{v} \).
Step 3: Applying the geometry of the focal point (where paraxial rays converge), we substitute \( f = R/2 \) and simplify to obtain \( \frac{1}{v} + \frac{1}{u} = \frac{1}{f} \). This is the standard mirror formula.
Complete Optics Formula Sheet
| Formula Name | Expression | Variables | SI Units | NCERT Chapter |
|---|---|---|---|---|
| Mirror Formula | \( \frac{1}{f} = \frac{1}{v} + \frac{1}{u} \) | f = focal length, v = image distance, u = object distance | m | Class 10, Ch 10; Class 12, Ch 9 |
| Lens Formula | \( \frac{1}{f} = \frac{1}{v} – \frac{1}{u} \) | f = focal length, v = image distance, u = object distance | m | Class 10, Ch 10; Class 12, Ch 9 |
| Magnification (Mirror) | \( m = -\frac{v}{u} = \frac{h_i}{h_o} \) | v = image distance, u = object distance, h_i = image height, h_o = object height | Dimensionless | Class 10, Ch 10 |
| Magnification (Lens) | \( m = \frac{v}{u} = \frac{h_i}{h_o} \) | v = image distance, u = object distance | Dimensionless | Class 10, Ch 10 |
| Refractive Index | \( n = \frac{c}{v_{medium}} \) | c = speed of light in vacuum, v = speed in medium | Dimensionless | Class 10, Ch 10; Class 12, Ch 9 |
| Snell's Law | \( n_1 \sin\theta_1 = n_2 \sin\theta_2 \) | n = refractive index, θ = angle with normal | Dimensionless / degrees | Class 10, Ch 10; Class 12, Ch 9 |
| Lensmaker's Equation | \( \frac{1}{f} = (n-1)\left(\frac{1}{R_1} – \frac{1}{R_2}\right) \) | n = refractive index, R1 and R2 = radii of curvature | m | Class 12, Ch 9 |
| Power of a Lens | \( P = \frac{1}{f} \) | f = focal length in metres | Dioptre (D) | Class 10, Ch 10; Class 12, Ch 9 |
| Combined Power of Lenses | \( P = P_1 + P_2 + P_3 + \ldots \) | P1, P2 = individual lens powers | Dioptre (D) | Class 12, Ch 9 |
| Critical Angle | \( \sin\theta_c = \frac{1}{n} \) | n = refractive index of denser medium w.r.t. rarer medium | degrees | Class 12, Ch 9 |
| Prism Formula (Minimum Deviation) | \( n = \frac{\sin\left(\frac{A+\delta_m}{2}\right)}{\sin\left(\frac{A}{2}\right)} \) | A = prism angle, δm = minimum deviation | Dimensionless | Class 12, Ch 9 |
| Young's Double Slit — Fringe Width | \( \beta = \frac{\lambda D}{d} \) | λ = wavelength, D = screen distance, d = slit separation | m | Class 12, Ch 10 |
| Condition for Constructive Interference | \( \Delta x = n\lambda \) | Δx = path difference, n = integer, λ = wavelength | m | Class 12, Ch 10 |
| Condition for Destructive Interference | \( \Delta x = \left(n + \frac{1}{2}\right)\lambda \) | Δx = path difference, n = integer | m | Class 12, Ch 10 |
| Resolving Power of a Microscope | \( RP = \frac{2n\sin\theta}{\lambda} \) | n = refractive index, θ = half-angle of cone, λ = wavelength | m⁻¹ | Class 12, Ch 9 |
| Brewster's Law | \( \tan\theta_B = n \) | θB = Brewster's angle, n = refractive index | degrees | Class 12, Ch 10 |
Optics Formula — Solved Examples
Example 1 (Class 10 Level) — Mirror Formula
Problem: An object is placed 30 cm in front of a concave mirror of focal length 10 cm. Find the image distance and state the nature of the image.
Given: u = −30 cm (object in front of mirror, negative by sign convention), f = −10 cm (concave mirror, negative focal length)
Step 1: Write the mirror formula: \( \frac{1}{f} = \frac{1}{v} + \frac{1}{u} \)
Step 2: Substitute values: \( \frac{1}{-10} = \frac{1}{v} + \frac{1}{-30} \)
Step 3: Rearrange: \( \frac{1}{v} = \frac{1}{-10} + \frac{1}{30} = \frac{-3 + 1}{30} = \frac{-2}{30} \)
Step 4: Solve: \( v = -15 \) cm
Step 5: Find magnification: \( m = -\frac{v}{u} = -\frac{-15}{-30} = -0.5 \)
Answer
Image distance v = −15 cm. The image is real, inverted, and diminished (formed in front of the mirror at 15 cm). Magnification = −0.5.
Example 2 (Class 12 Level) — Lensmaker's Equation and Power
Problem: A biconvex lens is made of glass with refractive index 1.5. Both surfaces have equal radii of curvature of 20 cm. Calculate (a) the focal length and (b) the power of the lens.
Given: n = 1.5, R1 = +20 cm = +0.20 m, R2 = −20 cm = −0.20 m (biconvex, so second surface curves away)
Step 1: Write the Lensmaker's equation: \( \frac{1}{f} = (n-1)\left(\frac{1}{R_1} – \frac{1}{R_2}\right) \)
Step 2: Substitute: \( \frac{1}{f} = (1.5 – 1)\left(\frac{1}{0.20} – \frac{1}{-0.20}\right) = 0.5 \times (5 + 5) = 0.5 \times 10 = 5 \)
Step 3: Solve for f: \( f = \frac{1}{5} = 0.20 \) m = 20 cm
Step 4: Calculate power: \( P = \frac{1}{f} = \frac{1}{0.20} = +5 \) D
Answer
Focal length f = 20 cm (0.20 m). Power P = +5 dioptres. The positive sign confirms it is a converging (convex) lens.
Example 3 (JEE/NEET Level) — Young's Double Slit Experiment
Problem: In a Young's double slit experiment, the slit separation is 0.5 mm and the screen is placed 1.0 m away. When light of wavelength 600 nm is used, find (a) the fringe width and (b) the position of the 3rd bright fringe from the central maximum.
Given: d = 0.5 mm = 5 × 10−4 m, D = 1.0 m, λ = 600 nm = 6 × 10−7 m
Step 1: Write the fringe width formula: \( \beta = \frac{\lambda D}{d} \)
Step 2: Substitute values: \( \beta = \frac{6 \times 10^{-7} \times 1.0}{5 \times 10^{-4}} = \frac{6 \times 10^{-7}}{5 \times 10^{-4}} \)
Step 3: Calculate: \( \beta = 1.2 \times 10^{-3} \) m = 1.2 mm
Step 4: Position of 3rd bright fringe: \( y_3 = n\beta = 3 \times 1.2 = 3.6 \) mm
Step 5: Verify using path difference condition: For the 3rd bright fringe, \( \Delta x = 3\lambda = 3 \times 600 = 1800 \) nm. This confirms constructive interference at that position.
Answer
Fringe width β = 1.2 mm. The 3rd bright fringe is located 3.6 mm from the central maximum.
CBSE Exam Tips 2025-26
- Always state the sign convention first. In CBSE 2025-26 board exams, marks are awarded for correctly applying the New Cartesian Sign Convention. Write it before solving any mirror or lens problem.
- Memorise the difference between mirror and lens formulas. The mirror formula uses \( \frac{1}{v} + \frac{1}{u} \), while the lens formula uses \( \frac{1}{v} – \frac{1}{u} \). This is the single most common error students make.
- Draw ray diagrams for full marks. CBSE awards separate marks for ray diagrams in optics questions. We recommend practising at least 10 ray diagrams before the exam.
- Power of a lens must be in dioptres. Always convert focal length to metres before calculating power. A focal length of 25 cm gives P = 1/0.25 = +4 D, not 1/25.
- Wave optics — remember path difference conditions. Constructive interference occurs at \( \Delta x = n\lambda \) and destructive at \( \Delta x = (n + \frac{1}{2})\lambda \). Write these clearly in answers.
- Revise the prism formula. The relation \( n = \frac{\sin\left(\frac{A+\delta_m}{2}\right)}{\sin\left(\frac{A}{2}\right)} \) is directly asked in Class 12 board exams. Practise its derivation thoroughly.
Common Mistakes to Avoid
- Confusing mirror and lens sign conventions for magnification. For mirrors, \( m = -v/u \). For lenses, \( m = v/u \). Many students use the same formula for both and lose marks.
- Using centimetres instead of metres for power. Power P = 1/f requires f in metres. Using centimetres gives an answer 100 times too large.
- Forgetting the negative sign for concave mirrors and diverging lenses. A concave mirror has a negative focal length. A diverging (concave) lens also has a negative focal length. Always assign signs before substituting.
- Mixing up critical angle and Brewster's angle. The critical angle formula is \( \sin\theta_c = 1/n \). Brewster's angle uses \( \tan\theta_B = n \). These are different phenomena with different formulas.
- Applying the thin lens formula to thick lenses. The standard lens formula \( \frac{1}{f} = \frac{1}{v} – \frac{1}{u} \) is valid only for thin lenses. For thick lenses, the Lensmaker's equation must account for the lens thickness.
JEE/NEET Application of Optics Formula
In our experience, JEE aspirants encounter optics in almost every paper — typically 2 to 4 questions in JEE Main and 1 to 2 in NEET. The most frequently tested optics formulas are the mirror formula, lens formula, Snell's law, and Young's double slit fringe width. Here are the key application patterns:
Pattern 1 — Combination of Lenses and Mirrors
JEE Advanced frequently tests problems where a lens and a mirror are placed coaxially. The image formed by the lens acts as the object for the mirror. You apply the lens formula first, then use that image as the object for the mirror formula. The combined power of lenses in contact is \( P = P_1 + P_2 \). This concept is tested in Class 12, Chapter 9.
Pattern 2 — Total Internal Reflection and Optical Fibres
NEET and JEE Main regularly ask about total internal reflection (TIR). The critical angle is given by \( \sin\theta_c = \frac{n_2}{n_1} \) where \( n_1 > n_2 \). TIR occurs only when light travels from a denser to a rarer medium and the angle of incidence exceeds the critical angle. Optical fibre problems use this concept directly.
Pattern 3 — Wave Optics: Interference and Diffraction
Young's double slit experiment is a guaranteed topic in JEE Main. Students must know the fringe width formula \( \beta = \frac{\lambda D}{d} \), the effect of changing wavelength, screen distance, or slit separation on fringe width, and the conditions for bright and dark fringes. Single slit diffraction — where the central maximum has width \( \frac{2\lambda D}{a} \) — is also tested. In our experience, students who practise numerical variations of these formulas score consistently in this section.
FAQs on Optics Formula
For a deeper understanding of related physics topics, explore our comprehensive guides on the Spring Constant Formula, the Angular Displacement Formula, and the Flow Rate Formula. For the full collection of physics formulas covered in NCERT, visit our Physics Formulas hub. For official NCERT textbook resources, refer to the NCERT official website.