CBSE Class 12 Physics (Code 042) • NCERT Statutory Blueprint
CBSE Class 12 Physics All Formulas, Derivations & SI Units Sheet
The definitive chapter-wise formula and derivation compilation for CBSE Class 12 Board candidates. Features complete mathematical definitions, SI units, dimensional formulas, statutory NCERT derivation starting assumptions, and common examiner traps across all 14 syllabus chapters.
📑 Quick Navigation / Jump to Chapter
1. Electric Charges & Fields
2. Electrostatic Potential & Capacitance
3. Current Electricity & Circuits
4. Moving Charges & Magnetism
5. Magnetism and Matter
6. Electromagnetic Induction (EMI)
7. Alternating Current (AC)
8. Electromagnetic Waves
9. Ray Optics & Optical Instruments
10. Wave Optics (Interference & Diffraction)
11. Dual Nature of Radiation & Matter
12. Atoms (Bohr Model & Spectra)
13. Nuclei (Binding Energy & Q-Value)
14. Semiconductor Electronics & Diodes
⭐ Top 15 Mandatory 5-Mark Derivations
Chapter 1: Electric Charges and Fields
Unit I • Electrostatics (Chapters 1 & 2 combine for 16 Marks)| Concept / Law | Mathematical Formula | SI Units & Key Variables |
|---|---|---|
| Coulomb's Law | F = (1 / 4πε₀) · (|q₁ q₂| / r²) Vector: F₁₂ = (1 / 4πε₀) · (q₁ q₂ / r²) · r̂₂₁ |
N (Newton) 1/(4πε₀) = 9 × 10⁹ N·m²/C² ε₀ = 8.854 × 10⁻¹² C²/N·m² |
| Electric Dipole Moment | p = q · 2a (Directed -q to +q) | C·m (Coulomb-meter) Vector quantity pointing from negative to positive charge. |
| Dipole Field: Axial Point | E_axial = (1 / 4πε₀) · (2pr / (r² - a²)²) For r >> a: E_axial ≈ (1 / 4πε₀) · (2p / r³) |
N/C or V/m Field points in the direction of dipole moment p. |
| Dipole Field: Equatorial Point | E_eq = (1 / 4πε₀) · (p / (r² + a²)^(3/2)) For r >> a: E_eq ≈ (1 / 4πε₀) · (p / r³) |
N/C Note: E_axial = 2 · E_eq. Equatorial field points anti-parallel to p. |
| Torque & Potential Energy on Dipole | τ = p × E = p E sin θ U = -p · E = -p E cos θ |
Torque: N·m; Energy: J (Joules) Stable equilibrium at θ = 0°; Unstable at θ = 180°. |
| Gauss's Law & Flux | Φ_E = ∮ E · dA = q_encl / ε₀ | N·m²/C or V·m Flux depends only on enclosed charge, independent of shape/size. |
| Field: Infinite Straight Wire | E = λ / (2πε₀ r) | λ = linear charge density (C/m). Field varies as 1/r. |
| Field: Infinite Plane Sheet | E = σ / (2ε₀) | σ = surface charge density (C/m²). Independent of distance r! |
| Field: Thin Spherical Shell | Inside (r < R): E = 0 Outside (r ≥ R): E = (1 / 4πε₀) · (q / r²) |
At surface: E_max = σ / ε₀. Inside a conductor or conducting shell, field is strictly zero. |
📌 Key 5-Mark Derivation Blueprint: Gauss's Law to Plane Sheet
1. Consider a thin infinite plane sheet with uniform surface charge density σ. Symmetry dictates E is perpendicular to the sheet pointing outward on both sides.
2. Choose a cylindrical Gaussian surface of cross-sectional area A and length 2r cutting symmetrically through the sheet.
3. Flux through curved cylindrical surface = ∮ E · dA = E · A · cos 90° = 0.
4. Flux through two end circular caps = E · A + E · A = 2EA.
5. Enclosed charge = q_encl = σ · A.
6. By Gauss's Law: 2EA = (σ · A) / ε₀ ⟹ E = σ / (2ε₀) (Independent of r).
⚠️ Examiner Pitfall Alert:
In dipole derivations, students often confuse E_axial and E_eq. Remember that E_axial = 2 · E_eq at large distances (r >> a), and E_eq points in the direction OPPOSITE to dipole moment vector p.
Chapter 2: Electrostatic Potential and Capacitance
Unit I • Conservative Fields, Dielectrics & Stored Energy| Concept | Formula | Remarks |
|---|---|---|
| Potential due to Point Charge | V = (1 / 4πε₀) · (q / r) | V (Volt = J/C). Scalar quantity. V varies as 1/r. |
| Potential due to Dipole | V = (1 / 4πε₀) · (p cos θ / r²) | Axial (θ = 0°): V = kp/r². Equatorial (θ = 90°): V = 0 everywhere! |
| Field-Potential Relation | E = -dV / dr | Field points in the direction in which potential decreases steepest. Equipotential surfaces are always perpendicular to electric field lines. |
| Parallel Plate Capacitor | Vacuum: C₀ = ε₀ A / d With Dielectric: C = K · C₀ = ε_r ε₀ A / d |
F (Farad). C depends only on geometric factors (A, d) and permittivity of medium, NOT on Q or V. |
| Capacitor with Slab (Thickness t) | Dielectric Slab: C = ε₀ A / (d - t + t/K) Conducting Slab (K = ∞): C = ε₀ A / (d - t) |
Always increases capacitance because effective separation between plates decreases. |
| Energy Stored in Capacitor | U = (1/2) C V² = (1/2) Q V = Q² / (2C) Energy Density: u = (1/2) ε₀ E² |
Energy is stored in the electrostatic field between the plates. Energy density unit: J/m³. |
| Combination of Capacitors | Series: 1/C_s = 1/C₁ + 1/C₂ + ... (Q same) Parallel: C_p = C₁ + C₂ + ... (V same) |
Series divides voltage; parallel divides charge. |
| Common Potential & Energy Loss | V_common = (C₁V₁ + C₂V₂) / (C₁ + C₂) ΔU = (1/2) [C₁C₂ / (C₁ + C₂)] · (V₁ - V₂)² |
Energy loss ΔU is always positive and dissipates as heat and EM radiation in connecting wires. |
Chapter 3: Current Electricity
Unit II • 7 Marks in Board Blueprints| Parameter / Law | Formula | Key Physical Constants |
|---|---|---|
| Drift Velocity & Current | v_d = -e E τ / m I = n e A v_d |
τ = relaxation time (s), m = electron mass (9.1 × 10⁻³¹ kg), e = 1.6 × 10⁻¹⁹ C, n = electron density (m⁻³). |
| Current Density & Ohm's Law (Microscopic) | j = I / A = σ E ρ = 1/σ = m / (n e² τ) |
j = current density (A/m²), σ = conductivity (S/m or Ω⁻¹m⁻¹), ρ = resistivity (Ω·m). |
| Temperature Dependence of Resistance | R_T = R₀ (1 + α ΔT) ρ_T = ρ₀ (1 + α ΔT) |
α = temperature coefficient of resistance (K⁻¹ or °C⁻¹). Metals: α > 0; Semiconductors/Insulators: α < 0. |
| Cell EMF, Terminal PD & Internal Resistance | Discharging: V = E - I r Charging: V = E + I r Internal Resistance: r = R (E/V - 1) |
E = open-circuit EMF (V), V = terminal potential difference (V), r = internal resistance (Ω). Maximum current I_max = E/r when external R = 0. |
| Kirchhoff's Laws | KCL (Junction Rule): Σ I = 0 KVL (Loop Rule): Σ ΔV = Σ E - Σ IR = 0 |
KCL is based on Conservation of Charge. KVL is based on Conservation of Energy. |
| Balanced Wheatstone Bridge | P / Q = R / S (when I_g = 0) | When bridge is balanced, no current flows through galvanometer; interchanging battery and galvanometer does not affect balance condition! |
Chapters 4 & 5: Moving Charges, Magnetism & Matter
Unit III • 8 Marks| Law / Configuration | Formula | Notes |
|---|---|---|
| Lorentz Force | F = q(E + v × B) Magnetic: F_m = q(v × B) = q v B sin θ |
Magnetic force does ZERO work because F_m is always perpendicular to velocity v (power P = F · v = 0). Kinetic energy remains constant. |
| Biot-Savart Law | dB = (μ₀ / 4π) · (I dl × r̂ / r²) | μ₀ / 4π = 10⁻⁷ T·m/A. Valid for steady currents in conductors. |
| Field: Axis of Circular Loop | B = (μ₀ I R²) / [2(R² + x²)^(3/2)] At Center (x = 0): B_center = μ₀ I / (2R) |
For N turns, multiply by N. At large distances (x >> R): B = (μ₀ / 4π) · (2m / x³) where magnetic dipole moment m = I A = I (π R²). |
| Ampere's Circuital Law | ∮ B · dl = μ₀ I_encl Long straight wire: B = μ₀ I / (2π r) Solenoid (inside): B = μ₀ n I |
n = N / L (turns per unit length). At ends of solenoid, field drops to B_end = (1/2) μ₀ n I. |
| Force between Parallel Currents | F / L = (μ₀ I₁ I₂) / (2π d) | Parallel currents in SAME direction ATTRACT; antiparallel currents REPEL. Defines 1 Ampere (F/L = 2 × 10⁻⁷ N/m at d = 1 m). |
| Moving Coil Galvanometer | Deflection: θ = (N A B / C) · I Current Sensitivity: I_s = θ / I = NAB / C Voltage Sensitivity: V_s = θ / V = NAB / (C R_g) |
Radial magnetic field produced using concave magnetic poles and soft iron core ensures θ is linearly proportional to I (sin 90° = 1 always). |
| Galvanometer Conversions | To Ammeter: Shunt S = I_g R_g / (I - I_g) in parallel. To Voltmeter: Resistor R = (V / I_g) - R_g in series. |
Ideal Ammeter has ZERO resistance. Ideal Voltmeter has INFINITE resistance. |
Chapters 6 & 7: Electromagnetic Induction & Alternating Current
Unit IV • 8 Marks| Concept | Formula | Key Variables |
|---|---|---|
| Faraday's & Lenz's Law | E = -dΦ_B / dt = -N · d(B A cos θ) / dt | Negative sign represents Lenz's Law, which is a consequence of Conservation of Energy. |
| Motional EMF | Translational: E = B v l Rotational: E = (1/2) B ω l² |
v = velocity perpendicular to B and length l. Rotational rod rotated about one end with angular speed ω. |
| Self & Mutual Inductance | Φ = L I ⟹ E = -L (dI / dt) Solenoid: L = μ₀ N² A / l Mutual: M = μ₀ N₁ N₂ A / l |
Unit: H (Henry). Stored magnetic energy: U_B = (1/2) L I². Energy density u_B = B² / (2μ₀). |
| AC Quantities (RMS Values) | I_rms = I₀ / √2 ≈ 0.707 I₀ V_rms = V₀ / √2 ≈ 0.707 V₀ |
Household supply (220 V AC) has peak voltage V₀ = 220 × √2 ≈ 311 V, making AC more lethal than DC of same nominal voltage! |
| Reactance & Series LCR Impedance | X_L = ω L = 2π f L X_C = 1 / (ω C) = 1 / (2π f C) Z = √[ R² + (X_L - X_C)² ] |
Units: Ω (Ohm). Phase angle: tan φ = (X_L - X_C) / R. Pure inductor: current lags voltage by 90°. Pure capacitor: current leads by 90°. |
| Resonance & Quality Factor (Q) | ω₀ = 1 / √(L C) ⟹ f₀ = 1 / [2π √(L C)] Q = (ω₀ L) / R = (1 / R) · √(L / C) |
At resonance: X_L = X_C, Z_min = R, current I_max = V/R, power factor cos φ = 1. High Q factor means sharp selectivity (essential in radio tuning). |
| AC Power & Power Factor | P_avg = V_rms · I_rms · cos φ | Power factor = cos φ = R / Z. In pure inductive or capacitive circuit, cos 90° = 0, so average power dissipated = 0 (Wattless current). |
| Ideal Transformer Equations | V_s / V_p = N_s / N_p = I_p / I_s = k | k > 1 (Step-up: V_s > V_p, I_s < I_p); k < 1 (Step-down). Core laminated to minimize eddy current losses. |
Chapters 9 & 10: Ray & Wave Optics
Unit VI • Optics is the heaviest unit with 18 Marks!| Optical Formula | Mathematical Expression | Sign Convention & Crucial Notes |
|---|---|---|
| Mirror & Lens Formulas | Mirror: 1/v + 1/u = 1/f (m = -v/u) Lens: 1/v - 1/u = 1/f (m = +v/u) |
Cartesian convention: light travels left to right. Real objects have u < 0. Convex lens f > 0; Concave lens f < 0. |
| Lens Maker's Formula | 1/f = (μ₂/μ₁ - 1) · [1/R₁ - 1/R₂] | μ₂ = lens medium, μ₁ = surrounding medium. If immersed in liquid with μ₁ > μ₂, a convex lens behaves as a concave lens! |
| Refraction at Spherical Surface | μ₂ / v - μ₁ / u = (μ₂ - μ₁) / R | Fundamental derivation stepping stone to Lens Maker's formula. |
| Prism Formula | μ = sin[(A + D_m)/2] / sin(A/2) | A = angle of prism, D_m = angle of minimum deviation. At minimum deviation: i = e, r₁ = r₂ = A/2, and refracted ray is parallel to prism base. |
| Compound Microscope Magnification | Normal Adjustment (image at ∞): m = -(v_o / u_o) · (D / f_e) Near Point (image at D): m = -(v_o / u_o) · (1 + D / f_e) |
D = least distance of distinct vision (25 cm). Both objective and eyepiece must have small focal lengths (f_o < f_e < D) with f_o very small for large m. Tube length L ≈ v_o + f_e. |
| Astronomical Telescope Magnification | Normal Adjustment: m = -f_o / f_e (L = f_o + f_e) Near Point: m = -(f_o / f_e) · (1 + f_e / D) |
Objective must have large focal length and large aperture (for high light-gathering power and resolution); eyepiece must have small focal length. |
| Wave Optics: YDSE Fringe Width | β = λ D / d Bright Fringe: y_n = n λ D / d (Path diff = nλ) Dark Fringe: y_n = (n - 1/2) λ D / d |
λ = wavelength, D = distance from slits to screen, d = slit separation. If whole apparatus is immersed in medium of index μ: β' = β / μ (fringes shrink). |
| Single Slit Diffraction | Minima Condition: a sin θ = n λ (n = 1, 2, ...) Central Maximum Width: 2β₀ = 2 λ D / a |
a = slit width. Note key difference from YDSE: In diffraction, a sin θ = nλ gives MINIMA, whereas in YDSE d sin θ = nλ gives MAXIMA! Central maximum is twice as wide as secondary maxima. |
Chapters 11–14: Modern Physics & Semiconductor Electronics
Units VII, VIII, IX • 20 Marks Combined| Phenomenon | Formula | Key Constant & Interpretation |
|---|---|---|
| Einstein's Photoelectric Equation | K_max = h ν - φ₀ = e V₀ h ν = h ν₀ + (1/2) m v_max² |
h = 6.626 × 10⁻³⁴ J·s, φ₀ = work function (eV), V₀ = stopping potential (V). Slope of V₀ vs ν graph is h/e (universal constant, independent of metal). |
| de Broglie Wavelength | λ = h / p = h / (m v) = h / √(2m E) Accelerated by V: λ = 1.227 / √V nm |
For thermal neutron at T Kelvin: λ = h / √(3m k_B T). |
| Bohr Model Formulas (Hydrogen-like) | Radius: r_n = 0.529 · (n² / Z) Å Velocity: v_n = 2.18 × 10⁶ · (Z / n) m/s Energy: E_n = -13.6 · (Z² / n²) eV |
Kinetic energy K = -E_n; Potential energy U = 2E_n = -2K. Negative total energy signifies that the electron is bound to the nucleus. |
| Rydberg Formula & Spectral Series | 1 / λ = R · Z² · [1/n₁² - 1/n₂²] | R = 1.097 × 10⁷ m⁻¹. Lyman (n₁=1, UV); Balmer (n₁=2, Visible); Paschen (n₁=3, IR); Brackett (n₁=4, IR); Pfund (n₁=5, IR). |
| Nuclear Radius, Density & Q-Value | R = R₀ · A^(1/3) (R₀ ≈ 1.2 fm) Nuclear Density: ρ_nuc ≈ 2.3 × 10¹⁷ kg/m³ Q = [m_reactants - m_products] · 931.5 MeV |
Nuclear density is INDEPENDENT of mass number A (constant for all nuclei). Q > 0 = Exothermic (fission/fusion); Q < 0 = Endothermic. |
| Semiconductor Mass Action Law | n_e · n_h = n_i² Conductivity: σ = e (n_e μ_e + n_h μ_h) |
Intrinsic: n_e = n_h = n_i. n-type: n_e ≈ N_D >> n_h. p-type: n_h ≈ N_A >> n_e. Depletion layer width decreases in forward bias and increases in reverse bias. |
⭐ Top 15 Mandatory 5-Mark Derivations Master Checklist
These 15 proofs appear every single year in CBSE Class 12 Board Exams Section D & E1. Axial & Equatorial Electric Dipole Field
Resolve components using Coulomb's law. Axial: E = 2kp/r³ along p; Equatorial: E = kp/r³ anti-parallel to p.
2. Gauss's Law: Field of Infinite Line Charge
Construct co-axial cylindrical Gaussian surface. Flux = E(2πrl) = q_encl/ε₀ = λl/ε₀ ⟹ E = λ / (2πε₀r).
3. Parallel Plate Capacitor with Dielectric Slab
V = E₀(d - t) + E_slab(t) = E₀(d - t + t/K). Substitute E₀ = σ/ε₀ = Q/(Aε₀) ⟹ C = ε₀A / (d - t + t/K).
4. Microscopic Ohm's Law (Drift Velocity)
Apply F = -eE = ma ⟹ v_d = -eEτ/m. Current I = neAv_d ⟹ I = (ne²Aτ/m)E ⟹ j = σE and ρ = m/(ne²τ).
5. Magnetic Field on Axis of Circular Loop
Biot-Savart dB = (μ₀/4π) · (Idl/r²). Diametrically opposite perpendicular components cancel; axial dB sin θ integrate to B = μ₀IR² / [2(R² + x²)^(3/2)].
6. Force Between Two Long Parallel Conductors
Field of wire 1 at wire 2: B₁ = μ₀I₁ / (2πd). Force on length L: F₂₁ = I₂L B₁ sin 90° = (μ₀I₁I₂L) / (2πd). Defines standard SI Ampere.
7. Moving Coil Galvanometer Torque Equation
Deflecting torque τ_d = NIAB sin θ = NIAB (radial field). Restoring torque τ_r = Cθ. Equating gives I = (C / NAB)θ.
8. Motional EMF in Sliding Conductor
Magnetic flux enclosed Φ = B · A = B l x. Rate of flux change dΦ/dt = B l (dx/dt) = B l v. By Faraday's Law, E = -dΦ/dt = Bvl.
9. Series LCR Impedance by Phasor Diagram
Phasors: V_R = IR in phase, V_L = IX_L leads by π/2, V_C = IX_C lags by π/2. Resultant V² = V_R² + (V_L - V_C)² ⟹ Z = √[R² + (X_L - X_C)²].
10. Refraction at Spherical Surface
Snell's law for small angles: n₁ i = n₂ r. Relate exterior angles to interior: i = α + γ, γ = r + β. Yields n₂/v - n₁/u = (n₂ - n₁)/R.
11. Lens Maker's Formula
Apply single spherical surface refraction to both faces of a thin lens and add equations to eliminate intermediate virtual image distance v₁.
12. Prism Refraction & Minimum Deviation
Geometric proof: A = r₁ + r₂ and δ = i + e - A. At minimum deviation D_m, i = e and r₁ = r₂ = A/2. Applying Snell's law yields μ = sin[(A+D_m)/2] / sin(A/2).
13. Huygens' Principle: Proof of Snell's Law
Consider incident plane wavefront AB on interface. Time taken to reach C is t = BC/v₁. Secondary wave from A travels distance AD = v₂t in medium 2. sin i / sin r = (BC/AC) / (AD/AC) = v₁/v₂ = n₂/n₁.
14. Young's Double Slit Fringe Width (β)
Path difference Δx = S₂P - S₁P = yd/D. For bright fringes yd/D = nλ ⟹ y_n = nλD/d. Fringe width β = y_{n+1} - y_n = λD/d.
15. Bohr Model: Radii & Energy Levels of Hydrogen
Equate Coulomb force to centripetal force: (1/4πε₀)(e²/r²) = mv²/r. Apply Bohr's quantization mvr = nh/(2π) to eliminate v and solve for r_n and E_n.