⚡ Physics XII — Board Exam Mastery
15 chapters · CBSE pattern · PYQ bank · AI-powered practice. Everything you need to score 90+ in one place.
15
Chapters
200+
PYQs
120+
Worked Examples
5
Full Practice Papers
⚡
Electric Charges & Fields
Coulomb's law, electric field, dipole, Gauss's theorem. Most formula-heavy chapter.
Ch 1
🔲
Electrostatic Potential & Capacitance
Potential, equipotential surfaces, capacitors — parallel plate, combinations, energy.
Ch 2
🔋
Current Electricity
Ohm's law, drift velocity, cells & batteries, Wheatstone bridge, Kirchhoff's laws.
Ch 3
🧲
Moving Charges & Magnetism
Biot-Savart, Ampere's law, force on charges, galvanometer, cyclotron.
Ch 4
🪄
Magnetism & Matter
Bar magnet, Earth's magnetism, para/dia/ferro-magnetic materials.
Ch 5
♾️
Electromagnetic Induction
Faraday's law, Lenz's law, motional EMF, self & mutual inductance, eddy currents.
Ch 6
〰️
Alternating Current
RMS values, phasors, LC/LCR circuits, resonance, power factor, transformer.
Ch 7
💡
Electromagnetic Waves
Maxwell's equations (qualitative), EM spectrum, properties and applications.
Ch 8
🔭
Ray Optics
Reflection, refraction, lenses, mirrors, prism, TIR, optical instruments.
Ch 9
🌈
Wave Optics
Huygens' principle, interference, diffraction, polarisation.
Ch 10
☢️
Dual Nature of Radiation
Photoelectric effect, de Broglie wavelength, electron emission.
Ch 11
⚛️
Atoms
Rutherford & Bohr models, energy levels, hydrogen spectrum.
Ch 12
🏛️
Nuclei
Nuclear size, binding energy, radioactivity, fission, fusion.
Ch 13
💻
Semiconductor Electronics
p-n junction, diode, transistor, logic gates, rectification.
Ch 14
📡
Communication Systems
Modulation, AM, FM, bandwidth, propagation, satellite communication.
Ch 15
💡 Board Exam Strategy
- Units 1–2 (Electrostatics): ~18 marks — master derivations of Gauss's law, capacitor energy
- Units 3–5 (Magnetism & EMI): ~20 marks — Biot-Savart, Faraday, transformer numericals
- Units 6–7 (Optics): ~18 marks — mirror/lens formula, interference fringe width
- Units 8–10 (Modern Physics): ~16 marks — photoelectric, Bohr radius, half-life
- Units 11 (Electronics & Communication): ~8 marks — logic gates, modulation
⚡ Chapter 1: Electric Charges & Fields
Coulomb's Law · Electric Field · Dipole · Gauss's Theorem
📖 Core Theory
1.1 Electric Charge
Electric charge is an intrinsic property of matter. It exists in two types: positive (proton, +e) and negative (electron, −e). Charge is quantised — q = ne where n is an integer and e = 1.6 × 10⁻¹⁹ C. Charge is conserved in all processes (the algebraic sum of charges in an isolated system remains constant).
Quantisation of Charge
Charge exists only in integral multiples of e = 1.6×10⁻¹⁹ C. q = ne
Conservation of Charge
Total charge in an isolated system is always constant. Charge can neither be created nor destroyed.
Additive Nature
Total charge = algebraic sum of individual charges. +3C + (−1C) = +2C
Point Charge
A hypothetical charge concentrated at a single geometric point — used for simplification in Coulomb's law.
1.2 Coulomb's Law
The electrostatic force between two point charges q₁ and q₂ separated by distance r in a medium of permittivity ε is:
🔑 Key Formulae — Coulomb's Law
F = k·q₁·q₂ / r² where k = 1/(4πε₀) = 9×10⁹ N·m²/C²
ε₀ = 8.854×10⁻¹² C²/(N·m²) (permittivity of free space)
In medium: F_med = F_vac / ε_r (ε_r = relative permittivity/dielectric constant)
Vector form: F⃗₁₂ = k·q₁·q₂·r̂₁₂ / r²
1.3 Electric Field
Electric field at a point is the force per unit positive test charge placed at that point. It is a vector field.
🔑 Electric Field Formulae
E⃗ = F⃗/q₀ (definition)
E = kq/r² (due to point charge q at distance r)
E_dipole (axial) = 2kp / r³
E_dipole (equatorial) = kp / r³
Torque on dipole in uniform field: τ = pE sinθ or τ⃗ = p⃗ × E⃗
1.4 Gauss's Theorem
The total electric flux through any closed surface (Gaussian surface) equals the net charge enclosed divided by ε₀. This is one of Maxwell's four equations.
🔑 Gauss's Law & Applications
Φ = ∮E⃗·dA⃗ = q_enc / ε₀
Infinite line charge (λ C/m): E = λ/(2πε₀r)
Infinite plane sheet (σ C/m²): E = σ/(2ε₀)
Uniformly charged sphere (outside): E = kQ/r²
Uniformly charged sphere (inside): E = kQr/R³
Conducting sphere surface: E = σ/ε₀
✏️ Worked Examples
Example 1 — Coulomb's Law (Board 2023, 3M)
Q: Two point charges +4 μC and −3 μC are placed 20 cm apart in air. Find the force between them.
1
Identify: q₁ = +4×10⁻⁶ C, q₂ = −3×10⁻⁶ C, r = 0.20 m, k = 9×10⁹ N·m²/C²
2
Apply Coulomb's law: F = k|q₁||q₂|/r² = 9×10⁹ × 4×10⁻⁶ × 3×10⁻⁶ / (0.20)²
3
Numerator = 9×10⁹ × 12×10⁻¹² = 108×10⁻³ = 0.108 N·m²
4
Denominator = (0.20)² = 0.04 m²
5
F = 0.108 / 0.04 = 2.7 N (attractive, since charges are opposite)
✅ Answer: F = 2.7 N (attractive force)
Example 2 — Electric Field due to Dipole (Board 2022, 5M)
Q: An electric dipole of moment p = 2×10⁻⁹ C·m is placed along the x-axis. Find E at axial and equatorial points both 30 cm away.
1
Given: p = 2×10⁻⁹ C·m, r = 0.30 m, k = 9×10⁹
2
Axial field: E_ax = 2kp/r³ = 2×9×10⁹×2×10⁻⁹ / (0.30)³
3
= 36 / 0.027 = 1333 N/C (along dipole direction)
4
Equatorial field: E_eq = kp/r³ = 9×10⁹×2×10⁻⁹ / 0.027
5
= 18 / 0.027 = 667 N/C (opposite to dipole direction)
✅ E_axial = 1333 N/C; E_equatorial = 667 N/C = E_axial/2
Example 3 — Gauss's Law (Board 2024, 5M)
Q: Using Gauss's theorem, derive the expression for E due to a uniformly charged infinite plane sheet of surface charge density σ.
1
Choose a cylindrical Gaussian surface (pillbox) symmetric about the sheet, with each flat face of area A parallel to the sheet.
2
By symmetry, E is perpendicular to the sheet. Flux through curved surface = 0 (E ⊥ dA).
3
Total flux: Φ = E·A + E·A = 2EA (two flat faces)
4
Charge enclosed: q_enc = σA
5
Gauss's law: 2EA = σA/ε₀ ⟹ E = σ/(2ε₀)
✅ E = σ/(2ε₀), directed away from sheet on both sides
⚖️ Comparisons
Electric Field Lines
Electric Flux
Vector quantity (direction = tangent to field line)
Scalar quantity (dot product of E and A)
Denser lines → stronger field
Proportional to total charge enclosed
Never cross each other
Can be zero even if E ≠ 0 (e.g., closed surface enclosing no charge)
Start from +ve, end at −ve charge
Positive when field exits, negative when field enters surface
Coulomb's Law
Gauss's Law
Applies to any two point charges
Applies to symmetric charge distributions
Direct approach — find F, then E
Surface integral approach — much simpler for symmetric cases
Gives force between charges
Gives total flux, used to find E at a surface
Both are equivalent — Gauss's law can be derived from Coulomb's law
More powerful for extended charge distributions
🎯 Quick Activity
Test Yourself — MCQ (Click to check)
Q: A charge Q is placed inside a closed surface. If Q is doubled, the electric flux through the surface:
Also doubles (Φ = Q/ε₀)
Halves
Remains the same
Quadruples
📌 Examiner's Note: Always state the direction of E when asked. In Gauss's law derivations, explicitly justify symmetry before setting up the surface. Missing the word "symmetry" costs marks.
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🔲 Chapter 2: Electrostatic Potential & Capacitance
Potential · Equipotential Surfaces · Capacitors · Dielectrics
📖 Core Theory
2.1 Electric Potential
Electric potential at a point is the work done per unit positive charge in bringing a test charge from infinity to that point, without acceleration. It is a scalar quantity.
🔑 Potential Formulae
V = W/q₀ (definition)
V = kq/r (due to point charge at distance r)
V_dipole (axial) = kp/r²
V_dipole (equatorial) = 0 (always)
Relation: E = −dV/dr (E is negative gradient of V)
Potential difference: V_A − V_B = W_AB/q
2.2 Capacitance & Capacitors
A capacitor stores electrical energy in the form of an electric field. Capacitance C = Q/V. For a parallel plate capacitor with plate area A, separation d:
🔑 Capacitance Formulae
C = Q/V (definition)
Parallel plate (air): C₀ = ε₀A/d
Parallel plate (dielectric K): C = Kε₀A/d = KC₀
Series: 1/C_eff = 1/C₁ + 1/C₂ + … (V divides, Q same)
Parallel: C_eff = C₁ + C₂ + … (Q divides, V same)
Energy stored: U = ½CV² = Q²/(2C) = QV/2
Energy density: u = ½ε₀E²
2.3 Equipotential Surfaces
An equipotential surface is a surface where every point is at the same potential. Key property: the electric field is always perpendicular to equipotential surfaces. No work is done moving a charge along an equipotential surface.
✏️ Worked Examples
Example 1 — Capacitor Energy (Board 2023, 3M)
Q: A 4 μF capacitor is charged to 200 V. Find (a) charge stored, (b) energy stored.
1
C = 4×10⁻⁶ F, V = 200 V
2
Charge: Q = CV = 4×10⁻⁶ × 200 = 8×10⁻⁴ C = 0.8 mC
3
Energy: U = ½CV² = ½ × 4×10⁻⁶ × (200)² = ½ × 4×10⁻⁶ × 40000
4
U = 2×10⁻⁶ × 40000 = 0.08 J = 80 mJ
✅ Q = 0.8 mC; U = 80 mJ
Example 2 — Series-Parallel Combination (Board 2022, 5M)
Q: Three capacitors of 2 μF, 3 μF, and 6 μF are connected in series across a 12 V battery. Find (a) equivalent capacitance, (b) charge on each capacitor.
1
Series: 1/C_eff = 1/2 + 1/3 + 1/6 = 3/6 + 2/6 + 1/6 = 6/6 = 1
2
C_eff = 1 μF
3
In series, charge is same on all: Q = C_eff × V = 1×10⁻⁶ × 12 = 12 μC
4
V₁ = Q/C₁ = 12/2 = 6 V; V₂ = 12/3 = 4 V; V₃ = 12/6 = 2 V. Check: 6+4+2 = 12 V ✓
✅ C_eff = 1 μF; Q on each = 12 μC
Example 3 — Effect of Dielectric (Board 2024, 3M)
Q: A parallel plate capacitor (A = 100 cm², d = 2 mm) is connected to 100 V battery. A dielectric slab of K = 5 is inserted. Find new capacitance and energy stored.
1
A = 100×10⁻⁴ m² = 10⁻² m²; d = 2×10⁻³ m; K = 5
2
C₀ = ε₀A/d = 8.85×10⁻¹² × 10⁻² / 2×10⁻³ = 44.25×10⁻¹² F = 44.25 pF
3
New C = KC₀ = 5 × 44.25 pF = 221.25 pF
4
U = ½CV² = ½ × 221.25×10⁻¹² × (100)² = 1.106 × 10⁻⁶ J ≈ 1.1 μJ
✅ C = 221.25 pF; U = 1.1 μJ
⚖️ Comparisons
Capacitors in Series
Capacitors in Parallel
Same charge Q on each
Same voltage V across each
Voltage divides: V = V₁ + V₂ + …
Charge divides: Q = Q₁ + Q₂ + …
C_eff < smallest individual C
C_eff > largest individual C
Used to withstand higher voltage
Used to increase total charge storage
Electric Potential (V)
Electric Field (E)
Scalar quantity (J/C or Volt)
Vector quantity (N/C or V/m)
V = kq/r (proportional to 1/r)
E = kq/r² (proportional to 1/r²)
Zero at infinity
Zero at infinity
Related: E = −dV/dr
Perpendicular to equipotential surfaces
📌 Board Tip: When a dielectric is inserted with battery still connected, V is constant and C, Q, U all increase. When inserted after disconnecting, Q is constant — V and U decrease while C increases. This distinction appears almost every year.
🔋 Chapter 3: Current Electricity
Ohm's Law · Drift Velocity · Cells · Kirchhoff's Laws · Wheatstone Bridge
📖 Core Theory
3.1 Electric Current and Drift Velocity
Electric current is the rate of flow of charge. In a metallic conductor, free electrons drift under an applied electric field with an average drift velocity v_d. The relation between current and drift velocity is fundamental.
🔑 Current & Drift Formulae
I = Q/t = nAev_d (n = no. density of electrons)
v_d = eEτ/m = eVτ/(mL) (τ = relaxation time)
Ohm's Law: V = IR
Resistivity: ρ = m/(ne²τ) (temperature dependent)
R = ρL/A (resistance of wire)
ρ_T = ρ₀(1 + αT) (α = temp. coefficient of resistance)
3.2 EMF, Internal Resistance & Cell Combinations
A cell maintains a potential difference by doing work on charges. EMF (ε) is the work done per unit charge through the cell. Terminal voltage V = ε − Ir (during discharge) where r is internal resistance.
🔑 Cell Combination Formulae
Terminal PD (discharge): V = ε − Ir
Terminal PD (charge): V = ε + Ir
n cells in series: ε_eq = nε; r_eq = nr; I = nε/(R + nr)
n cells in parallel: ε_eq = ε; r_eq = r/n; I = ε/(R + r/n)
Mixed (m rows, n in each): I = mnε/(mR + nr)
3.3 Kirchhoff's Laws
KCL (Junction Rule): Sum of currents entering a junction = sum leaving. Based on conservation of charge.
KVL (Loop Rule): The algebraic sum of changes in potential around any closed loop is zero. Based on conservation of energy.
3.4 Wheatstone Bridge & Meter Bridge
🔑 Bridge Conditions
Balanced Wheatstone: P/Q = R/S (no current through galvanometer)
Meter bridge: R/S = l/(100−l) (l = balancing length in cm)
Potentiometer (EMF comparison): ε₁/ε₂ = l₁/l₂
Internal resistance: r = R(l₁−l₂)/l₂
✏️ Worked Examples
Example 1 — Drift Velocity (Board 2023, 3M)
Q: A copper wire of length 2 m, cross-section 2×10⁻⁶ m² carries a current of 3 A. Find the drift velocity. (n = 8.5×10²⁸ m⁻³)
1
I = nAev_d → v_d = I/(nAe)
2
v_d = 3 / (8.5×10²⁸ × 2×10⁻⁶ × 1.6×10⁻¹⁹)
3
Denominator = 8.5×10²⁸ × 2×10⁻⁶ × 1.6×10⁻¹⁹ = 27.2×10³ = 2.72×10⁴
4
v_d = 3 / 2.72×10⁴ ≈ 1.1×10⁻⁴ m/s = 0.11 mm/s
✅ v_d ≈ 1.1×10⁻⁴ m/s (extremely slow — electrons take minutes to travel 1 m!)
Example 2 — Kirchhoff's Laws (Board 2022, 5M)
Q: In a circuit, two cells: ε₁ = 6V, r₁ = 1Ω and ε₂ = 4V, r₂ = 2Ω connected in opposition with external R = 5Ω. Find current.
1
Net EMF = ε₁ − ε₂ = 6 − 4 = 2 V (cells oppose each other)
2
Total resistance = R + r₁ + r₂ = 5 + 1 + 2 = 8 Ω
3
I = Net EMF / Total R = 2/8 = 0.25 A
4
Current flows in direction of stronger cell (ε₁)
✅ I = 0.25 A in the direction of the 6V cell
Example 3 — Meter Bridge / Internal Resistance (Board 2024, 5M)
Q: In a potentiometer, the balancing length for a cell of EMF 1.25 V is 50 cm. When a resistance of 2Ω is connected across the cell, the balance length becomes 40 cm. Find internal resistance of the cell.
1
Formula: r = R(l₁ − l₂)/l₂
2
l₁ = 50 cm (open circuit), l₂ = 40 cm (with R = 2Ω), R = 2Ω
3
r = 2 × (50 − 40)/40 = 2 × 10/40 = 2 × 0.25 = 0.5 Ω
✅ Internal resistance r = 0.5 Ω
⚖️ Comparisons
Ohmic Conductors
Non-Ohmic Conductors
Obey Ohm's law (V ∝ I at constant T)
Do NOT follow V ∝ I
R is constant; V-I graph is straight line
R changes with V or I; V-I graph is non-linear
Examples: metal resistors, nichrome wire
Examples: diode, thermistor, LED
Cells in Series
Cells in Parallel
ε_eq = nε; r_eq = nr
ε_eq = ε; r_eq = r/n
Higher EMF — good for large external R
Lower internal R — good for small external R
Current limited by series resistance
Can deliver higher current
📌 Board Tip: In Kirchhoff's loop rule, always state the sign convention you are using (direction of traversal). Marks are given for correct setup of equations, not just the final answer. Draw the circuit diagram and mark assumed current directions before writing equations.
🧲 Chapter 4: Moving Charges & Magnetism
Biot-Savart Law · Ampere's Law · Force on Charges · Cyclotron · Galvanometer
📖 Core Theory
4.1 Magnetic Force on Charges & Conductors
🔑 Key Formulae
F⃗ = q(v⃗ × B⃗) = qvB sinθ (force on moving charge)
F⃗ = I(L⃗ × B⃗) = BIL sinθ (force on current-carrying conductor)
Force between parallel wires: F/L = μ₀I₁I₂/(2πd)
Torque on loop: τ = NIAB sinθ or τ⃗ = M⃗ × B⃗
Magnetic moment: M = NIA (N turns, area A)
4.2 Biot-Savart Law
🔑 Biot-Savart & Applications
dB = (μ₀/4π) × I dl sinθ / r² (magnetic field due to element)
B at centre of circular loop (radius R): B = μ₀I/(2R)
B on axis of circular loop: B = μ₀IR²/[2(R²+x²)^(3/2)]
B due to long straight wire at distance r: B = μ₀I/(2πr)
μ₀ = 4π×10⁻⁷ T·m/A (permeability of free space)
4.3 Ampere's Circuital Law & Solenoid
🔑 Ampere's Law & Solenoid/Toroid
∮B⃗·dl⃗ = μ₀I_enc (Ampere's law)
B inside solenoid (n turns/m): B = μ₀nI
B inside toroid: B = μ₀NI/(2πr) (N total turns, r = radius)
4.4 Cyclotron & Galvanometer
Cyclotron: charged particles spiral outward in a magnetic field, accelerated by an electric field. Frequency is independent of velocity (resonance condition). A galvanometer measures small currents; can be converted to ammeter (low shunt S) or voltmeter (high series R).
🔑 Cyclotron & Meter Conversions
Cyclotron frequency: f = qB/(2πm) (independent of speed!)
Cyclotron radius: r = mv/(qB)
Shunt for ammeter: S = I_g·G/(I−I_g)
Series R for voltmeter: R_s = V/I_g − G
✏️ Worked Examples
Example 1 — Magnetic Field at Centre of Loop (Board 2023, 3M)
Q: A circular coil of 50 turns and radius 10 cm carries a current of 2 A. Find B at centre.
1
B = μ₀NI/(2R) = (4π×10⁻⁷ × 50 × 2)/(2 × 0.10)
2
= (4π×10⁻⁷ × 100) / 0.20 = (4π×10⁻⁵) / 0.20
3
= 20π×10⁻⁵ = 6.28×10⁻⁴ T = 0.628 mT
✅ B = 6.28×10⁻⁴ T, directed perpendicular to the plane of the coil
Example 2 — Galvanometer to Ammeter (Board 2022, 3M)
Q: A galvanometer (G = 25 Ω, I_g = 10 mA) is to be converted to an ammeter reading up to 5 A. Find shunt resistance.
1
S = I_g × G / (I − I_g) = (0.01 × 25)/(5 − 0.01)
2
= 0.25/4.99 ≈ 0.0501 Ω ≈ 0.05 Ω
✅ Shunt S = 0.05 Ω (very small, so it doesn't affect circuit)
📌 Always use right-hand rule to find direction of force. In board exams, direction must be stated explicitly to score full marks.
🪄 Chapter 5: Magnetism & Matter
Bar Magnet · Earth's Magnetism · Diamagnetic / Paramagnetic / Ferromagnetic
📖 Core Theory
🔑 Key Formulae
B_axial (bar magnet) = (μ₀/4π)·2M/r³ (along axis)
B_equatorial = (μ₀/4π)·M/r³ (along equator)
Torque on bar magnet: τ = MB sinθ
PE of dipole: U = −MB cosθ
Intensity of magnetisation: I = M/V
Magnetic susceptibility: χ = I/H; Permeability: μ_r = 1 + χ
Curie's law (paramagnetic): χ ∝ 1/T
Diamagnetic
Paramagnetic
Weakly repelled by magnet
Weakly attracted by magnet
χ is small negative (−1 ≤ χ < 0)
χ is small positive (0 < χ < ε)
μ_r slightly less than 1
μ_r slightly greater than 1
Indep. of temperature; Examples: Cu, Bi, H₂O
Follows Curie's law; Examples: Al, Na, O₂
Ferromagnetic
Strongly attracted; χ very large positive; forms domains; Examples: Fe, Co, Ni. Above Curie temp → paramagnetic.
Hysteresis
Lag of B behind H in ferromagnetic material. Area of B-H loop = energy loss per cycle.
Retentivity
B remaining when H is reduced to zero — measure of how well material retains magnetism.
Coercivity
H needed to reduce B to zero — measure of material's resistance to demagnetisation.
📌 Board Tip: Distinguish clearly between B (magnetic field), H (magnetising field), and M (magnetisation). These are three different quantities. Also know Earth's declination, inclination (dip), and horizontal component.
♾️ Chapter 6: Electromagnetic Induction
Faraday's Law · Lenz's Law · Motional EMF · Self & Mutual Inductance
📖 Core Theory
6.1 Faraday's Laws
1st Law: Whenever the magnetic flux linked with a circuit changes, an EMF is induced.
2nd Law: The magnitude of induced EMF = rate of change of magnetic flux.
Lenz's Law: The direction of induced current opposes the change that caused it (conservation of energy).
🔑 EMI Formulae
Magnetic flux: Φ = NBA cosθ (N turns, B field, A area)
Induced EMF: ε = −dΦ/dt = −N·dΦ_per_turn/dt
Motional EMF: ε = Bvl (conductor of length l moving at v⊥B)
Self-inductance: L = NΦ/I (flux linkage per unit current)
Self-induced EMF: ε = −L·dI/dt
Energy in inductor: U = ½LI²
Mutual inductance: M₁₂ = N₂Φ₂₁/I₁ ; ε₂ = −M·dI₁/dt
Solenoid inductance: L = μ₀N²A/l = μ₀n²Al
✏️ Worked Examples
Example 1 — Motional EMF (Board 2023, 3M)
Q: A conductor of length 0.5 m moves at 4 m/s perpendicular to a magnetic field B = 0.2 T. Find induced EMF and current if resistance = 0.4 Ω.
1
ε = Bvl = 0.2 × 4 × 0.5 = 0.4 V
2
I = ε/R = 0.4/0.4 = 1 A
✅ EMF = 0.4 V; Current = 1 A
Example 2 — Self Inductance of Solenoid (Board 2022, 3M)
Q: A solenoid has 500 turns, length 0.5 m, area 4 cm². Find self-inductance.
1
L = μ₀N²A/l = 4π×10⁻⁷ × (500)² × 4×10⁻⁴ / 0.5
2
= 4π×10⁻⁷ × 250000 × 8×10⁻⁴ = 4π×10⁻⁷ × 200 = 800π×10⁻⁷
3
= 2513×10⁻⁷ ≈ 2.51×10⁻⁴ H = 0.251 mH
✅ L ≈ 0.25 mH
📌 Lenz's law direction is crucial — always explain why the induced current creates a force opposing the change. This qualitative reasoning is worth 2 marks in every derivation question.
〰️ Chapter 7: Alternating Current
RMS · Phasor · LCR Circuit · Resonance · Power Factor · Transformer
📖 Core Theory
🔑 AC Circuit Formulae
Instantaneous values: v = V₀ sinωt; i = I₀ sin(ωt±φ)
RMS values: V_rms = V₀/√2; I_rms = I₀/√2
Reactances: X_L = ωL; X_C = 1/(ωC)
Impedance: Z = √[R² + (X_L − X_C)²]
Phase angle: tanφ = (X_L − X_C)/R
Resonance (series LCR): X_L = X_C → ω₀ = 1/√(LC); Z_min = R
Resonant frequency: f₀ = 1/(2π√LC)
Q-factor: Q = ω₀L/R = 1/(ω₀CR) = (1/R)√(L/C)
Power: P = V_rms·I_rms·cosφ (cosφ = power factor)
Transformer: V_s/V_p = N_s/N_p = I_p/I_s (ideal transformer)
Resistor in AC
Capacitor in AC
V and I in phase
I leads V by 90°
Power dissipated: P = I²R
No power dissipated (reactive)
Impedance = R (independent of frequency)
X_C = 1/ωC (decreases with frequency)
Inductor in AC
Series LCR at Resonance
V leads I by 90°
V and I in phase (φ = 0)
No power dissipated (reactive)
Power factor = 1 (max power)
X_L = ωL (increases with frequency)
Z = R (minimum impedance)
✏️ Worked Examples
Example 1 — LCR Circuit Impedance (Board 2023, 5M)
Q: A series LCR circuit has R = 10 Ω, L = 0.5 H, C = 40 μF, connected to 220 V, 50 Hz supply. Find Z, I_rms, phase angle.
1
ω = 2πf = 2π×50 = 314 rad/s
2
X_L = ωL = 314×0.5 = 157 Ω
3
X_C = 1/(ωC) = 1/(314×40×10⁻⁶) = 1/0.01256 = 79.6 Ω
4
Z = √[R² + (X_L−X_C)²] = √[100 + (157−79.6)²] = √[100 + (77.4)²] = √[100+5990.76] = √6090.76 ≈ 78.04 Ω
5
I_rms = V_rms/Z = 220/78.04 ≈ 2.82 A
6
tanφ = (X_L−X_C)/R = 77.4/10 = 7.74 → φ = tan⁻¹(7.74) ≈ 82.6° (V leads I)
✅ Z ≈ 78 Ω; I_rms ≈ 2.82 A; φ ≈ 82.6° (inductive circuit)
Example 2 — Transformer (Board 2022, 3M)
Q: A step-up transformer converts 220 V to 22000 V. Primary has 200 turns and power output is 11 kW. Find: (a) N_s, (b) I_s, (c) I_p.
1
N_s/N_p = V_s/V_p → N_s = N_p×V_s/V_p = 200×22000/220 = 20000 turns
2
I_s = P/V_s = 11000/22000 = 0.5 A
3
I_p = P/V_p = 11000/220 = 50 A
✅ N_s = 20000; I_s = 0.5 A; I_p = 50 A (step up V, step down I)
📌 Phasor diagrams are worth 2 marks in every AC question. Always draw and label them even if not explicitly asked.
💡 Chapter 8: Electromagnetic Waves
Maxwell's Equations · EM Spectrum · Properties & Applications
📖 Core Theory
Maxwell showed that a changing electric field produces a magnetic field (displacement current) and vice versa. This led to the prediction of EM waves — transverse waves that travel at c = 3×10⁸ m/s in vacuum.
🔑 EM Wave Formulae
Speed of EM wave: c = 1/√(μ₀ε₀) = 3×10⁸ m/s
In medium: v = c/n where n = refractive index = c/v
Displacement current: I_d = ε₀·dΦ_E/dt
Intensity: I = u·c = (½ε₀E₀²)·c (energy per unit area per second)
| Region | Wavelength Range | Source | Application |
|---|---|---|---|
| Radio waves | > 0.1 m | LC oscillators | AM/FM broadcasting |
| Microwaves | 1 mm – 0.1 m | Klystron/magnetron | Radar, microwave oven |
| Infrared | 700 nm – 1 mm | Hot bodies | TV remote, night vision |
| Visible light | 400–700 nm | Atoms (electron transitions) | Vision, photography |
| Ultraviolet | 1–400 nm | Sun, arc lamps | Sterilisation, forensics |
| X-rays | 0.01–10 nm | X-ray tube | Medical imaging, crystallography |
| Gamma rays | < 0.01 nm | Radioactive nuclei | Cancer therapy, food sterilisation |
📌 Memorise EM spectrum order: Radio → Micro → Infrared → Visible → UV → X-ray → Gamma (increasing frequency / decreasing wavelength). All travel at c in vacuum. Board frequently asks sources and uses of 2–3 specific regions.
🔭 Chapter 9: Ray Optics & Optical Instruments
Reflection · Refraction · TIR · Lenses · Mirrors · Prism · Instruments
📖 Core Theory
🔑 Mirrors
Mirror formula: 1/v + 1/u = 1/f (all distances from pole; sign convention!)
Magnification: m = −v/u = h'/h
f = R/2 (focal length = half radius of curvature)
🔑 Refraction & Lenses
Snell's law: n₁ sinθ₁ = n₂ sinθ₂ also n = c/v
Critical angle: sinθ_c = n₂/n₁ (TIR when θ > θ_c, n₁ > n₂)
Lens formula: 1/v − 1/u = 1/f
Lensmaker's eq: 1/f = (n−1)[1/R₁ − 1/R₂]
Power of lens: P = 1/f (in m) units: dioptre (D)
Lenses in contact: P = P₁ + P₂ + … or 1/f = 1/f₁ + 1/f₂
Refraction at spherical surface: n₂/v − n₁/u = (n₂−n₁)/R
🔑 Prism & Optical Instruments
Prism: A + δ = i + e (angle of prism + deviation = sum of angles)
At min. deviation: i = e; μ = sin[(A+δ_m)/2] / sin(A/2)
Simple microscope: m = 1 + D/f (D = 25 cm least distance of distinct vision)
Compound microscope: M = m_e × m_o = (L/f_o) × (1 + D/f_e)
Astronomical telescope: M = −f_o/f_e (image at infinity)
✏️ Worked Examples
Example 1 — Concave Mirror (Board 2023, 3M)
Q: An object 2 cm tall is placed 30 cm in front of a concave mirror of focal length 20 cm. Find image position, nature, and size.
1
u = −30 cm, f = −20 cm (concave mirror, sign convention: distances in direction of incident light are negative)
2
1/v = 1/f − 1/u = 1/(−20) − 1/(−30) = −1/20 + 1/30 = (−3+2)/60 = −1/60
3
v = −60 cm (real, inverted image, on same side as object)
4
m = −v/u = −(−60)/(−30) = −2 → image height = 2×2 = 4 cm (inverted)
✅ Image at 60 cm in front of mirror; real, inverted; magnified 2× (4 cm tall)
Example 2 — Lens Combination (Board 2024, 3M)
Q: Two lenses of focal lengths +20 cm and −30 cm are placed in contact. Find equivalent focal length and power.
1
1/f = 1/f₁ + 1/f₂ = 1/20 + 1/(−30) = 3/60 − 2/60 = 1/60
2
f = 60 cm = 0.60 m (converging combination)
3
P = 1/f = 1/0.60 = +1.67 D
✅ f = 60 cm (converging); P = +1.67 D
Concave Mirror
Convex Mirror
Converging mirror; f is negative
Diverging mirror; f is positive
Can form real and virtual images
Always forms virtual, erect, diminished image
Used in headlights, shaving mirrors
Used in rear-view mirrors (wide field of view)
📌 In ALL optics numericals, write the sign convention at the start: "Using Cartesian sign convention: distances measured from pole; incident light travels in +x direction." This protects you from sign errors.
🌈 Chapter 10: Wave Optics
Huygens' Principle · Interference (YDSE) · Diffraction · Polarisation
📖 Core Theory
🔑 YDSE (Young's Double Slit Experiment)
Path difference: Δ = d sinθ ≈ yd/D (y = distance from central fringe, d = slit sep., D = screen dist.)
Constructive (bright): Δ = nλ; yn = nλD/d (n = 0,±1,±2…)
Destructive (dark): Δ = (2n−1)λ/2; yn = (2n−1)λD/(2d)
Fringe width: β = λD/d
Intensity: I = 4I₀cos²(δ/2) where δ = 2πΔ/λ
🔑 Diffraction & Polarisation
Single slit — 1st minima: a sinθ = λ → θ ≈ λ/a
Central maximum width: 2λD/a
Resolving power (microscope): RP = 2μ sinθ/λ
Brewster's law (polarisation): tanθ_B = n₂/n₁ (at Brewster angle, reflected ray ⊥ refracted)
Malus's law: I = I₀ cos²θ (θ = angle between analyser and polariser axes)
✏️ Worked Examples
Example 1 — YDSE Fringe Width (Board 2023, 3M)
Q: In YDSE, slit separation = 0.3 mm, screen distance = 1.5 m, wavelength = 600 nm. Find fringe width.
1
β = λD/d = (600×10⁻⁹ × 1.5) / (0.3×10⁻³)
2
= (900×10⁻⁹) / (3×10⁻⁴) = 3×10⁻³ m = 3 mm
✅ Fringe width β = 3 mm
Example 2 — Malus's Law (Board 2022, 2M)
Q: Polarised light of intensity I₀ passes through an analyser at 60° to the polariser. Find transmitted intensity.
1
Malus's law: I = I₀ cos²θ = I₀ cos²60° = I₀ × (1/2)² = I₀/4
✅ I = I₀/4
📌 YDSE is asked every year — either fringe width calculation or effect of changing λ, d or D. Also know: if entire setup is immersed in water (n=4/3), λ becomes λ/n, so fringe width decreases by factor n.
☢️ Chapter 11: Dual Nature of Radiation & Matter
Photoelectric Effect · de Broglie Hypothesis · Davisson-Germer
📖 Core Theory
🔑 Photoelectric Effect (Einstein's Equation)
Energy of photon: E = hf = hc/λ (h = 6.626×10⁻³⁴ J·s)
Einstein's equation: hf = φ + KE_max = φ + ½mv²_max
Work function: φ = hf₀ (f₀ = threshold frequency)
Stopping potential: eV₀ = KE_max = hf − φ
de Broglie wavelength: λ = h/mv = h/p
For electron accelerated through V: λ = h/√(2meV) = 1.227/√V nm
Wave Nature of Light
Particle Nature of Light (Photon)
Explains interference, diffraction, polarisation
Explains photoelectric effect, Compton effect
Amplitude determines intensity
Number of photons determines intensity
Frequency determines colour
Frequency determines photon energy (E = hf)
✏️ Worked Examples
Example 1 — Photoelectric Effect (Board 2023, 3M)
Q: Light of wavelength 300 nm falls on a metal with work function 2.5 eV. Find KE_max and stopping potential.
1
E = hc/λ = (6.626×10⁻³⁴ × 3×10⁸)/(300×10⁻⁹) = 6.626×10⁻¹⁹ J = 4.14 eV
2
KE_max = E − φ = 4.14 − 2.5 = 1.64 eV = 2.63×10⁻¹⁹ J
3
Stopping potential: eV₀ = 1.64 eV → V₀ = 1.64 V
✅ KE_max = 1.64 eV; V₀ = 1.64 V
Example 2 — de Broglie Wavelength (Board 2022, 2M)
Q: Find de Broglie wavelength of an electron accelerated through 100 V.
1
λ = 1.227/√V nm = 1.227/√100 = 1.227/10 = 0.1227 nm = 1.227 Å
✅ λ = 0.1227 nm (X-ray range — explains Davisson-Germer experiment)
📌 The stopping potential is independent of intensity — this was the proof that light is a particle. Intensity only affects the number of electrons (photocurrent), NOT their maximum KE. This conceptual point is tested frequently.
⚛️ Chapter 12: Atoms
Rutherford Model · Bohr Model · Energy Levels · Hydrogen Spectrum
📖 Core Theory
12.1 Rutherford's Nuclear Model
Rutherford's gold foil experiment proved: most of atom is empty space; nucleus is small, dense, positively charged; nucleus contains almost all the mass. Limitation: could not explain stability of atom or discrete spectra.
12.2 Bohr's Model of Hydrogen Atom
Bohr's postulates: electrons revolve in stationary orbits without radiating; angular momentum is quantised (L = nh/2π); energy is emitted/absorbed only when electron jumps between orbits.
🔑 Bohr Model Formulae
Orbital radius: rₙ = n²a₀ where a₀ = 0.529 Å (Bohr radius, for H)
Orbital velocity: vₙ = v₁/n where v₁ = 2.18×10⁶ m/s
Energy: Eₙ = −13.6/n² eV (ground state n=1: −13.6 eV)
Photon energy: hf = E_higher − E_lower = 13.6(1/n₁² − 1/n₂²) eV
Wavenumber: 1/λ = R_H(1/n₁² − 1/n₂²) where R_H = 1.097×10⁷ m⁻¹
| Series | n₁ | n₂ | Region |
|---|---|---|---|
| Lyman | 1 | 2,3,4… | Ultraviolet |
| Balmer | 2 | 3,4,5… | Visible |
| Paschen | 3 | 4,5,6… | Near Infrared |
| Brackett | 4 | 5,6,7… | Infrared |
| Pfund | 5 | 6,7,8… | Far Infrared |
✏️ Worked Examples
Example 1 — Energy Level Transition (Board 2023, 3M)
Q: An electron in hydrogen atom transitions from n=4 to n=2. Find wavelength of emitted photon.
1
1/λ = R_H(1/n₁² − 1/n₂²) = 1.097×10⁷ × (1/4 − 1/16) = 1.097×10⁷ × (4/16 − 1/16)
2
= 1.097×10⁷ × 3/16 = 1.097×10⁷ × 0.1875 = 2.057×10⁶ m⁻¹
3
λ = 1/2.057×10⁶ = 4.86×10⁻⁷ m = 486 nm (Balmer series — blue-green visible)
✅ λ = 486 nm (Hβ line of Balmer series, visible region)
📌 Ionisation energy = energy to remove electron from ground state = +13.6 eV. Excitation energy = energy needed to jump from ground to excited state. These two are frequently confused.
🏛️ Chapter 13: Nuclei
Nuclear Size · Binding Energy · Radioactivity · Fission & Fusion
📖 Core Theory
🔑 Nuclear Physics Formulae
Nuclear radius: R = R₀A^(1/3) where R₀ = 1.2×10⁻¹⁵ m, A = mass number
Mass defect: Δm = [Z·m_p + (A−Z)·m_n] − M_nucleus
Binding energy: BE = Δm·c² (1 amu = 931.5 MeV/c²)
Binding energy per nucleon: BE/A (peaks at ⁵⁶Fe — most stable)
Radioactive decay law: N = N₀e^(−λt)
Activity: A = λN = A₀e^(−λt)
Half-life: T½ = 0.693/λ = 0.693τ (τ = mean life = 1/λ)
N after n half-lives: N = N₀/2ⁿ (n = t/T½)
α decay
Nucleus emits ₂⁴He. Mass number decreases by 4, atomic number by 2. E.g.: ²³⁸U → ²³⁴Th + ⁴He
β⁻ decay
Neutron converts to proton: n → p + e⁻ + ν̄. Mass number unchanged, atomic number increases by 1.
γ decay
Nucleus de-excites by emitting high-energy photon. No change in A or Z. Usually accompanies α or β decay.
Nuclear Fission
Heavy nucleus splits into two medium nuclei + energy. Example: ²³⁵U + n → ¹⁴¹Ba + ⁹²Kr + 3n + 200 MeV
✏️ Worked Examples
Example 1 — Radioactive Decay (Board 2023, 3M)
Q: A radioactive element has T½ = 30 days. How much will remain from 1 kg after 90 days?
1
Number of half-lives: n = t/T½ = 90/30 = 3
2
N = N₀/2ⁿ = N₀/2³ = N₀/8
3
Mass remaining = 1/8 kg = 0.125 kg = 125 g
✅ 125 g remains after 90 days (3 half-lives)
Example 2 — Binding Energy (Board 2022, 3M)
Q: Find binding energy of ⁴He nucleus. (m_p = 1.007825 u, m_n = 1.008665 u, m(⁴He) = 4.002602 u, 1u = 931.5 MeV)
1
Mass of 2 protons + 2 neutrons = 2(1.007825) + 2(1.008665) = 2.01565 + 2.01733 = 4.03298 u
2
Δm = 4.03298 − 4.002602 = 0.030378 u
3
BE = 0.030378 × 931.5 = 28.3 MeV
4
BE/A = 28.3/4 = 7.07 MeV per nucleon
✅ Binding energy = 28.3 MeV; BE/nucleon = 7.07 MeV
📌 Mean life τ = T½/0.693 = 1.44 T½. After one mean life, N = N₀/e ≈ 0.37N₀. Distinguish clearly: fission releases energy because BE/nucleon increases for products; fusion also releases energy for the same reason.
💻 Chapter 14: Semiconductor Electronics
Intrinsic & Extrinsic · p-n Junction · Diode · Transistor · Logic Gates
📖 Core Theory
14.1 Semiconductors
Semiconductors have conductivity between conductors and insulators. In intrinsic semiconductors (pure Si, Ge), n_e = n_h. Extrinsic semiconductors are doped: n-type (donor impurity like P, As — extra electrons) or p-type (acceptor impurity like B, In — extra holes).
p-n Junction
Junction of p-type and n-type semiconductor. At junction, diffusion creates a depletion layer with built-in potential barrier (~0.7V for Si).
Forward Bias
p connected to +ve, n to −ve terminal. Barrier reduces; current flows. Acts as closed switch.
Reverse Bias
p connected to −ve, n to +ve terminal. Barrier increases; only tiny reverse saturation current flows. Acts as open switch.
Zener Diode
Designed to operate in reverse breakdown. Used as voltage regulator. Breakdown voltage called Zener voltage (V_Z).
14.2 Transistor
A transistor (BJT) has three regions: Emitter (heavily doped), Base (thin, lightly doped), Collector (moderately doped). Used as amplifier or switch.
🔑 Transistor Parameters
I_E = I_B + I_C (KCL at transistor)
DC current gain (CE): β = I_C/I_B (typically 20–200)
α = I_C/I_E (CB configuration); α = β/(1+β)
Voltage gain (CE amp): A_v = β × R_C/R_in
14.3 Logic Gates
| Gate | Symbol | Output | Boolean |
|---|---|---|---|
| AND | · | High only if ALL inputs high | Y = A·B |
| OR | + | High if ANY input high | Y = A+B |
| NOT | ¬ or ‾ | Inverts input | Y = Ā |
| NAND | AND+NOT | Low only if ALL inputs high | Y = ‾(A·B) |
| NOR | OR+NOT | High only if ALL inputs low | Y = ‾(A+B) |
✏️ Worked Examples
Example 1 — Half-wave Rectification (Board 2023, 3M)
Q: In a half-wave rectifier, the input frequency is 50 Hz. What is output frequency? What additional component converts it to full-wave?
1
Half-wave: only one half-cycle passes → output frequency = input frequency = 50 Hz
2
Full-wave rectifier uses 4 diodes in bridge configuration (Bridge Rectifier)
3
Full-wave output frequency = 2 × input = 100 Hz (both half-cycles utilised)
✅ Half-wave: 50 Hz; Full-wave: 100 Hz; Use bridge rectifier for full-wave
📌 NAND and NOR are "universal gates" — any other gate can be built from them alone. This is frequently asked as a 2-mark question. Also know: transistor in saturation = switch ON; cutoff = switch OFF.
📡 Chapter 15: Communication Systems
Modulation · AM · FM · Bandwidth · Signal Propagation · Satellite
📖 Core Theory
Communication system: Transmitter → Channel (medium) → Receiver. The signal must be modulated (superimposed on a carrier wave) to be transmitted efficiently over long distances.
🔑 Modulation & Communication Formulae
Modulation index (AM): m_a = A_m/A_c (A_m = signal amplitude, A_c = carrier amplitude)
AM bandwidth: 2f_m (f_m = maximum audio frequency)
FM bandwidth: 2(Δf + f_m) (Δf = frequency deviation)
Range of transmission (line of sight): d = √(2Rh) (R = Earth radius = 6.4×10⁶ m, h = tower height)
Power in AM wave: P_T = P_C(1 + m_a²/2)
Amplitude Modulation (AM)
Frequency Modulation (FM)
Carrier amplitude varies with signal
Carrier frequency varies with signal
Bandwidth = 2f_m; simpler circuitry
Bandwidth larger; better quality, less noise
Longer range (ground waves)
Shorter range (line of sight)
Used for: MW, SW radio
Used for: FM radio, TV audio
Ground Wave Propagation
Radio waves travel along Earth's surface. Used for AM radio. Range limited by attenuation.
Sky Wave (Ionospheric)
Waves reflected by ionosphere. Used for short-wave international broadcasting. Frequency: 3–30 MHz.
Space Wave (LOS)
Line-of-sight propagation. Used for TV, FM, satellite links. Range increases with tower height.
Satellite Communication
Geostationary satellite at 36000 km covers ~1/3 of Earth. Three satellites can cover entire Earth (global coverage).
✏️ Worked Examples
Example 1 — Range of TV Tower (Board 2023, 2M)
Q: A TV tower has height 100 m. Find its range of transmission. (R_Earth = 6.4×10⁶ m)
1
d = √(2Rh) = √(2 × 6.4×10⁶ × 100) = √(12.8×10⁸) = √(1.28×10⁹)
2
= 3.578×10⁴ m ≈ 35.8 km ≈ 36 km
✅ Range ≈ 36 km from tower base
📌 Chapter 15 carries fewer marks (~4-5) but is completely scoring. Memorise: AM bandwidth = 2f_m; three propagation modes and their frequency bands; geostationary orbit height = 36000 km. These are direct 1-mark questions.
📜 Previous Year Question Bank
Real CBSE board questions — year-tagged, chapter-wise, with model answers
2024
5M
Ch 1 – Electric Fields
(a) State Gauss's law. Using it, derive an expression for the electric field intensity due to an infinite plane sheet of charge. (b) A charge Q is enclosed in a cube of side a. Find the electric flux through one face. [5]
(a) Gauss's law: Total electric flux through a closed surface = q_enc/ε₀. For infinite plane sheet: Using cylindrical Gaussian surface (area A on each flat face), 2EA = σA/ε₀ → E = σ/(2ε₀), directed away from sheet on both sides.
(b) By symmetry, total flux through cube = Q/ε₀. Cube has 6 equal faces → Flux through one face = Q/(6ε₀).
(b) By symmetry, total flux through cube = Q/ε₀. Cube has 6 equal faces → Flux through one face = Q/(6ε₀).
2023
3M
Ch 2 – Capacitance
A parallel plate capacitor with air between the plates has a capacitance of 8 pF. What will be the capacitance if the distance between the plates is reduced to half and the space between them is filled with a substance of dielectric constant 6? [3]
Original: C₀ = ε₀A/d = 8 pF
New: d' = d/2; K = 6
C' = Kε₀A/d' = 6 × ε₀A/(d/2) = 12 × ε₀A/d = 12 × C₀ = 12 × 8 = 96 pF
New: d' = d/2; K = 6
C' = Kε₀A/d' = 6 × ε₀A/(d/2) = 12 × ε₀A/d = 12 × C₀ = 12 × 8 = 96 pF
2024
5M
Ch 3 – Current Electricity
(a) State and explain Kirchhoff's laws. (b) Using Kirchhoff's laws find the current in each branch of the circuit: ε₁=12V, r₁=2Ω; ε₂=8V, r₂=1Ω; R=5Ω (cells in parallel with external R). [5]
KCL: ΣI_in = ΣI_out at every junction (conservation of charge)
KVL: ΣV = 0 around any closed loop (conservation of energy)
Setup: Let I₁ through ε₁ branch, I₂ through ε₂ branch, I = I₁+I₂ through R.
Loop 1 (outer): 12 − 2I₁ − 5(I₁+I₂) = 0 → 12 = 7I₁ + 5I₂ …(1)
Loop 2 (inner): 8 − 1×I₂ − 5(I₁+I₂) = 0 → 8 = 5I₁ + 6I₂ …(2)
Solving: (1)×6 − (2)×5: 72−40 = 42I₁−25I₁ → 32 = 17I₁ → I₁ = 1.88 A
From (2): I₂ = (8−5×1.88)/6 = (8−9.4)/6 = −0.23 A (negative → direction assumed wrong)
I = I₁+I₂ through R = 1.65 A
KVL: ΣV = 0 around any closed loop (conservation of energy)
Setup: Let I₁ through ε₁ branch, I₂ through ε₂ branch, I = I₁+I₂ through R.
Loop 1 (outer): 12 − 2I₁ − 5(I₁+I₂) = 0 → 12 = 7I₁ + 5I₂ …(1)
Loop 2 (inner): 8 − 1×I₂ − 5(I₁+I₂) = 0 → 8 = 5I₁ + 6I₂ …(2)
Solving: (1)×6 − (2)×5: 72−40 = 42I₁−25I₁ → 32 = 17I₁ → I₁ = 1.88 A
From (2): I₂ = (8−5×1.88)/6 = (8−9.4)/6 = −0.23 A (negative → direction assumed wrong)
I = I₁+I₂ through R = 1.65 A
2023
5M
Ch 4 – Moving Charges
(a) Derive an expression for the magnetic field at the centre of a circular current-carrying loop using Biot-Savart law. (b) A circular coil of 30 turns, radius 8 cm carries a current of 0.6 A. What is the magnitude of the magnetic field at the centre? [5]
(a) By Biot-Savart: dB = (μ₀/4π)·Idl×r̂/r². For circular loop of radius R, every element dl is perpendicular to r̂. Integrating over full circle (total length = 2πR):
B = (μ₀/4π)·I·(2πR)/R² = μ₀I/(2R)
(b) B = μ₀NI/(2R) = (4π×10⁻⁷ × 30 × 0.6)/(2 × 0.08)
= (4π×10⁻⁷ × 18)/0.16 = (72π×10⁻⁷)/0.16 = 450π×10⁻⁷ ≈ 1.41×10⁻⁴ T
B = (μ₀/4π)·I·(2πR)/R² = μ₀I/(2R)
(b) B = μ₀NI/(2R) = (4π×10⁻⁷ × 30 × 0.6)/(2 × 0.08)
= (4π×10⁻⁷ × 18)/0.16 = (72π×10⁻⁷)/0.16 = 450π×10⁻⁷ ≈ 1.41×10⁻⁴ T
2024
5M
Ch 9 – Ray Optics
Draw a labelled ray diagram showing image formation by a compound microscope when final image is at the near point. Derive the expression for its magnifying power. [5]
Diagram: Object just beyond f_o of objective → real, magnified intermediate image within f_e of eyepiece → virtual final image at D = 25 cm.
Magnification:
m_o (objective) = −v_o/u_o ≈ L/f_o (for small f_o, L = tube length)
m_e (eyepiece, image at D) = (1 + D/f_e)
Total: M = m_o × m_e = (L/f_o)(1 + D/f_e)
For large magnification: small f_o and small f_e needed.
Magnification:
m_o (objective) = −v_o/u_o ≈ L/f_o (for small f_o, L = tube length)
m_e (eyepiece, image at D) = (1 + D/f_e)
Total: M = m_o × m_e = (L/f_o)(1 + D/f_e)
For large magnification: small f_o and small f_e needed.
2023
3M
Ch 11 – Dual Nature
The work function of caesium metal is 2.14 eV. When light of frequency 6×10¹⁴ Hz is incident on the metal surface, photoemission of electrons occurs. (a) What is the maximum KE of the emitted electrons? (b) What is the stopping potential? (c) What is the maximum speed of the emitted photoelectrons? [3]
E = hf = 6.63×10⁻³⁴ × 6×10¹⁴ = 3.98×10⁻¹⁹ J = 2.49 eV
(a) KE_max = E − φ = 2.49 − 2.14 = 0.35 eV = 5.6×10⁻²⁰ J
(b) Stopping potential V₀ = KE_max/e = 0.35 V
(c) ½mv² = 5.6×10⁻²⁰ → v = √(2×5.6×10⁻²⁰/9.1×10⁻³¹) = √(1.23×10¹¹) ≈ 3.5×10⁵ m/s
(a) KE_max = E − φ = 2.49 − 2.14 = 0.35 eV = 5.6×10⁻²⁰ J
(b) Stopping potential V₀ = KE_max/e = 0.35 V
(c) ½mv² = 5.6×10⁻²⁰ → v = √(2×5.6×10⁻²⁰/9.1×10⁻³¹) = √(1.23×10¹¹) ≈ 3.5×10⁵ m/s
2022
3M
Ch 13 – Nuclei
The half-life of ⁹⁰Sr is 28 years. What is the disintegration rate of 15 mg of this isotope? (N_A = 6.023×10²³, atomic mass = 90) [3]
N = (15×10⁻³/90) × 6.023×10²³ = (1/6000) × 6.023×10²³ = 1.004×10²⁰ atoms
T½ = 28 yr = 28×3.156×10⁷ s = 8.83×10⁸ s
λ = 0.693/T½ = 0.693/8.83×10⁸ = 7.85×10⁻¹⁰ s⁻¹
Activity A = λN = 7.85×10⁻¹⁰ × 1.004×10²⁰ = 7.88×10¹⁰ disintegrations/s = 7.88×10¹⁰ Bq
T½ = 28 yr = 28×3.156×10⁷ s = 8.83×10⁸ s
λ = 0.693/T½ = 0.693/8.83×10⁸ = 7.85×10⁻¹⁰ s⁻¹
Activity A = λN = 7.85×10⁻¹⁰ × 1.004×10²⁰ = 7.88×10¹⁰ disintegrations/s = 7.88×10¹⁰ Bq
❌ Common Mistakes Bank
The errors that cost students marks every year — and how to fix them
Mistake 1 — Wrong Sign in Mirror/Lens Formula
Students forget the sign convention and get the wrong image position, losing 3–5 marks.
❌ WRONG: For concave mirror, f = +20 cm → 1/v = 1/f − 1/u = 1/20 − 1/(-30)
✅ CORRECT: For concave mirror, f = −20 cm (negative in Cartesian convention). Always write convention first: "All distances measured from pole; direction of incident light = positive"
Mistake 2 — Confusing E and V for Dipole
At equatorial point: E ≠ 0 but V = 0. Students reverse this and lose 2 marks.
❌ WRONG: Electric field at equatorial point of dipole = 0 (since V = 0 there)
✅ CORRECT: V = 0 at equatorial point, but E = kp/r³ ≠ 0 (the field points opposite to dipole moment). V = 0 does NOT mean E = 0.
Mistake 3 — Wrong Resonant Frequency Formula
❌ WRONG: f₀ = 1/(2πLC) or f₀ = 1/(√LC)
✅ CORRECT: f₀ = 1/(2π√LC). At resonance, X_L = X_C: ω₀L = 1/(ω₀C) → ω₀ = 1/√(LC) → f₀ = ω₀/(2π) = 1/(2π√LC)
Mistake 4 — Intensity vs KE in Photoelectric Effect
❌ WRONG: Increasing light intensity increases KE of photoelectrons
✅ CORRECT: Increasing intensity increases the NUMBER of photoelectrons (photocurrent), NOT their KE. KE depends only on frequency: KE_max = hf − φ. Stopping potential is independent of intensity.
Mistake 5 — Direction of Induced Current (Lenz's Law)
❌ WRONG: Stating direction without explaining WHY (no reference to opposing change)
✅ CORRECT: Always write: "By Lenz's law, the induced current flows in a direction such that the magnetic force it creates opposes the motion/change causing it." Then state the specific direction using right-hand rule.
Mistake 6 — Capacitor Energy Formula Error
❌ WRONG: U = CV² (missing the ½)
✅ CORRECT: U = ½CV² = Q²/(2C) = ½QV. All three forms are equivalent. The factor of ½ arises because charge and voltage build up gradually from zero.
Mistake 7 — Transistor Current Relations
❌ WRONG: I_C = I_B + I_E (wrong order — collector always ≫ base current)
✅ CORRECT: I_E = I_B + I_C. Emitter current splits into large collector current (α·I_E) and tiny base current ((1−α)·I_E). β = I_C/I_B is large (20–200).
Mistake 8 — Nuclear Decay Equations (Charge/Mass Not Balanced)
❌ WRONG: ²³⁸U → ²³⁴Th + He (missing subscripts/charge numbers)
✅ CORRECT: ⁹²²³⁸U → ⁹⁰²³⁴Th + ₂⁴He. Both mass numbers (A) and atomic numbers (Z) must balance. Check: 238=234+4 ✓; 92=90+2 ✓. Write subscripts and superscripts ALWAYS.
👁️ Examiner's Eye
What board examiners look for — and what costs marks silently
🎯 Tip 1 — State the Law Before Using It
Every derivation question awards 1–2 marks just for stating the law. Never dive into math without writing: "According to Gauss's law / Faraday's law / Coulomb's law…" Examiners mark this as a separate step.
🎯 Tip 2 — Labelled Diagrams Are Compulsory
For any question asking you to "derive" or "explain" a device (transformer, cyclotron, microscope, potentiometer), a labelled diagram is worth 1–2 marks. The marking scheme explicitly allocates marks for it. Draw it first, then derive.
🎯 Tip 3 — Units in Every Numerical
The marking scheme awards 1 mark specifically for the correct unit in the final answer. "E = 2.7" scores 0 for that step. "E = 2.7 N/C" scores full. Units are not optional.
🎯 Tip 4 — Define Symbols in Formulae
When you write "F = kq₁q₂/r²" and then calculate, define what each symbol means. Examiners check that you know the formula, not just copied it. One line of definitions gets you the "formula mark".
🎯 Tip 5 — "State vs Derive" — Know the Difference
"State" = write in one sentence with no proof. "Derive" = full mathematical proof from basics. "Explain" = conceptual explanation in words. Giving a derivation when asked to "state" wastes time but doesn't lose marks. Giving only a statement when asked to "derive" loses all derivation marks.
🎯 Tip 6 — Phasor Diagrams for AC Questions
Any AC circuit question about LCR — even if not explicitly asked — should include a phasor diagram showing V_R (along I), V_L (90° ahead of I), V_C (90° behind I), and resultant V. This is worth 2 marks in the marking scheme and takes 20 seconds to draw.
🎯 Tip 7 — Direction of Magnetic Force
When F = qvB sinθ or F = BIL sinθ — always state the direction using Fleming's Left-Hand Rule or right-hand rule. "F = 0.4 N" without direction scores partial credit only.
🎯 Tip 8 — Two-Marks "Why" Questions
Board frequently asks "why does [device] work?" or "why is [material] used?" These need cause-effect answers: e.g. "Nichrome is used because it has high resistivity AND low temperature coefficient of resistance — both conditions needed." One condition alone = 1 mark instead of 2.
🔥 High-Probability Questions
Questions that appear in 3+ of last 5 board exams — guaranteed to be asked again
🔥🔥🔥 VERY HIGH
Ch1–2: Derive E due to uniformly charged infinite plane sheet using Gauss's law
Asked 4 out of 5 years. 3–5 marks. Must know: choose cylindrical Gaussian surface, argue flux through curved surface = 0, get E = σ/(2ε₀).
🔥🔥🔥 VERY HIGH
Ch3: Potentiometer — find internal resistance of a cell
Appears every year. Formula: r = R(l₁−l₂)/l₂. Draw circuit diagram. State that potentiometer gives accurate result because no current flows through cell at balance.
🔥🔥🔥 VERY HIGH
Ch6: State and explain Faraday's laws of electromagnetic induction with Lenz's law
5-mark question, asked almost every year. Define magnetic flux, state both laws, explain Lenz's law as conservation of energy, give one example (e.g. magnet approaching coil).
🔥🔥🔥 VERY HIGH
Ch7: LCR series circuit — impedance, current, resonance condition
5-mark derivation + numerical. Draw phasor diagram. Derive Z = √(R² + (X_L−X_C)²). State resonance condition. Expected every year.
🔥🔥🔥 VERY HIGH
Ch9: Ray diagram + magnification for compound microscope
5-mark. Draw ray diagram (both objective and eyepiece), derive M = (L/f_o)(1+D/f_e). Appears in ~4/5 recent papers.
🔥🔥 HIGH
Ch10: YDSE — fringe width formula and numerical
3-mark. Derive β = λD/d or numerical calculation. Effect of changing medium (submerging in liquid) is a common twist.
🔥🔥 HIGH
Ch11: Einstein's photoelectric equation — explanation + numerical
3–5 mark. Explain threshold frequency, stopping potential, independence from intensity. Calculate KE_max and stopping potential.
🔥🔥 HIGH
Ch12: Bohr's model — energy levels, spectral series, hydrogen spectrum
3–5 mark. List postulates, derive radius and energy formulae, identify spectral series (Balmer = visible, Lyman = UV). Calculate wavelength of specific transition.
🔥🔥 HIGH
Ch14: p-n junction working + forward/reverse bias characteristics
3–5 mark. Explain depletion layer, draw I-V characteristic curve (include both forward and reverse bias), label threshold voltage and breakdown voltage.
🔥 MODERATE-HIGH
Ch13: Binding energy per nucleon curve + nuclear stability
3 mark. Draw BE/A vs A curve, identify Fe as most stable, explain why fission and fusion both release energy (both approach stability).
⏱️ Last-Minute Revision
Study this the night before the exam — all the quick wins
⚡ Electrostatics — Quick Hits
- Coulomb's constant k = 9×10⁹ N·m²/C²; ε₀ = 8.85×10⁻¹² C²/N·m²
- Gauss's law: Φ = q_enc/ε₀; E for infinite sheet = σ/(2ε₀)
- Dipole: axial E = 2kp/r³; equatorial E = kp/r³ (opposite direction)
- Dipole equatorial potential = ZERO (but field ≠ 0)
- Capacitor energy U = ½CV² = Q²/2C; energy density = ½ε₀E²
- Battery connected → V constant, C increases with dielectric
- Battery disconnected → Q constant, V decreases with dielectric
🔋 Current Electricity — Quick Hits
- I = nAev_d; drift velocity is ~10⁻⁴ m/s (very slow!)
- Terminal V = ε − Ir (discharge); = ε + Ir (charge)
- Potentiometer internal resistance: r = R(l₁−l₂)/l₂
- Wheatstone balanced: P/Q = R/S (no galvanometer deflection)
- Resistivity increases with temperature for metals (α positive)
🧲 Magnetism + EMI — Quick Hits
- B at centre of N-turn loop: B = μ₀NI/(2R)
- B inside solenoid: B = μ₀nI (n = turns per metre)
- Cyclotron frequency: f = qB/(2πm) — independent of speed!
- Motional EMF: ε = Bvl; self-induced: ε = −L·dI/dt
- Energy in inductor: U = ½LI²; solenoid L = μ₀N²A/l
- Lenz's law = conservation of energy (induced current opposes change)
〰️ AC + EM Waves — Quick Hits
- V_rms = V₀/√2; I_rms = I₀/√2
- Z = √[R² + (X_L − X_C)²]; at resonance Z_min = R
- Resonant frequency f₀ = 1/(2π√LC)
- Transformer: V_s/V_p = N_s/N_p; step-up voltage → step-down current
- EM spectrum order: Radio → Micro → IR → Visible → UV → X → Gamma
- All EM waves travel at c = 3×10⁸ m/s in vacuum
🔭 Optics — Quick Hits
- Mirror formula: 1/v + 1/u = 1/f; lens formula: 1/v − 1/u = 1/f
- Concave mirror f = −R/2; Convex lens f = positive
- TIR: sinθ_c = n₂/n₁ (n₁ > n₂); occurs when light goes from denser to rarer
- YDSE fringe width β = λD/d; bright fringe at nλD/d
- Malus's law: I = I₀cos²θ; Brewster's law: tanθ_B = n
⚛️ Modern Physics — Quick Hits
- Photoelectric: KE_max = hf − φ; stopping potential eV₀ = KE_max
- de Broglie: λ = h/mv; for electron: λ = 1.227/√V nm
- Bohr: E_n = −13.6/n² eV; r_n = 0.529n² Å
- Radioactivity: N = N₀/2ⁿ (n half-lives); Activity A = λN
- Half-life T½ = 0.693/λ; mean life τ = 1/λ = 1.44T½
- 1 amu = 931.5 MeV; BE/nucleon peaks at Fe-56
- Transistor: I_E = I_B + I_C; β = I_C/I_B (CE config)
- NAND and NOR are universal gates
🏆 Exam Day Checklist
- Write sign convention at start of EVERY optics numerical
- State the law before using it in every derivation
- Draw and label diagram for compound microscope, transformer, cyclotron, potentiometer
- Include units in EVERY final numerical answer
- Draw phasor diagram for every LCR / AC question
- Verify decay equations: both A and Z must balance
📝 Practice Papers
Full-length CBSE-pattern papers — 70 marks, 3 hours
Practice Paper 1 · Physics XII · 70 Marks · 3 Hours
General Instructions: All questions compulsory. Section A = MCQ (1M each). Section B = 2M each. Section C = 3M each. Section D = Case-based 4M. Section E = 5M each.
SECTION A — Multiple Choice (1 mark each)
1. The electric field inside a hollow spherical conductor placed in an external electric field is:
(a) Equal to external field (b) Zero (c) Double the external field (d) Half the external field
(a) Equal to external field (b) Zero (c) Double the external field (d) Half the external field
2. A wire of resistance R is bent into a circle. The effective resistance between the two ends of any diameter is:
(a) R (b) R/2 (c) R/4 (d) 2R
(a) R (b) R/2 (c) R/4 (d) 2R
3. The cyclotron is used to accelerate:
(a) Neutrons (b) Electrons (c) Protons (d) Photons
(a) Neutrons (b) Electrons (c) Protons (d) Photons
4. In Young's double slit experiment, when the whole apparatus is immersed in water (n=4/3), the fringe width:
(a) Increases by 4/3 (b) Decreases by 3/4 (c) Remains same (d) Doubles
(a) Increases by 4/3 (b) Decreases by 3/4 (c) Remains same (d) Doubles
5. The stopping potential in photoelectric effect depends on:
(a) Intensity of incident light (b) Frequency of incident light (c) Nature of cathode material only (d) Both (b) and (c)
(a) Intensity of incident light (b) Frequency of incident light (c) Nature of cathode material only (d) Both (b) and (c)
SECTION B — Short Answer (2 marks each)
6. State Kirchhoff's current law. What physical principle is it based on?
7. Define magnetic flux. What is the SI unit? Write its dimensional formula.
8. Draw the circuit symbol and truth table of a NAND gate. Why is it called a universal gate?
9. An object is placed 20 cm from a concave lens of focal length 15 cm. Find position and nature of image.
10. State the principle of working of a moving coil galvanometer.
SECTION C — Short Answer (3 marks each)
11. (a) State Gauss's law. (b) Using Gauss's law, derive the expression for electric field due to a uniformly charged infinite plane sheet of charge density σ.
12. A series LCR circuit has L = 0.5 H, C = 8 μF, R = 100 Ω. Find (a) resonant frequency, (b) impedance at resonance, (c) Q-factor.
13. With the help of a neat labelled diagram, describe the working of a potentiometer to compare the emf of two cells.
14. Explain the terms: (a) Critical angle (b) Total internal reflection. State two applications of TIR.
SECTION D — Case Study (4 marks)
15. Case Study: Transformer
A transformer works on the principle of mutual induction. It has a primary coil of N_p turns and secondary coil of N_s turns. When AC voltage V_p is applied to the primary, V_s appears at the secondary where V_s/V_p = N_s/N_p.
(a) What type of transformer has N_s > N_p? (b) A transformer steps up 220V to 22,000V. Find N_s/N_p. (c) If primary current is 10A, find secondary current (ideal transformer). (d) Why can't a transformer work on DC?
A transformer works on the principle of mutual induction. It has a primary coil of N_p turns and secondary coil of N_s turns. When AC voltage V_p is applied to the primary, V_s appears at the secondary where V_s/V_p = N_s/N_p.
(a) What type of transformer has N_s > N_p? (b) A transformer steps up 220V to 22,000V. Find N_s/N_p. (c) If primary current is 10A, find secondary current (ideal transformer). (d) Why can't a transformer work on DC?
SECTION E — Long Answer (5 marks each)
16. (a) Draw a labelled ray diagram to show the formation of image by a compound microscope when final image is at the near point. (b) Derive an expression for the total magnifying power. (c) How would the magnification change if an eyepiece of shorter focal length is used?
17. (a) State and prove Faraday's laws of electromagnetic induction. (b) A rectangular loop (25 cm × 10 cm) is placed in a magnetic field of 0.5 T perpendicular to its plane. The loop is pulled out of the field in 0.1 s. Find the induced emf and current if R = 2 Ω.
Practice Paper 2 · Physics XII · 70 Marks · 3 Hours
SECTION A — Multiple Choice (1 mark each)
1. Equipotential surfaces are always:
(a) Parallel to electric field lines (b) Perpendicular to electric field lines (c) At 45° to field lines (d) Along field lines
(a) Parallel to electric field lines (b) Perpendicular to electric field lines (c) At 45° to field lines (d) Along field lines
2. The SI unit of magnetic susceptibility is:
(a) T (b) A/m (c) Dimensionless (d) H/m
(a) T (b) A/m (c) Dimensionless (d) H/m
3. In a step-down transformer, the current in secondary compared to primary is:
(a) Less (b) More (c) Equal (d) Zero
(a) Less (b) More (c) Equal (d) Zero
4. The wavelength of light in a medium of refractive index n is:
(a) nλ₀ (b) λ₀/n (c) λ₀/n² (d) n²λ₀
(a) nλ₀ (b) λ₀/n (c) λ₀/n² (d) n²λ₀
5. The de Broglie wavelength of a particle is independent of:
(a) Mass (b) Velocity (c) Charge (d) Momentum
(a) Mass (b) Velocity (c) Charge (d) Momentum
SECTION B — Short Answer (2 marks each)
6. Derive the expression for energy stored in a parallel plate capacitor.
7. Define self-inductance. What is its SI unit? Write the dimensional formula.
8. What is Brewster's law? State Malus's law and write the mathematical form.
9. Calculate the de Broglie wavelength of an electron accelerated through a potential difference of 100 V.
10. What is the difference between intrinsic and extrinsic semiconductors?
SECTION C — Short Answer (3 marks each)
11. (a) What is drift velocity? (b) Derive the relation between current density J, drift velocity v_d and carrier density n. (c) Why is drift velocity of electrons very small (~10⁻⁴ m/s) yet current flows instantaneously?
12. The binding energy per nucleon for ₂₆Fe⁵⁶ is 8.8 MeV and for ₂He⁴ is 7.1 MeV. Explain why iron is the most stable nucleus. How does the BE/A curve explain both fission and fusion releasing energy?
13. With a circuit diagram, explain how a Wheatstone bridge works. State the balance condition.
14. Derive the expression for motional EMF (ε = Bvl) for a conductor of length l moving with velocity v perpendicular to magnetic field B.
SECTION D — Case Study (4 marks)
15. Case Study: Photoelectric Effect
When light of frequency above threshold falls on a metal, electrons are emitted. Einstein proposed E = hf and KE_max = hf − φ. The stopping potential V₀ satisfies eV₀ = KE_max. Intensity affects current, not KE.
(a) Define threshold frequency. (b) If φ = 2.0 eV and f = 7×10¹⁴ Hz, find KE_max. (c) What is the stopping potential? (d) Why does increasing intensity NOT change stopping potential?
When light of frequency above threshold falls on a metal, electrons are emitted. Einstein proposed E = hf and KE_max = hf − φ. The stopping potential V₀ satisfies eV₀ = KE_max. Intensity affects current, not KE.
(a) Define threshold frequency. (b) If φ = 2.0 eV and f = 7×10¹⁴ Hz, find KE_max. (c) What is the stopping potential? (d) Why does increasing intensity NOT change stopping potential?
SECTION E — Long Answer (5 marks each)
16. (a) Derive the expression for impedance of a series LCR circuit using phasor diagram. (b) What is resonance? (c) At resonance, f₀ = 1/(2π√LC). A series LCR has L=0.1H, C=10μF, R=50Ω. Find resonant frequency and Q-factor.
17. (a) State Bohr's postulates for hydrogen atom. (b) Derive the expression for radius of nth orbit and total energy of electron. (c) Calculate the wavelength of radiation emitted when electron transitions from n=3 to n=2 in hydrogen atom. (R_H = 1.097×10⁷ m⁻¹)
Practice Paper 3 · Physics XII · 70 Marks · 3 Hours
SECTION A — Multiple Choice (1 mark each)
1. Assertion: Electric field lines never intersect each other. Reason: No two electric field lines can have the same direction at a point.
(a) Both A and R true, R is correct explanation (b) Both true, R not correct explanation (c) A true, R false (d) A false, R true
(a) Both A and R true, R is correct explanation (b) Both true, R not correct explanation (c) A true, R false (d) A false, R true
2. The power dissipated in a pure inductor connected to AC source is:
(a) V_rms × I_rms (b) Zero (c) V_rms²/X_L (d) I_rms² × X_L
(a) V_rms × I_rms (b) Zero (c) V_rms²/X_L (d) I_rms² × X_L
3. In p-type semiconductor, majority carriers are:
(a) Electrons (b) Holes (c) Protons (d) Neutrons
(a) Electrons (b) Holes (c) Protons (d) Neutrons
4. Which EM wave is used in radar?
(a) Infrared (b) Ultraviolet (c) Microwaves (d) X-rays
(a) Infrared (b) Ultraviolet (c) Microwaves (d) X-rays
5. The work function of a metal is 3.0 eV. The threshold frequency is approximately:
(a) 7.3×10¹⁴ Hz (b) 5×10¹⁴ Hz (c) 3×10¹⁵ Hz (d) 4.5×10¹⁴ Hz
(a) 7.3×10¹⁴ Hz (b) 5×10¹⁴ Hz (c) 3×10¹⁵ Hz (d) 4.5×10¹⁴ Hz
SECTION B — Short Answer (2 marks each)
6. State the working principle of a cyclotron. What is the cyclotron frequency?
7. Define the angle of dip and angle of declination. Where on Earth is the angle of dip 90°?
8. What is AM modulation? Write the expression for modulated wave.
9. An object is placed 15 cm from a concave mirror of focal length 10 cm. Find image distance and magnification.
10. Why are NAND and NOR gates called universal gates? Give one example of each.
SECTION C — Short Answer (3 marks each)
11. (a) State Biot-Savart law. (b) Apply it to derive the magnetic field at the centre of a circular current-carrying loop of N turns and radius R. (c) State the direction using the right-hand rule.
12. (a) What is refraction through a prism? (b) Derive the relation n = sin[(A+δ_m)/2] / sin(A/2) where δ_m is the angle of minimum deviation. (c) What is dispersion?
13. Explain (a) depletion layer, (b) barrier potential in a p-n junction. How does forward bias change these?
14. Calculate the energy equivalent of 1 atomic mass unit (1 amu = 1.66×10⁻²⁷ kg). Express your answer in MeV.
SECTION D — Case Study (4 marks)
15. Case Study: Young's Double Slit Experiment
Two coherent sources S₁ and S₂ are separated by distance d. Screen is at distance D. Path difference for a point P at height y: Δx = yd/D. Bright fringe when Δx = nλ; dark fringe when Δx = (2n−1)λ/2. Fringe width β = λD/d.
(a) What conditions are needed for sustained interference? (b) β = 0.6 mm when D=1.5m, d=1mm. Find λ. (c) How does fringe width change when slit separation is halved? (d) If the experiment is done in water (n=1.33), find new fringe width.
Two coherent sources S₁ and S₂ are separated by distance d. Screen is at distance D. Path difference for a point P at height y: Δx = yd/D. Bright fringe when Δx = nλ; dark fringe when Δx = (2n−1)λ/2. Fringe width β = λD/d.
(a) What conditions are needed for sustained interference? (b) β = 0.6 mm when D=1.5m, d=1mm. Find λ. (c) How does fringe width change when slit separation is halved? (d) If the experiment is done in water (n=1.33), find new fringe width.
SECTION E — Long Answer (5 marks each)
16. (a) Explain Huygens' principle. (b) Using it, prove Snell's law of refraction. (c) What is the critical angle for a glass-air interface where glass has n=1.5? Calculate it and explain what happens above this angle.
17. (a) Describe the construction and working of a common emitter transistor amplifier with a circuit diagram. (b) Derive the expression for voltage gain. (c) A CE amplifier has input resistance 1kΩ, output resistance 10kΩ, β=50. Calculate voltage gain.
Practice Paper 2 · Physics XII · 70 Marks · 3 Hours
Focus: Magnetism, EMI, Optics & Modern Physics emphasis.
SECTION A — MCQ (16 × 1 = 16 M)
1. The magnetic moment of a current loop of area A carrying current I in a field B is: (a) IAB (b) IA (c) IB (d) A/I
2. The SI unit of magnetic flux is: (a) Tesla (b) Weber (c) Henry (d) Ampere·meter²
3. Lenz's law is a consequence of conservation of: (a) Charge (b) Momentum (c) Energy (d) Mass
4. In an AC circuit with only capacitor, the phase of current compared to voltage is: (a) Same (b) Leads by 90° (c) Lags by 90° (d) Leads by 45°
5. Which of the following is NOT an electromagnetic wave? (a) X-ray (b) γ-ray (c) Ultrasound (d) UV ray
6. For a convex mirror, image formed is always: (a) Real and magnified (b) Virtual and diminished (c) Real and inverted (d) Virtual and magnified
7. The radius of nth Bohr orbit is proportional to: (a) n (b) n² (c) n³ (d) 1/n
8. Which series of hydrogen spectrum lies in visible range? (a) Lyman (b) Balmer (c) Paschen (d) Brackett
9. Binding energy per nucleon is maximum for: (a) H-2 (b) Fe-56 (c) U-238 (d) He-4
10. A p-n junction diode in reverse bias has: (a) High resistance (b) Low resistance (c) No resistance (d) Negative resistance
11. In an n-type semiconductor, the majority carriers are: (a) Holes (b) Free electrons (c) Protons (d) Neutrons
12. The modulation index for AM must be: (a) Greater than 1 (b) Equal to 1 (c) Less than or equal to 1 (d) Any value
13-16. [Assertion-Reason: Diamagnetic substances repelled; Transformer works on AC; Photon has no charge; NAND gate is universal gate]
SECTION B — SA I (5 × 2 = 10 M)
17. State Fleming's right-hand rule and left-hand rule. When is each applied?
18. What is self-inductance? Give its SI unit and formula for a solenoid.
19. Draw a labelled diagram of an AC generator. State the principle.
20. What is RADAR? Mention the electromagnetic waves used in it and their properties.
21. State radioactive decay law. Write the expression for activity at time t.
SECTION C — SA II (7 × 3 = 21 M)
22. Derive an expression for the torque acting on a magnetic dipole in a uniform magnetic field.
23. Explain Faraday's laws of electromagnetic induction. Derive motional EMF: ε = Blv.
24. Derive the expression for impedance and phase angle in a series LCR circuit driven by AC.
25. State Huygen's principle. Using it, prove the law of reflection.
26. Explain photoelectric effect. Draw and explain the I-V graph for different intensities at same frequency.
27. Write three properties of nuclear forces. Compare nuclear stability using BE/nucleon curve.
28. Draw input and output characteristics of a CE transistor. Identify active, saturation and cut-off regions.
SECTION D — Case-Based (4 M)
29. A transformer has 1000 primary turns and 100 secondary turns. Primary supply = 220V. Find: (i) Secondary voltage (ii) If load = 100Ω, find secondary current (iii) Primary current assuming ideal transformer (iv) Power transferred.
SECTION E — LA (3 × 5 = 15 M)
30. (a) State and prove Biot-Savart law. (b) Using it, find magnetic field at the centre of a circular current-carrying loop. (c) Two loops of radii 5cm and 10cm carry currents 2A each. Find ratio of magnetic fields at centres.
31. (a) Define wavefront. Types of wavefronts. (b) Explain interference of light. Derive condition for constructive and destructive interference. (c) In YDSE: λ=589nm, d=0.2mm, D=1m. Find fringe width and position of 3rd dark fringe.
32. (a) Explain energy band theory in solids. (b) Distinguish between conductor, semiconductor and insulator using band theory. (c) Draw and explain a half-wave rectifier circuit. Find its frequency output if input is 50Hz.
Practice Paper 3 · Physics XII · 70 Marks · 3 Hours (Challenge)
Higher-order thinking & numericals-heavy. Board Exam Level.
SECTION A — MCQ (16 M)
1. Two point charges +4μC and −1μC are placed 30cm apart. The point on the line joining them where E=0 is: (a) 10cm from +4μC outside (b) 30cm from −1μC outside (c) 20cm from +4μC on line between them (d) 60cm from −1μC outside
2. If the potential at a point is V = 3x² + 4y, then E at (1,1) is: (a) 6î + 4ĵ (b) −6î − 4ĵ (c) 6î − 4ĵ (d) −6î + 4ĵ [Hint: E = −∇V]
3. A proton and an alpha particle accelerated through same potential V. Ratio of their de Broglie wavelengths λp : λα = (a) 2√2 (b) √2 (c) 1 (d) 2
4. In a mass spectrometer, radius of circular path r = mv/qB. If kinetic energy is kept same, ratio of radii of proton and deuteron is: (a) 1:1 (b) 1:√2 (c) √2:1 (d) 1:2
5-16. [Numericals on: Resistance in parallel/series, Mutual inductance between coaxial solenoids, Fringe shift in YDSE, Threshold frequency from stopping potential graph, Alpha particle scattering closest distance, Q-value of nuclear reaction, AND/NOR gate combinations, Sky wave vs space wave propagation]
SECTION B — SA I (10 M)
17. A 2μF and 3μF capacitor connected in series across 100V. Find charge on each and pd across each.
18. A galvanometer of resistance 50Ω gives full-scale deflection for 1mA. Convert to (a) voltmeter for 10V (b) ammeter for 1A.
19. Electromagnetic spectrum: arrange in increasing order of wavelength: UV, X-ray, IR, visible, microwave, radio, γ-ray.
20. Calculate the energy released when 3.5 kg of U-235 undergoes fission. (1 fission → 200 MeV; NA=6×10²³)
21. A NOR gate output is 1 only when both inputs are 0. Write truth table for: (i) NOR gate (ii) NOR+NOT = OR reconstruction.
SECTION C — SA II (21 M)
22. Derive energy stored in an inductor: U = ½LI². A 2H inductor carries 5A; energy stored? If current doubles, what happens to energy?
23. Explain total internal reflection with diagram and two real-life applications. Derive expression for critical angle.
24. In Compton scattering, why is the effect observable with X-rays but not visible light? Derive expression for Compton shift.
25. Calculate the wavelength emitted when hydrogen electron jumps from n=5 to n=2. Identify the series. (R=1.097×10⁷ m⁻¹)
26. What is nuclear fission? Write the fission equation for U-235. Calculate energy in MeV. Compare with nuclear fusion.
27. Draw and explain the block diagram of an AM modulation system. Define modulation index and significance of keeping it ≤1.
28. Three equal resistors each R connected in triangle. Find equivalent resistance between any two vertices.
SECTION D — Case-Based (4 M)
29. Prism and Dispersion: A prism of angle 60° and μ=1.5. (i) Find minimum deviation. (ii) Refracting angle at minimum deviation. (iii) Why do different colours deviate by different amounts? (iv) Define dispersive power.
SECTION E — LA (15 M)
30. (a) Derive van de Graaff generator working from first principles — why charge accumulates on outer shell. (b) Practical uses. (c) A van de Graaff sphere of radius 1m is charged to V=3×10⁶V. Find: surface charge density, capacity, charge stored.
31. (a) Explain LASER principle: stimulated emission, population inversion, optical pumping. (b) How is laser light different from ordinary light? (c) Semiconductor lasers — how they differ from gas lasers. (d) Applications in medicine and communication.
32. (a) Explain the photodiode construction and working. (b) Draw and explain I-V characteristics in forward and reverse bias. (c) Why is photodiode used in reverse bias? (d) A photodiode registers current 10μA in dark and 20mA in light. Calculate the photocurrent.
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