CBSE Class 12 Physics: Most Important Derivations & Numericals for Board Exams 2027

Master Every High-Scoring Derivation & Numerical in Physics

CBSE Class 12 Physics derivations carry 3 to 5 marks each and appear in nearly every long-answer question. The 2025-26 exam pattern allocates 30% marks to constructed-response questions including derivations and numericals. This chapter-wise guide covers the most frequently asked derivations from Coulomb's law to photoelectric effect, must-do numericals from every unit, and a proven scoring strategy. Stop guessing what will come in the exam. Prepare what actually does.

In This Article

• Chapter-Wise Marks Weightage (2025-26)
• Electrostatics: Derivations & Numericals
• Current Electricity: Key Derivations
• Magnetic Effects: Biot-Savart & Ampere's Law
• EMI & AC Circuits: Must-Know Derivations
• Optics: Lens Formula, Mirror Formula & More
• Modern Physics: Photoelectric Effect & de Broglie
• Must-Do Numericals: Chapter-Wise List
• Derivation Writing & Scoring Strategy
• Frequently Asked Questions

1. Coulomb's Law in vector form — Force between two point charges, consistency with Newton's Third Law. Draw a clear diagram showing charge positions and force vectors. (2-3 marks)
2. Electric field due to an electric dipole — (a) On the axial line and (b) On the equatorial plane. Both cases are frequently asked. Show the superposition principle clearly. (3-5 marks)
3. Gauss's Theorem (statement and proof) — Derive the expression for electric flux through a closed surface enclosing charge q. This is the foundation for the next three derivations. (3 marks)
4. Gauss's Law application: Infinitely long straight charged wire — Choose a cylindrical Gaussian surface, show flux through curved surface only. Final expression: E = λ/2πε₀r. (3-5 marks)
5. Gauss's Law application: Uniformly charged infinite plane sheet — Use a cylindrical Gaussian surface perpendicular to the sheet. Final expression: E = σ/2ε₀. (3-5 marks)
6. Gauss's Law application: Uniformly charged thin spherical shell — Field at a point (a) outside, (b) on the surface, and (c) inside the shell. This has appeared repeatedly. (3-5 marks)
7. Electric potential due to a point charge — Derive V = q/4πε₀r using work-energy theorem. (2-3 marks)
8. Electric potential due to a dipole — At any general point, then simplify for axial and equatorial positions. (3 marks)
9. Capacitance of a parallel plate capacitor — With and without dielectric slab. Derive C = ε₀A/d and show how inserting a dielectric of thickness t changes the result. (3-5 marks)
10. Energy stored in a capacitor — Derive U = ½CV² = Q²/2C = ½QV. Show the step-by-step work done in charging. (2-3 marks)

• Force between two charges in vacuum and in a medium (Coulomb's Law)
• Electric field and potential at a point due to multiple charges (superposition)
• Capacitors in series and parallel combinations — finding equivalent capacitance, charge, and voltage
• Energy stored before and after connecting charged capacitors
• Effect of inserting a dielectric on charge, voltage, and capacitance
• Electric flux through a surface placed in a non-uniform field

1. Ohm's Law from drift velocity — Derive the relation j = σE using the expression for drift velocity (v_d = eEτ/m). Show how V = IR follows naturally. (3 marks)
2. Resistivity and its temperature dependence — Derive ρ = m/ne²τ and explain why resistivity increases with temperature for conductors and decreases for semiconductors. (2-3 marks)
3. EMF and internal resistance of a cell — Derive the terminal voltage relation V = E - Ir. Show the condition for maximum current. (2-3 marks)
4. Kirchhoff's rules and Wheatstone Bridge condition — Apply Kirchhoff's junction and loop rules to derive the balance condition P/Q = R/S. (3 marks)
5. Metre bridge working principle — Derive the expression for unknown resistance using the Wheatstone bridge principle. (2-3 marks)
6. Potentiometer: Comparison of EMFs — Derive E₁/E₂ = l₁/l₂ and explain why a potentiometer is preferred over a voltmeter for measuring EMF. (3 marks)

1. Biot-Savart Law: Magnetic field due to a current-carrying circular loop — Derive the expression for the magnetic field at the centre and at an axial point. At centre: B = μ₀I/2R. At axial point: B = μ₀IR²/2(R² + x²)^3/2. Draw the diagram showing the current element and field direction. (3-5 marks)
2. Ampere's Circuital Law: Statement and proof — State the law and derive ∮B·dl = μ₀I for a long straight conductor. (2-3 marks)
3. Ampere's Law application: Magnetic field inside a solenoid — Choose a rectangular Amperian loop and derive B = μ₀nI, where n is the number of turns per unit length. (3-5 marks)
4. Ampere's Law application: Magnetic field inside a toroid — Derive B = μ₀NI/2πr. Show that the field is zero outside and inside the core. (3 marks)
5. Force between two parallel current-carrying conductors — Derive the expression and use it to define one ampere. Like currents attract, unlike currents repel. (3 marks)
6. Torque on a current loop in a magnetic field — Derive τ = NIAB sinθ = M × B. This connects to the magnetic dipole moment concept. (3 marks)

1. Faraday's Law of Electromagnetic Induction — Derive the expression for induced EMF: ε = -dΦ_B/dt. Explain Lenz's law as a consequence of conservation of energy. (3 marks)
2. Motional EMF — Derive ε = Blv for a straight conductor moving in a uniform magnetic field. Show the direction of induced current using Fleming's right-hand rule. (3 marks)
3. Self-inductance of a long solenoid — Derive L = μ₀n²Al. Define the henry as the SI unit of inductance. (3 marks)
4. Mutual inductance between two co-axial solenoids — Derive M = μ₀n₁n₂Al. Show that M₁₂ = M₂₁. (3 marks)
5. AC generator (alternator) working principle — Derive the expression for instantaneous EMF: ε = NBAω sinωt. Draw the labelled diagram and explain the role of slip rings. (5 marks)
6. Impedance of a series LCR circuit — Derive Z = √(R² + (X_L - X_C)²) using the phasor diagram method. Derive the resonance condition X_L = X_C and show that resonant frequency f₀ = 1/2π√(LC). (5 marks)
7. Power in an AC circuit — Derive P = V_rmsI_rmscosφ, where cosφ is the power factor. Discuss the special cases: purely resistive (φ = 0), purely inductive (φ = 90°), and at resonance. (3 marks)
8. Transformer: Working principle and EMF equation — Derive V_s/V_p = N_s/N_p and explain step-up vs step-down transformers. Discuss energy losses in transformers. (3 marks)

1. Mirror formula — Derive 1/v + 1/u = 1/f for a concave mirror using geometry and sign convention. Draw a clear ray diagram showing object, image, centre of curvature, and focal point. (3-5 marks)
2. Refraction at a spherical surface — Derive μ₁/u + μ₂/v = (μ₂ - μ₁)/R. This is the foundation for the lens maker's formula. (3-5 marks)
3. Lens Maker's Formula — Derive 1/f = (μ - 1)(1/R₁ - 1/R₂) using refraction at two spherical surfaces. This is one of the most asked 5-mark derivations. (3-5 marks)
4. Thin lens formula — Derive 1/v - 1/u = 1/f and the magnification expression m = v/u. (2-3 marks)
5. Refraction through a prism — Derive A + δ = i + e and μ = sin[(A + δ_m)/2] / sin(A/2) at minimum deviation. Draw the prism diagram with angles clearly labelled. (3-5 marks)
6. Total Internal Reflection — Derive the critical angle condition sin C = 1/μ and explain applications (optical fibre, mirage, diamond brilliance). (2-3 marks)

1. Young's Double Slit Experiment — Derive the fringe width expression β = λD/d. Show conditions for constructive and destructive interference. Draw the experimental setup diagram. (5 marks)
2. Huygens' Principle: Refraction at a plane surface — Use Huygens' wave theory to derive Snell's law (sin i / sin r = v₁/v₂). (3 marks)
3. Single slit diffraction — Derive the condition for first minimum: a sinθ = λ. Explain the central maximum width. (3 marks)

1. Einstein's Photoelectric Equation — Derive KE_max = hν - φ (or ½mv²_max = hν - hν₀). Explain the three observations that classical wave theory could not explain: threshold frequency, instantaneous emission, and intensity independence of KE_max. (3-5 marks)
2. de Broglie Wavelength — Derive λ = h/mv = h/p. Show the wave-particle duality concept. Derive the expression for de Broglie wavelength of an electron accelerated through potential V: λ = h/√(2meV) = 1.227/√V nm. (3 marks)
3. Bohr's Model: Radius and energy of hydrogen atom — Derive r_n = n²h²ε₀/πme² and E_n = -13.6/n² eV. Explain the quantisation condition and the origin of spectral lines. (5 marks)
4. Hydrogen spectrum series — Derive the wave number relation 1/λ = R(1/n₁² - 1/n₂²). Identify Lyman, Balmer, Paschen, Brackett, and Pfund series. (2-3 marks)
5. Nuclear binding energy and mass defect — Define mass defect Δm = [Zm_p + (A-Z)m_n] - M. Derive binding energy BE = Δmc². Explain the BE per nucleon curve and its significance for fission and fusion. (3 marks)
6. Radioactive decay law — Derive N = N₀e^-λt and the relation between half-life and decay constant: T_½ = 0.693/λ. (2-3 marks)

1. Draw a neat, labelled diagram first — CBSE awards 1 mark separately for diagrams in most derivation questions. Use a pencil and ruler for straight lines.
2. State the law or principle clearly — Begin with the formal statement (e.g., "By Gauss's theorem, the total electric flux through a closed surface is 1/ε₀ times the net charge enclosed").
3. Define all variables and symbols — Examiners look for this. Write "where λ = linear charge density, r = perpendicular distance" etc.
4. Show every algebraic step — Never skip intermediate steps. CBSE follows step-marking: each logical step earns partial credit even if the final answer is wrong.
5. Box or underline the final formula — Make the final expression visually distinct so the examiner can spot it immediately.
6. Write units where applicable — Especially in numericals. Missing units cost marks even when the calculation is correct.
7. Practice writing under timed conditions — A 5-mark derivation should take 8-10 minutes. A 3-mark derivation should take 5-6 minutes. Time yourself during practice.

Q: How many derivations are asked in the CBSE Class 12 Physics board exam?

Typically 3 to 5 derivation-based questions appear in the paper, carrying a total of 12 to 18 marks. These include 2-3 long-answer questions (5 marks each) and 1-2 short-answer questions (3 marks each). In the 2025-26 pattern, derivations may come split as Part (a) being the derivation (3 marks) and Part (b) being a related conceptual or numerical question (2 marks).

Q: Which derivation is most frequently asked in CBSE 12 Physics?

Gauss's Law applications (electric field due to infinitely long wire, infinite plane sheet, or spherical shell) are the most repeated derivations, appearing in 4 out of the last 6 years. Lens Maker's formula, Young's Double Slit Experiment fringe width, and LCR circuit impedance are the next most frequent. Preparing these four derivations thoroughly guarantees you can answer at least one long-answer question.

Q: Do I get marks for diagrams in derivation questions?

Yes. CBSE typically awards 1 mark separately for a correct, well-labelled diagram in derivation questions. Even if you cannot complete the full derivation, drawing an accurate diagram with proper labels earns you partial credit. Always use a pencil and ruler for straight lines, label all angles and distances clearly, and indicate the direction of vectors with arrows.

Q: How should I prepare numericals for CBSE 12 Physics?

Follow the formula-substitute-calculate-units method for every numerical. First write the relevant formula, then substitute known values with correct units, show the calculation steps, and always write the final answer with proper units. Practise at least 30 numericals per week, focusing on Electrostatics, Current Electricity, AC circuits, and Ray Optics. NCERT textbook examples and exercises should be your first priority, followed by CBSE previous year numericals.

Q: Can I score full marks in Physics without memorising all derivations?

You can score 55-60 out of 70 without memorising every derivation, but not 65+. Derivations carry 12-18 marks directly, and understanding them also helps answer competency-based questions that test the same concepts in applied form. The most efficient approach is to master the top 15 derivations (listed in our Priority 1 and Priority 2 categories) rather than trying to memorise all 30+. Focus on understanding the logical flow rather than rote memorisation.

Q: What are the most important chapters for numericals?

Current Electricity and Electrostatics are the most numerical-intensive chapters, with Kirchhoff's rules, Wheatstone bridge, potentiometer, capacitor combinations, and Coulomb's Law problems appearing every year. AC circuits (LCR impedance, resonant frequency, power factor) and Ray Optics (lens/mirror formula, prism deviation) are equally important. Modern Physics numericals on photoelectric effect and de Broglie wavelength are shorter but carry 2-3 marks and are easy to score.

Q: How do I handle case-based questions on derivations?

Case-based questions present a real-world scenario (e.g., a capacitor in a defibrillator or electromagnetic induction in a power plant) and ask 4-5 sub-questions based on it. These test conceptual understanding rather than derivation skills. The strategy is: read the passage twice, identify which chapter concept is being tested, and answer each sub-question independently. Even if you cannot answer one part, attempt the others since marks are awarded separately for each sub-question.

Q: Is NCERT enough for Physics derivations and numericals?

NCERT is the primary source and covers about 85% of what is asked. Every derivation in the CBSE paper comes directly from NCERT or is a slight variation. For numericals, start with all NCERT solved examples and exercise problems. Once NCERT is complete, solve CBSE previous year papers (2020-2025) to cover the remaining patterns. Supplementary books are optional and only needed if you are targeting 68+ out of 70.

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