NCERT Physics (Class 11–12) Cheatsheet

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Mechanics

CLASS 11

Units & Measurements

SI Base Unitsm (length), kg (mass), s (time), A (current), K (temp), mol (amount), cd (luminous)

Supplementaryrad (plane angle), sr (solid angle)

Derived: Newton1 N = 1 kg·m/s²

Derived: Joule1 J = 1 kg·m²/s² = 1 N·m

Derived: Watt1 W = 1 J/s = 1 kg·m²/s³

Derived: Pascal1 Pa = 1 N/m² = 1 kg/(m·s²)

Dimensional Analysis

PrinciplePhysical quantities with same dimensions can be added/subtracted

UseCheck correctness of equations, derive relations, convert units

LimitationCannot determine dimensionless constants (π, 2, ½, etc.)

Example: T of pendulumT ∝ √(L/g) → [T] = [L]¹/²[M⁻¹/²][T⁻¹/²⁻¹] = T ✓

Significant Figures Rules

Rule	Example
Non-zero digits are significant	345 → 3 sig figs
Zeros between non-zeros are significant	105 → 3 sig figs
Leading zeros are NOT significant	0.0045 → 2 sig figs
Trailing zeros after decimal are significant	2.50 → 3 sig figs
Exact numbers have infinite sig figs	π = 3.14159...
Round up if digit ≥ 5	2.346 → 2.35 (3 sf)

Errors in Measurements

Error Type	Definition	Formula
Absolute Error	Magnitude of difference between true & measured value	Δxᵢ = \|xₜᵣᵤₑ − xᵢ\|
Mean Absolute Error	Mean of all absolute errors	Δx_mean = Σ\|Δxᵢ\| / n
Relative Error	Ratio of mean absolute error to true value	δx = Δx_mean / xₜᵣᵤₑ
Percentage Error	Relative error expressed as percentage	% error = δx × 100%

Combination of Errors

Sum/DifferenceIf z = a ± b, then Δz = Δa + Δb

ProductIf z = a × b, then Δz/z = Δa/a + Δb/b

QuotientIf z = a/b, then Δz/z = Δa/a + Δb/b

PowerIf z = aⁿ, then Δz/z = |n| × Δa/a

Motion in a Straight Line

DisplacementΔx = x₂ − x₁ (vector, shortest path)

DistanceTotal path length (scalar, always ≥ |displacement|)

Average Speedv̄ = Total distance / Total time

Average Velocityv̄ = Δx / Δt (vector)

Instantaneous Velocityv = dx/dt = lim(Δt→0) Δx/Δt

Accelerationa = dv/dt

Kinematic Equations (Uniform Acceleration)

v = u + atVelocity-time relation

s = ut + ½at²Displacement-time relation

v² = u² + 2asVelocity-displacement relation (no t)

s = ½(u + v)tDisplacement (average velocity)

sₙ = u + ½a(2n − 1)Displacement in n-th second

Graphs — Key Relations

Graph	Slope Gives	Area Gives
Position vs Time	Velocity	—
Velocity vs Time	Acceleration	Displacement
Acceleration vs Time	Jerk (da/dt)	Change in velocity

Free Fall (g = 9.8 m/s²)

v = u + gtDownward positive

s = ut + ½gt²Displacement in time t

v² = u² + 2gsNo time needed

Time to reach max heightt = u/g

Max heightH = u²/(2g)

Time of flightT = 2u/g

Relative Velocity

1Dv_AB = v_A − v_B (velocity of A relative to B)

2Dv_AB = v_A − v_B (vector subtraction)

Rain problemv_man, rain = v_rain − v_man → angle of umbrella

River-boatv_boat,ground = v_boat,river + v_river (vector sum)

Crossing (min time)Boat heads perpendicular: t = d/v_boat,river

Crossing (shortest)Drift = 0: v_boat,river cos θ = v_river

Motion in a Plane — Vectors

MagnitudeIf A = Aₓî + Aᵧĵ, then |A| = √(Aₓ² + Aᵧ²)

Directiontan θ = Aᵧ/Aₓ

Unit VectorÂ = A/|A|

Dot ProductA·B = |A||B|cos θ = AₓBₓ + AᵧBᵧ + A_zB_z

Cross ProductA×B = |A||B|sin θ n̂ (perpendicular to both)

ResolutionAₓ = A cos θ, Aᵧ = A sin θ

Projectile Motion (g downward)

Horizontalx = (u cos θ)·t, vₓ = u cos θ (constant)

Verticaly = (u sin θ)t − ½gt², vᵧ = u sin θ − gt

Time of FlightT = 2u sin θ / g

Max HeightH = u² sin²θ / (2g)

Horizontal RangeR = u² sin 2θ / g

Max RangeR_max = u²/g at θ = 45°

Trajectoryy = x tan θ − gx²/(2u² cos²θ) (parabola)

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Complementary angles: For θ and (90° − θ), the range is the same: R = u² sin 2θ/g. However, the time of flight and max height differ.

Circular Motion

Angular Velocityω = dθ/dt, v = rω

PeriodT = 2π/ω = 2πr/v

Centripetal Accelerationa_c = v²/r = ω²r = 4π²r/T² (towards center)

Centripetal ForceF_c = mv²/r = mω²r (always towards center)

Tangential Accelerationa_t = dv/dt (changes speed, not direction)

Total Accelerationa = √(a_c² + a_t²)

Projectile Motion — Special Cases

Angle for max rangeθ = 45° (R = u²/g)

θ and (90°−θ)Same range, different H and T

Horizontal projectionθ = 0: R = u√(2H/g), T = √(2H/g)

Trajectory equationy = x tan θ − gx²/(2u² cos²θ)

Velocity at any pointv = √(vₓ² + vᵧ²), tan α = vᵧ/vₓ

Radius of curvature at topR_top = u² cos²θ / g

Equilibrium — Key Concepts

Translational EquilibriumΣF = 0 (net force zero → a = 0)

Rotational EquilibriumΣτ = 0 (net torque zero → no angular acceleration)

Stable EquilibriumSmall displacement → restoring force (PE increases)

Unstable EquilibriumSmall displacement → force away from equilibrium (PE decreases)

Neutral EquilibriumDisplacement → no restoring force (PE constant)

Ladder problemUse both ΣF = 0 and Στ = 0; choose convenient pivot point

Newton's Laws of Motion

Law	Statement	Key Formula
1st (Inertia)	A body continues in its state of rest or uniform motion unless acted upon by external force	ΣF = 0 → v = const
2nd (F = ma)	Rate of change of momentum is proportional to applied force	F = dp/dt = ma
3rd (Action-Reaction)	Every action has an equal and opposite reaction	F_AB = −F_BA

Force & Momentum

Linear Momentump = mv (vector, kg·m/s)

ImpulseJ = F·Δt = Δp = mΔv (change in momentum)

Impulse-MomentumF·Δt = m(v₂ − v₁)

Newton's 2nd LawF = dp/dt = d(mv)/dt = ma (constant mass)

WeightW = mg (force due to gravity)

Normal ForceN = mg cos θ (on inclined plane)

Free Body Diagrams — Common Forces

Force	Direction	Formula / Notes
Tension (T)	Along rope, away from body	Same throughout massless, inextensible string
Normal (N)	Perpendicular to surface	N = mg cos θ on incline
Friction (f)	Opposite to motion/tendency	f ≤ μN
Spring Force	Towards equilibrium	F = −kx (Hooke's law)
Weight (mg)	Vertically downward	mg = W
Applied Force	Direction of push/pull	Given in problem

Friction

Type	Formula	Condition
Static Friction (f_s)	f_s ≤ μ_s · N	Body at rest, no sliding
Limiting Friction	f_max = μ_s · N	Just about to slide
Kinetic Friction (f_k)	f_k = μ_k · N	Body in motion, μ_k < μ_s
Rolling Friction	f_r = μ_r · N	μ_r << μ_k

Angle of Reposetan θ = μ_s → θ is the angle of friction = angle of repose

Banking of Roadstan θ = v²/(rg) → ideal speed on banked road (no friction needed)

Banking with frictionv_max = √[rg(tan θ + μ)/(1 − μ tan θ)]

Connected Bodies & Pulleys

Atwood Machinea = (m₁ − m₂)g/(m₁ + m₂), T = 2m₁m₂g/(m₁ + m₂)

Block on inclinea = g(sin θ − μ cos θ) (down the plane)

Hanging block + table blocka = m₂g/(m₁ + m₂), T = m₁m₂g/(m₁ + m₂)

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Common mistake: When solving pulley problems, remember that if the string is inextensible, the magnitude of acceleration is the same for all connected bodies. Also, tension is not always equal to mg — it changes based on the acceleration of the system.

Work, Energy & Power

Work by constant FW = F·d·cos θ = F · d (dot product)

Work by variable FW = ∫F·dx (area under F-x curve)

Kinetic EnergyKE = ½mv²

Work-Energy TheoremW_net = ΔKE = ½mv₂² − ½mv₁²

PowerP = W/t = F·v (instantaneous)

Power (force at angle)P = F·v·cos θ

Potential Energy

Gravitational PEU = mgh (near Earth surface)

Spring PEU = ½kx² (elastic potential energy)

Gravitational PE (general)U = −GMm/r

Force-PE relationF = −dU/dx (conservative force)

Conservative vs Non-Conservative

Property	Conservative	Non-Conservative
Work done	Path independent	Path dependent
Closed loop work	Zero	Non-zero
Example	Gravity, Spring force	Friction, Air drag
PE defined?	Yes	No

Collisions

Type	KE Conserved?	Momentum?
Perfectly Elastic	Yes	Yes
Inelastic	No	Yes
Perfectly Inelastic	No (max loss)	Yes — bodies stick

Elastic Collision — 1D Formulas

Conservation of Momentumm₁u₁ + m₂u₂ = m₁v₁ + m₂v₂

Final velocitiesv₁ = [(m₁−m₂)u₁ + 2m₂u₂]/(m₁+m₂)

Final velocitiesv₂ = [(m₂−m₁)u₂ + 2m₁u₁]/(m₁+m₂)

Equal masses (m₁=m₂)Velocities exchange: v₁ = u₂, v₂ = u₁

Light body hits heavy (m₂>>m₁)v₁ ≈ −u₁, v₂ ≈ u₂ (light rebounds)

Heavy hits light (m₁>>m₂)v₁ ≈ u₁, v₂ ≈ 2u₁ − u₂ (heavy continues)

Rotational Motion — Centre of Mass

CM (discrete)x_cm = Σmᵢxᵢ / Σmᵢ

CM (continuous)x_cm = (1/M)∫x dm

2D CMx_cm = Σmᵢxᵢ/M, y_cm = Σmᵢyᵢ/M

Moment of Inertia (I)

DefinitionI = Σmᵢrᵢ² or I = ∫r² dm (analogue of mass)

Parallel Axis TheoremI = I_cm + Md² (d = distance from CM axis)

Perpendicular Axis TheoremI_z = I_x + I_y (only for planar bodies)

Radius of GyrationI = Mk² → k = √(I/M)

Moments of Inertia — Common Bodies

Body	Axis	I
Solid cylinder/disc	Through center, ⊥ to plane	½MR²
Hollow cylinder/ring	Through center, ⊥ to plane	MR²
Solid sphere	Through diameter	⅖MR²
Hollow sphere (thin)	Through diameter	⅔MR²
Thin rod	Through center, ⊥ to length	ML²/12
Thin rod	Through end, ⊥ to length	ML²/3
Rectangular plate	Through center, ⊥ to plane	M(a²+b²)/12

Rotational Dynamics

Torqueτ = r × F = rF sin θ (vector, N·m)

Newton's 2nd (rotation)τ = Iα (α = angular acceleration)

Angular MomentumL = Iω = r × p (vector, kg·m²/s)

ConservationI₁ω₁ = I₂ω₂ (if τ_ext = 0)

Rotational KEKE_rot = ½Iω²

Rolling (no slip)v = Rω, a = Rα, a_cm = F_net/(M+I/R²)

Rolling KE totalKE = ½Mv² + ½Iω² = ½Mv²(1 + I/MR²)

Acceleration down inclinea = g sin θ / (1 + I/MR²)

Moment of Inertia — Additional Bodies

Body	Axis	I
Solid cylinder (about diameter)	Through center, along diameter	(1/4)MR² + (1/12)ML²
Hollow cylinder (thin wall)	Through center, ⊥ to axis	½MR² + (1/12)ML²
Solid cone (about axis)	Through apex, along axis	(3/10)MR²
Solid cone (about base diameter)	Through base center	(3/20)M(R² + 4H²)
Ring (about diameter)	Through center, along diameter	½MR²

Rolling Motion — Comparison

Object	I (about CM)	a (down incline)	Speed at bottom
Solid cylinder	½MR²	(5/7)g sin θ	√(10gh/7)
Hollow cylinder	MR²	½g sin θ	√(gh)
Solid sphere	⅖MR²	(5/7)g sin θ	√(10gh/7)
Hollow sphere	⅔MR²	(3/5)g sin θ	√(6gh/5)
Ring	MR²	½g sin θ	√(gh)
Solid disc	½MR²	(2/3)g sin θ	√(4gh/3)

Key insightObject with lower I/MR² ratio reaches bottom faster

Gravitation

Newton's LawF = GMm/r² (attractive, along line joining)

G (universal)6.674 × 10⁻¹¹ N·m²/kg²

Acceleration due to gravityg = GM/R² (on surface)

g at height hg' = g·R²/(R+h)² ≈ g(1 − 2h/R) for h << R

g at depth dg' = g(1 − d/R) (decreases linearly)

Gravitational Potential & Energy

Potential (V)V = −GM/r (scalar, J/kg)

Potential on surfaceV_surface = −GM/R

PEU = −GMm/r = mV

PE change (surface to h)ΔU = mgh (for h << R)

Escape Velocityvₑ = √(2GM/R) = √(2gR) ≈ 11.2 km/s

Orbital Velocityvₒ = √(GM/R) = √(gR) ≈ 7.9 km/s

Satellites & Kepler's Laws

Kepler's Law	Statement
1st (Law of Orbits)	Every planet revolves in an elliptical orbit with Sun at one focus
2nd (Law of Areas)	Radius vector sweeps equal areas in equal times → dA/dt = L/(2m) = const
3rd (Law of Periods)	T² ∝ R³ → T² = (4π²/GM)R³

Satellite PeriodT = 2π√(R³/GM) = 2π√((R+h)³/(gR²))

GeostationaryT = 24 hrs, h ≈ 35,786 km above surface, orbits west→east in equatorial plane

Orbital EnergyE = −GMm/(2r) = ½KE + U (KE = −U/2)

Binding EnergyE_b = GMm/(2r) = |E|

Variation of g — Comprehensive

Effect	Formula	Result
At height h	g' = g(R/(R+h))² ≈ g(1−2h/R)	Decreases with height
At depth d	g' = g(1−d/R)	Decreases linearly (zero at center)
Due to rotation	g' = g − ω²R cos²φ	Max at poles, min at equator
Due to latitude φ	g' = g − ω²R cos²φ	g_pole > g_equator by ~0.034 m/s²
On a mountain	g' = g(R/(R+h))²	Same as height formula
Effect of shape (Earth)	g_pole > g_eq (oblate spheroid)	REquator > RPole by ~21 km

g at polesg ≈ 9.832 m/s² (maximum, no centrifugal effect)

g at equatorg ≈ 9.780 m/s² (minimum, maximum centrifugal effect)

g at standardg = 9.80665 m/s² (standard value)

Mechanical Properties of Solids

Stressσ = F/A (N/m² = Pa) — restoring force per unit area

Strainε = ΔL/L (dimensionless) — fractional change

Young's Modulus (Y)Y = (F/A)/(ΔL/L) = stress/strain (longitudinal)

Bulk Modulus (B)B = −P/(ΔV/V) — volume stress/strain, B = 1/κ

Shear Modulus (G)G = (F/A)/θ — tangential stress/strain

Compressibilityκ = 1/B = ΔV/(PV) (Pa⁻¹)

Poisson's Ratioσ = −(lateral strain)/(longitudinal strain), range: −1 to 0.5

Elastic EnergyU = ½ × stress × strain × volume = ½Yε²V

Mechanical Properties of Fluids

PressureP = F/A (Pascal, 1 atm = 1.013 × 10⁵ Pa)

Pascal's LawPressure applied to enclosed fluid is transmitted undiminished to all parts

Hydraulic liftF₁/A₁ = F₂/A₂ → F₂ = F₁(A₂/A₁)

Pressure at depth hP = P₀ + ρgh

Buoyancy (Archimedes)F_b = ρ_fluid × V_displaced × g

Fluid Dynamics

Equation of ContinuityA₁v₁ = A₂v₂ (incompressible fluid)

Bernoulli's EquationP + ½ρv² + ρgh = constant

VenturimeterP₁ + ½ρv₁² = P₂ + ½ρv₂² → flow rate

Speed of effluxv = √(2gh) (Torricelli's theorem)

Viscosity & Surface Tension

Viscous Force (Newton)F = ηA(dv/dx) — η = coefficient of viscosity (Pa·s)

Stokes' LawF = 6πηrv (sphere in viscous fluid)

Terminal Velocityv_t = 2r²(ρ − σ)g/(9η)

Poiseuille's EquationQ = πPr⁴/(8ηl) (flow through pipe)

Surface TensionT = F/l (N/m) — force per unit length

Excess Pressure (bubble)ΔP = 4T/R (soap bubble), ΔP = 2T/R (droplet)

Capillary Riseh = 2T cos θ/(ρgr)

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Surface tension: Temperature decreases surface tension. Adding surfactants (like soap) also reduces it. Surface tension is the property of the liquid surface that acts as a stretched elastic membrane.

Dimensional Formulas — Quick Reference

Quantity	Formula	Dimensions
Velocity	v = dx/dt	[LT⁻¹]
Acceleration	a = dv/dt	[LT⁻²]
Force	F = ma	[MLT⁻²]
Work / Energy	W = Fd	[ML²T⁻²]
Power	P = W/t	[ML²T⁻³]
Pressure	P = F/A	[ML⁻¹T⁻²]
Momentum	p = mv	[MLT⁻¹]
Impulse	J = Ft	[MLT⁻¹]
Torque	τ = rF	[ML²T⁻²]
Angular Momentum	L = mvr	[ML²T⁻¹]
Moment of Inertia	I = mr²	[ML²]
Surface Tension	T = F/l	[MT⁻²]
Universal Grav. Constant	G = Fr²/m₁m₂	[M⁻¹L³T⁻²]
Planck Constant	h = E/f	[ML²T⁻¹]
Coefficient of Viscosity	η = F/(A·dv/dx)	[ML⁻¹T⁻¹]
Bulk Modulus	B = −P/(ΔV/V)	[ML⁻¹T⁻²]
Young's Modulus	Y = stress/strain	[ML⁻¹T⁻²]
Wavelength	λ = v/f	[L]
Frequency	f = 1/T	[T⁻¹]
Angular frequency	ω = 2πf	[T⁻¹]

Important Dimensionless Quantities

Quantity	Formula / Value
Strain	ΔL/L (dimensionless)
Poisson's ratio	σ = −lateral strain/longitudinal strain
Refractive index	n = c/v
Magnification	m = v/u or h'/h
Coefficient of friction	μ = f/N
Mechanical advantage	MA = load/effort
Efficiency	η = W_out/W_in
Relative density	ρ_substance/ρ_water
Reynolds number	Re = ρvD/η (determines flow type)
Mach number	Ma = v/v_sound (subsonic < 1, supersonic > 1)

Dimensional Analysis — Applications

Convert unitsUse dimensional formula: 1 N = 1 kg·m/s² = 10⁵ dyne

Check correctnessLHS and RHS must have same dimensions

Derive relationsFind how physical quantities relate (e.g., T of pendulum)

LimitationCannot find dimensionless constants (π, e, 2, etc.)

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Thermodynamics

CLASS 11

Thermal Properties of Matter

Temperature ScalesT(K) = T(°C) + 273.15, T(°F) = (9/5)T(°C) + 32

Linear ExpansionΔL = L₀ α ΔT → L = L₀(1 + α ΔT)

Area ExpansionΔA = A₀ β ΔT, β ≈ 2α

Volume ExpansionΔV = V₀ γ ΔT, γ ≈ 3α

Expansion of waterWater has anomalous expansion — max density at 4°C

Heat & Calorimetry

Specific HeatQ = mcΔT (c = specific heat, J/kg·K)

Molar Specific HeatC = Q/(nΔT) (J/mol·K)

Latent Heat (fusion)Q = mL_f (melting/solidifying at constant T)

Latent Heat (vaporization)Q = mL_v (boiling/condensing at constant T)

CalorimetryHeat lost = Heat gained (no loss to surroundings)

Water equivalentW = mc (mass of water with same heat capacity)

Heat Transfer

Mode	Mechanism	Key Formula
Conduction	Direct molecular collision	dQ/dt = −kA(dT/dx) (Fourier's law)
Convection	Bulk movement of fluid	Natural: buoyancy driven, Forced: pump/fan
Radiation	EM wave emission	P = eσAT⁴ (Stefan-Boltzmann)

Thermal Radiation Laws

Stefan-BoltzmannP = eσAT⁴, σ = 5.67 × 10⁻⁸ W/m²·K⁴

Wien's Displacementλ_max × T = b = 2.898 × 10⁻³ m·K

Newton's Law of CoolingdT/dt = −k(T − T_s), dT/dt ∝ (T − T_s)

Cooling (full formula)T(t) = T_s + (T₀ − T_s)e^(−kt)

Perfect BlackbodyAbsorbs all radiation, e = 1 (ideal emitter/absorber)

Emissivity0 ≤ e ≤ 1; good absorber = good emitter (Kirchhoff's)

Laws of Thermodynamics

Law	Statement
Zeroth	If A is in thermal equilibrium with B and C, then B is in equilibrium with C → defines temperature
First	ΔU = Q − W → energy cannot be created or destroyed, only converted
Second	Heat cannot flow from cold to hot without external work; entropy of isolated system always increases
Third	As T → 0 K, entropy → 0 for a perfect crystal (unattainable absolute zero)

Thermodynamic Processes

Process	Condition	Key Relation	Work Done
Isothermal	ΔT = 0	PV = const	W = nRT ln(V₂/V₁)
Adiabatic	Q = 0	PV^γ = const	W = (P₁V₁ − P₂V₂)/(γ−1) = nRΔT/(γ−1)
Isobaric	ΔP = 0	V/T = const	W = PΔV
Isochoric	ΔV = 0	P/T = const	W = 0

Carnot Engine & Refrigerator

Carnot CycleTwo isothermal + two adiabatic processes (most efficient)

Efficiency (Carnot)η = 1 − T₂/T₁ = (T₁ − T₂)/T₁ = W/Q₁

Work per cycleW = Q₁ − Q₂ = nR(T₁ − T₂) ln(V₂/V₁)

Coefficient of Performance (refrigerator)β = Q₂/W = T₂/(T₁ − T₂)

COP (heat pump)β_HP = Q₁/W = T₁/(T₁ − T₂) = β + 1

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Second Law consequences:No engine can be 100% efficient (η < 1 always). Heat cannot spontaneously flow from cold to hot. Entropy of the universe always increases in irreversible processes.

Thermodynamic Potentials & Maxwell Relations

Internal Energy (U)U = f(T, V) — dU = TdS − PdV

Enthalpy (H)H = U + PV — dH = TdS + VdP

Helmholtz Free Energy (F)F = U − TS — dF = −SdT − PdV

Gibbs Free Energy (G)G = H − TS — dG = −SdT + VdP

Gibbs & EquilibriumAt equilibrium (const T, P): ΔG = 0; ΔG < 0 → spontaneous

Entropy (S)dS = δQ_rev/T — state function, path independent

Kinetic Theory of Gases — Postulates

Postulate	Description
1	Gas consists of large number of tiny particles (molecules)
2	Molecules are in continuous, random, rapid motion
3	Molecular collisions are perfectly elastic (KE conserved)
4	No intermolecular forces except during collisions
5	Volume of molecules is negligible compared to container volume
6	Time of collision is negligible compared to time between collisions

Kinetic Theory — Key Formulas

PressureP = ⅓ρv²_rms = ⅓(mN/V)v²_rms = (2/3)(KE/V)

Ideal Gas LawPV = nRT = NkT (R = 8.314 J/mol·K, k = 1.38 × 10⁻²³ J/K)

RMS Speedv_rms = √(3RT/M) = √(3kT/m) (M = molar mass)

Average Speedv_avg = √(8RT/πM)

Most Probable Speedv_mp = √(2RT/M)

Orderv_rms > v_avg > v_mp

KE per molecule½mv²_rms = (3/2)kT

Total KEE = (3/2)nRT = (f/2)nRT

Degrees of Freedom & Specific Heats

EquipartitionEnergy = ½kT per degree of freedom per molecule

Monatomic (f=3)Cv = (3/2)R, Cp = (5/2)R, γ = Cp/Cv = 5/3

Diatomic (f=5)Cv = (5/2)R, Cp = (7/2)R, γ = 7/5

Triatomic (non-linear, f=6)Cv = 3R, Cp = 4R, γ = 4/3

Mayer's RelationCp − Cv = R

γγ = Cp/Cv = 1 + 2/f

Mean Free Path

DefinitionAverage distance between successive collisions

Formulaλ = 1/(√2 πd²n) — d = molecular diameter, n = number density

In terms of T, Pλ = kT/(√2 πd²P)

Collision frequencyz = v_avg/λ = √2 πd²n v_avg

Gas Laws — Quick Summary

Law	Relation	Constant Parameter
Boyle's Law	PV = const	T (isothermal)
Charles's Law	V/T = const	P (isobaric)
Gay-Lussac's Law	P/T = const	V (isochoric)
Avogadro's Law	V/n = const	P, T
Ideal Gas	PV = nRT	All variables related
Dalton's Partial Pressure	P_total = P₁ + P₂ + ...	T, V
Graham's Law	r₁/r₂ = √(M₂/M₁)	P, T (diffusion)

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At STP: v_rms for O₂ ≈ 461 m/s, for H₂ ≈ 1920 m/s, for N₂ ≈ 517 m/s. RMS speed is proportional to √(T/M) — lighter molecules move faster at the same temperature.

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Oscillations & Waves

CLASS 11

Simple Harmonic Motion (SHM)

DefinitionMotion where acceleration ∝ −displacement from equilibrium

ForceF = −kx (Hooke's law — restoring force)

Accelerationa = −ω²x = −(k/m)x

Displacementx = A sin(ωt + φ) or x = A cos(ωt + φ)

Velocityv = Aω cos(ωt + φ) = ω√(A² − x²)

Accelerationa = −Aω² sin(ωt + φ)

Max velocityv_max = Aω (at x = 0)

Max accelerationa_max = Aω² (at x = ±A)

SHM — Time Period & Frequency

Angular frequencyω = 2π/T = 2πf = √(k/m)

Time Period (spring)T = 2π√(m/k)

Time Period (pendulum)T = 2π√(L/g)

Pendulum (effective)T = 2π√(L_eff/g), where L_eff = distance to CM

Frequencyf = 1/T = ω/(2π)

Pendulum in lift (up, a)T = 2π√(L/(g+a))

Pendulum in lift (down, a)T = 2π√(L/(g−a))

Pendulum (free fall)T → ∞ (no oscillation, g_eff = 0)

SHM — Energy

Total EnergyE = ½kA² = ½mω²A² (constant)

Kinetic EnergyKE = ½mω²(A² − x²) = ½k(A² − x²)

Potential EnergyPE = ½mω²x² = ½kx²

KE at mean positionKE_max = ½kA² = E (PE = 0)

PE at extremePE_max = ½kA² = E (KE = 0)

Average KE<KE> = ¼kA² = E/2

Average PE<PE> = ¼kA² = E/2

Damped & Forced Oscillations

Type	Damping Force	Frequency	Amplitude
Undamped	F = 0	ω₀ = √(k/m)	Constant (A)
Under-damped	F = −bv (small)	ω_d < ω₀	Decays exponentially
Critically damped	b = 2√(km)	ω = 0	Returns fastest to rest
Over-damped	b > 2√(km)	ω = 0	Slow return to rest

Damped equationma + bv + kx = 0 → x = Ae^(−bt/2m)cos(ω't + φ)

Damped frequencyω' = √(ω₀² − b²/4m²)

Resonanceω_driving = ω₀ → amplitude maximum (forced oscillation)

⚠️

Resonance applications: Tuning radio, swinging on a swing, musical instruments, MRI machines, bridges (Tacoma Narrows — destructive resonance). Always avoid resonance in structural engineering.

Musical Instruments & Harmonics

Instrument	Type	Harmonics	Notes
Flute (open)	Open pipe	All harmonics: f₁, 2f₁, 3f₁...	f₁ = v/(2L)
Clarinet (closed)	Closed pipe	Odd harmonics: f₁, 3f₁, 5f₁...	f₁ = v/(4L)
Guitar string	Fixed-fixed string	All harmonics: f₁, 2f₁, 3f₁...	f₁ = v/(2L), v = √(T/μ)
Violin	Fixed-fixed string	All harmonics + overtones	Helmholtz motion of bow
Tabla	Membrane	Complex modes	Circular membrane Bessel fn

OvertonesFrequencies above fundamental: 2nd harmonic = 1st overtone, 3rd = 2nd overtone, etc.

TimbreQuality of sound depends on relative amplitudes of harmonics

ConsonancePleasant sound: frequency ratios like 1:2 (octave), 2:3 (fifth), 3:4 (fourth)

Sound — Characteristics & Level

IntensityI = P/A = 2π²ρf²A²v (W/m²)

Loudness Levelβ = 10 log₁₀(I/I₀) dB, I₀ = 10⁻¹² W/m² (threshold of hearing)

Threshold of hearing0 dB (I₀ = 10⁻¹² W/m²)

Normal conversation~60 dB

Pain threshold~120 dB (I = 1 W/m²)

PitchDetermined by frequency — higher f = higher pitch

Quality/TimbreDepends on overtones and harmonics present

Infrasonicf < 20 Hz (elephants, earthquakes)

Ultrasonicf > 20 kHz (bats, medical imaging)

Wave Motion — Fundamentals

Type	Particles	Example	Wave Types
Transverse	⊥ to propagation	Light, EM waves, string	Plane, Circular, Elliptical
Longitudinal	‖ to propagation	Sound, spring	Compression, Rarefaction

Wave Speedv = fλ = ω/k (k = 2π/λ = wave number)

Displacement relationy = A sin(kx − ωt + φ) (travelling wave)

Angular wave numberk = 2π/λ

String wave speedv = √(T/μ) — T = tension, μ = mass/length

Sound in gasv = √(γP/ρ) = √(γRT/M)

Sound in solidv = √(Y/ρ) — Y = Young's modulus

Sound in liquidv = √(B/ρ) — B = bulk modulus

Speed in air (0°C)v ≈ 331 m/s

Superposition & Interference

Principley_net = y₁ + y₂ + y₃ + ... (vector sum)

ConstructivePath diff = nλ → A_net = A₁ + A₂ (max intensity)

DestructivePath diff = (n + ½)λ → A_net = |A₁ − A₂| (min intensity)

Phase differenceΔφ = (2π/λ) × path difference

Standing Waves

FormationTwo identical waves travelling in opposite directions

Equationy = 2A sin(kx) cos(ωt) — nodes and antinodes

Nodessin(kx) = 0 → x = nλ/2 (zero amplitude)

Antinodessin(kx) = ±1 → x = (2n+1)λ/4 (max amplitude)

Standing Waves — String (both ends fixed)

Fundamental (1st harmonic)L = λ/2 → f₁ = v/(2L)

2nd harmonicL = λ → f₂ = 2v/(2L) = 2f₁

nth harmonicfₙ = nv/(2L) = nf₁ (n = 1, 2, 3, ...)

Overtonesf₂, f₃, f₄... are 1st, 2nd, 3rd... overtones

Standing Waves — Pipe (organ)

Type	Ends	Fundamental	Harmonics
Open pipe	Both open	λ = 2L, f₁ = v/(2L)	fₙ = nv/(2L), n = 1,2,3...
Closed pipe	One closed	λ = 4L, f₁ = v/(4L)	fₙ = nv/(4L), n = 1,3,5... (odd only)

Beats

PhenomenonPeriodic variation in intensity when two nearby frequencies superpose

Beat Frequencyf_beat = |f₁ − f₂|

Amplitude modulationA(t) = 2A cos(2π(f₁−f₂)t/2)

Doppler Effect

General Formulaf' = f(v ± v₀)/(v ∓ vₛ)

Source towards observerf' = fv/(v − vₛ) → f' > f (higher pitch)

Source away from observerf' = fv/(v + vₛ) → f' < f (lower pitch)

Observer towards sourcef' = f(v + v₀)/v → f' > f

Observer away from sourcef' = f(v − v₀)/v → f' < f

Both moving towardsf' = f(v + v₀)/(v − vₛ)

💡

Doppler effect applications: RADAR speed guns, weather radar, medical ultrasound (blood flow), astronomical redshift/blueshift to determine if stars/galaxies are moving towards or away from us.

🔍

Optics

CLASS 12

Ray Optics — Reflection

Law of Reflection∠i = ∠r (angle of incidence = angle of reflection)

Plane MirrorImage: virtual, erect, same size, distance = object distance

Mirror Formula1/v + 1/u = 1/f

Magnificationm = h'/h = −v/u (negative = inverted)

Concave Mirrorf < 0, converging mirror (parallel rays converge)

Convex Mirrorf > 0, diverging mirror (always virtual, erect, diminished)

Ray Optics — Refraction

Snell's Lawn₁ sin i = n₂ sin r

Refractive Indexn = c/v = speed of light in vacuum / speed in medium

Lateral Shiftd = t sin(i−r)/cos r (for slab of thickness t)

Apparent Depthh' = h/n (real depth / refractive index)

Total Internal Reflection (TIR)

Condition 1Light travels from denser to rarer medium (n₁ > n₂)

Condition 2Angle of incidence > critical angle

Critical Anglesin θ_c = n₂/n₁ (for n₁ > n₂)

Glass-airθ_c = sin⁻¹(1/1.5) ≈ 41.8°

Water-airθ_c = sin⁻¹(1/1.33) ≈ 48.75°

ApplicationsOptical fibers, mirages, diamonds (sparkle), prisms

Lenses

Lens Maker's Eq.1/f = (n−1)[1/R₁ − 1/R₂]

Thin Lens Formula1/v − 1/u = 1/f (sign convention)

Power of LensP = 1/f (in meters) → unit: Dioptre (D)

ConvergingConvex lens (f > 0, converges light)

DivergingConcave lens (f < 0, diverges light)

Magnificationm = v/u = h'/h

Combination of Lenses

Thin lenses in contact1/F = 1/f₁ + 1/f₂ + ... (equivalent focal length)

Combined PowerP = P₁ + P₂ + ... (Dioptres add directly)

Separated by d1/F = 1/f₁ + 1/f₂ − d/(f₁f₂)

Prisms & Dispersion

Prism Formulaμ = sin((A+δ_m)/2) / sin(A/2)

Angle of Deviationδ = i + e − A (i = incident, e = emergent, A = prism angle)

Minimum Deviationδ_m occurs when i = e (ray passes symmetrically)

DispersionSplitting of white light into constituent colors (VIBGYOR)

Angular Dispersionδ_V − δ_R (deviation of violet − red)

Dispersive Powerω = (δ_V − δ_R)/δ_Y

Scattering & Optical Instruments

Rayleigh ScatteringI ∝ 1/λ⁴ — shorter wavelengths scatter more

Blue skyBlue light (shorter λ) scatters more → sky appears blue

Red sunsetLong path → blue scattered away → red/orange reaches eye

Microscope (simple)M = L/fₒ × D/fₑ (L = tube length, D = least distance)

Telescope (refracting)M = fₒ/fₑ (angular magnification)

Reflecting telescopeUses concave mirror — no chromatic aberration

Resolving Power (eye)≈ 1 arc minute (1/60°)

Wave Optics — Huygens' Principle

PrincipleEach point on a wavefront is a source of secondary wavelets

WavefrontLocus of points with same phase (plane, spherical, cylindrical)

ConstructionNew wavefront = forward envelope of secondary wavelets

Laws from HuygensReflection (i = r) and Refraction (Snell's law) can be derived

Interference — Young's Double Slit

SetupCoherent source → double slit → screen (fringes observed)

Path DifferenceΔx = d sin θ ≈ d·y/D (for small θ)

Bright Fringed sin θ = nλ → y_n = nλD/d (n = 0, ±1, ±2...)

Dark Fringed sin θ = (n+½)λ → y_n = (n+½)λD/d

Fringe Widthβ = λD/d (uniform spacing for small θ)

Angular Fringe Widthβ/D = λ/d

IntensityI = I₁ + I₂ + 2√(I₁I₂) cos φ

Equal sources (I₁=I₂)I_max = 4I₀, I_min = 0

Diffraction

Single Slita sin θ = nλ (minima), a sin θ = (2n+1)λ/2 (secondary maxima)

Central MaximumWidth = 2λD/a (twice the secondary maxima width)

Resolving PowerR = λ/(dλ) = nN (grating: N slits, order n)

Rayleigh CriterionMinimum angular separation = 1.22 λ/D (circular aperture)

Polarisation

DefinitionRestriction of wave vibration to a single plane

By reflectionBrewster's angle: tan i_B = n₂/n₁

Malus's LawI = I₀ cos²θ (θ = angle between polarizer and analyzer)

Crossed polarizersθ = 90° → I = 0 (no light passes)

Polarization byReflection, refraction, double refraction (calcite), scattering

Optically ActiveSubstances rotate plane of polarization (sugar solution)

💡

Interference vs Diffraction: Interference requires two coherent sources; diffraction occurs from a single slit/aperture. Both are superposition phenomena. In YDSE, each slit produces diffraction and the two slits together produce interference — the observed pattern is actually a combination.

Optical Instruments — Detailed

Instrument	Type	Magnification	Key Feature
Simple Microscope	Convex lens	M = 1 + D/f (at D) or M = D/f (at ∞)	Single converging lens
Compound Microscope	2 convex lenses	M = (L/fₒ)(D/fₑ)	Objective (short f) + Eyepiece
Refracting Telescope	2 lenses (objective + eyepiece)	M = fₒ/fₑ (normal adjustment)	Large objective, small eyepiece
Reflecting Telescope	Concave mirror + eyepiece	M = fₒ/fₑ	No chromatic aberration
Cassegrain Telescope	Primary + secondary mirror	M = fₒ/fₑ	Compact design (used in observatories)

Near point (D)D = 25 cm (least distance of distinct vision)

Resolving power (microscope)RP = 2μ sin β / λ (β = half-angle of cone of light)

Resolving power (telescope)RP = D/(1.22λ) — D = aperture diameter

Chromatic aberrationLens focuses different colors at different points (refractive index varies with λ)

Achromatic doubletCombines convex (crown glass) + concave (flint glass) to eliminate chromatic aberration

Spherical aberrationRays far from axis focus closer — use paraxial rays or stops

Dispersion — Prism Details

Dispersive Powerω = (μ_V − μ_R)/(μ_Y − 1) — μ_V, μ_R, μ_Y for violet, red, yellow

Angular Dispersionδ = δ_V − δ_R = A(μ_V − μ_R) (small angle approximation)

Rainbow FormationDispersion + total internal reflection in water droplets

Primary Rainbow42° from anti-solar point, red outside, violet inside

Secondary Rainbow51° from anti-solar point, violet outside, red inside (2 TIRs, dimmer)

Reflection & Refraction — Sign Conventions (New Cartesian)

Quantity	Sign Convention
Object distance (u)	Always negative (left of mirror/lens)
Image distance (v)	Positive if right of lens/mirror, negative if left
Focal length (f)	Positive for convex lens/concave mirror, negative for concave lens/convex mirror
Height (h, h')	Positive above principal axis, negative below
Magnification (m)	Positive = erect, Negative = inverted
R (radius)	Same sign as f (R = 2f)

🧲

Electromagnetism

CLASS 12

Electric Charges & Fields

Quantisationq = ne (e = 1.6 × 10⁻¹⁹ C, n = integer)

AdditivityCharges add algebraically (scalar)

ConservationTotal charge in isolated system is constant

Coulomb's LawF = (1/4πε₀)(q₁q₂/r²) = kq₁q₂/r²

k = 1/4πε₀k = 9 × 10⁹ N·m²/C²

ε₀ (permittivity)8.854 × 10⁻¹² C²/N·m²

Electric Field (E)

DefinitionE = F/q₀ (force per unit positive test charge)

Point ChargeE = kQ/r² (radial, away from +Q)

Dipole Field (axial)E = 2kp/r³ (along axis)

Dipole Field (equatorial)E = kp/r³ (opposite to dipole moment direction)

Dipole Momentp = q × 2a (from −q to +q, vector)

Torque on dipoleτ = p × E = pE sin θ

PE of dipoleU = −p·E = −pE cos θ

Electric FluxΦ_E = E·A = EA cos θ (N·m²/C)

Gauss's Law

Statement∮E·dA = Q_enclosed/ε₀

Infinite line chargeE = λ/(2πε₀r) (λ = linear charge density)

Infinite planeE = σ/(2ε₀) (σ = surface charge density, uniform)

Parallel platesE = σ/ε₀ (between plates, outside E = 0)

Solid sphere (inside)E = Qr/(4πε₀R³) = ρr/(3ε₀) (r < R)

Solid sphere (outside)E = Q/(4πε₀r²) (r > R, behaves as point charge)

⚠️

Gauss's law tip: Choose a Gaussian surface where E is either constant (parallel to surface normal) or zero (perpendicular). For spherical symmetry use a sphere, cylindrical use a cylinder, planar use a cylinder or pillbox.

Electrostatic Potential & Capacitance

Potential (V)V = kQ/r (scalar, J/C = Volt)

V from EE = −dV/dr (potential gradient)

V at infinityV = 0 (reference point)

Equipotential SurfaceSurface where V = const (E ⊥ to surface)

Work to move chargeW = q(V_B − V_A) = −q∫E·dr

PE of systemU = kq₁q₂/r (for two charges)

Capacitors

CapacitanceC = Q/V (Farad, F)

Parallel PlateC = ε₀A/d

With dielectricC' = εᵣC = ε₀εᵣA/d (εᵣ = dielectric constant)

SphericalC = 4πε₀R (isolated sphere)

Series1/C_eq = 1/C₁ + 1/C₂ + ...

ParallelC_eq = C₁ + C₂ + ...

Energy storedU = ½CV² = ½QV = Q²/(2C)

Energy densityu = ½ε₀E² = ½(σ²/ε₀) (J/m³)

Current Electricity

CurrentI = dQ/dt (Ampere)

Ohm's LawV = IR

ResistanceR = ρl/A (ρ = resistivity, Ω·m)

Temperature dependenceR_T = R₀[1 + α(T − T₀)] (α = temp coefficient)

PowerP = VI = I²R = V²/R

EnergyW = VIt = I²Rt (Joule heating / heating effect)

Drift Velocityv_d = eEτ/m = I/(neA) — n = carrier density

Mobilityμ = v_d/E = eτ/m

Current Electricity — Key Relationships

Cells in seriesSame current, EMFs add (if same direction): E_eq = E₁ + E₂ + ..., r_eq = r₁ + r₂ + ...

Cells in parallelSame EMF across each (must be identical): E_eq = E, 1/r_eq = 1/r₁ + 1/r₂ + ...

Series-parallelCombine step by step: first parallel groups, then series combinations

Grouping for max currentm rows × n columns of cells: I = mE/(R + mr/n)

Max power transferPower max when R_external = r_internal → P_max = E²/(4r)

Kirchhoff tipAssign current directions arbitrarily; if answer is negative, actual direction is opposite

Kirchhoff's Laws

Junction Rule (KCL)ΣI_in = ΣI_out (conservation of charge)

Loop Rule (KVL)ΣV = 0 around any closed loop (conservation of energy)

Circuit Analysis — Bridges & Meters

Wheatstone BridgeP/Q = R/S → galvanometer shows zero deflection

Metre BridgeS = R(l₂/l₁) — balanced at l₁ from left end

PotentiometerV = kl (k = potential gradient), E₁/E₂ = l₁/l₂

Internal Resistancer = R(l₁ − l₂)/l₂ ( potentiometer method)

Moving Charges & Magnetism

Biot-Savart LawdB = (μ₀/4π)(Idl × r̂)/r²

μ₀ (permeability)4π × 10⁻⁷ T·m/A

Straight wireB = μ₀I/(2πr) (circular field around wire)

Circular loop (center)B = μ₀I/(2R)

Circular loop (axis)B = μ₀IR²/(2(R²+x²)^(3/2))

Solenoid (inside)B = μ₀nI (n = N/L, turns per unit length)

ToroidB = μ₀NI/(2πr)

Force & Torque — Magnetic

Lorentz ForceF = q(E + v × B)

Force on chargeF = qvB sin θ (⊥ to both v and B)

Force on wireF = IlB sin θ = I(l × B)

Circular motionr = mv/(qB), T = 2πm/(qB), f = qB/(2πm)

Torque on loopτ = NIAB sin θ = m × B (m = NIA = magnetic moment)

Moving coil galvanometerI = (k/NBA)θ (θ = deflection)

Ampere's Circuital Law

Statement∮B·dl = μ₀I_enclosed

Long solenoidB = μ₀nI (derived from Ampere)

Hollow/inside wireB = 0 (no enclosed current)

Inside solid wireB = μ₀Ir/(2πR²) (r < R)

Outside wireB = μ₀I/(2πr) (r > R)

Galvanometer → Ammeter/Voltmeter

Conversion	Connection	Formula	Resistance
Ammeter	Series	S = I_g·R_g/(I − I_g)	R_A = R_g + S (very low)
Voltmeter	Parallel	R = V/I_g − R_g	R_V = R_g + R (very high)

Ideal ammeterR_A → 0 (zero resistance, no voltage drop)

Ideal voltmeterR_V → ∞ (infinite resistance, no current drawn)

Magnetism

Bar magnet as dipolem = NIA — field identical to current loop

Magnetic field linesContinuous loops from N to S outside, S to N inside

Earth's B fieldB ≈ 3.6 × 10⁻⁵ T (varies with location)

Elements of Earth's fieldB_H = B cos δ, B_V = B sin δ, tan δ = B_V/B_H

Magnetic MeridianVertical plane containing B and geographic N-S

Electromagnetic Induction (EMI)

Faraday's 1st LawChange in magnetic flux induces EMF

Faraday's 2nd Lawε = −dΦ/dt (magnitude = rate of change of flux)

Lenz's LawInduced EMF opposes the cause of induction (conservation)

Motional EMFε = Blv (conductor moving ⊥ to B)

Rotating coilε = NBAω sin(ωt) (peak ε₀ = NBAω)

Self Inductanceε = −L(dI/dt), L = NBA/I (Henry)

Mutual Inductanceε₂ = −M(dI₁/dt), M = Φ₂/I₁

Energy storedU = ½LI² = ½ΦI = Φ²/(2L)

Solenoid LL = μ₀n²Al (A = area, l = length, n = turns/length)

AC Circuits

AC VoltageV = V₀ sin(ωt) = V₀ sin(2πft)

RMS ValuesV_rms = V₀/√2, I_rms = I₀/√2

Pure RI = V/R, V and I in phase

Pure LI = V/(ωL), I lags V by 90°

Pure CI = VωC, I leads V by 90°

Inductive ReactanceX_L = ωL = 2πfL (Ω)

Capacitive ReactanceX_C = 1/(ωC) = 1/(2πfC) (Ω)

LCR Series Circuit

ImpedanceZ = √(R² + (X_L − X_C)²)

CurrentI = V/Z

Phase angletan φ = (X_L − X_C)/R

Resonance (X_L = X_C)f₀ = 1/(2π√(LC)), Z_min = R, I_max = V/R

Power Factorcos φ = R/Z (at resonance cos φ = 1)

Average PowerP = V_rms × I_rms × cos φ = I²_rms × R

Wattless CurrentComponent due to L/C, contributes zero power

LC Oscillations & Transformer

LC Oscillationq = q₀ cos(ωt), ω = 1/√(LC)

Frequencyf = 1/(2π√(LC))

Transformer equationV_s/V_p = N_s/N_p = I_p/I_s

Efficiencyη = P_output/P_input (ideal: 100%, real: 90-99%)

Energy loss sourcesCopper loss, flux leakage, hysteresis, eddy currents

Electromagnetic Waves

Maxwell's PredictionChanging E produces B and vice versa → self-sustaining EM wave

Speedc = 1/√(μ₀ε₀) = 3 × 10⁸ m/s

In mediumv = c/n = 1/√(με) (n = refractive index)

E/B ratioE₀/B₀ = c

TransverseE and B ⊥ to each other and ⊥ to direction of propagation

Energy densityu = ½ε₀E² + B²/(2μ₀) = ε₀E² (in vacuum)

Poynting VectorS = E × B/μ₀ (W/m² — energy flux)

EM Spectrum

Type	Wavelength Range	Source	Use
Gamma Rays	< 10⁻¹² m	Nuclear decay	Cancer treatment, sterilization
X-Rays	10⁻¹² − 10⁻⁹ m	Inner electron transition	Medical imaging, security
Ultraviolet	10⁻⁹ − 4×10⁻⁷ m	Hot bodies, sun	Sterilization, counterfeit detection
Visible Light	4×10⁻⁷ − 7×10⁻⁷ m	Sun, bulbs	Human vision, optical fiber
Infrared	7×10⁻⁷ − 10⁻³ m	Warm bodies	Remote control, thermal imaging
Microwaves	10⁻³ − 10⁻¹ m	Magnetron, Klystron	Cooking, radar, WiFi
Radio Waves	> 10⁻¹ m	LC oscillation	Radio, TV, mobile communication

💡

Maxwell's equations (integral form): (1) ∮E·dA = Q/ε₀ (Gauss for E), (2) ∮B·dA = 0 (Gauss for B — no magnetic monopoles), (3) ∮E·dl = −dΦ_B/dt (Faraday), (4) ∮B·dl = μ₀I + μ₀ε₀dΦ_E/dt (Ampere-Maxwell).

Maxwell's Equations — Complete Summary

Equation	Name	Integral Form	Meaning
(1)	Gauss (Electric)	∮E·dA = Q_enc/ε₀	Electric charges produce E field
(2)	Gauss (Magnetic)	∮B·dA = 0	No magnetic monopoles exist
(3)	Faraday	∮E·dl = −dΦ_B/dt	Changing B produces E
(4)	Ampere-Maxwell	∮B·dl = μ₀I + μ₀ε₀(dΦ_E/dt)	Current + changing E produces B

ConsequenceThese 4 equations + Lorentz force (F = qE + qv×B) describe ALL of classical electromagnetism

Displacement currentI_d = ε₀(dΦ_E/dt) — Maxwell's addition to complete Ampere's law

EM wave existenceFrom these equations: E and B fields can sustain each other as waves

Magnetic Materials

Type	Susceptibility (χ_m)	Permeability (μ_r)	Behavior
Diamagnetic	Small negative (~−10⁻⁵)	μ_r < 1 (slightly)	Weakly repelled by magnet
Paramagnetic	Small positive (~10⁻³)	μ_r > 1 (slightly)	Weakly attracted
Ferromagnetic	Large positive	μ_r >> 1	Strongly attracted, retains magnetism

Curie TemperatureAbove T_c, ferromagnet loses permanent magnetism → becomes paramagnetic

HysteresisB lags behind H — area of B-H loop = energy lost per cycle as heat

RetentivityB remaining when H = 0 (residual magnetism)

Coercivity−H needed to demagnetize (H to make B = 0)

Soft ironLow retentivity, low coercivity → used for electromagnets, transformer cores

Hard steelHigh retentivity, high coercivity → used for permanent magnets

Electromagnetic Induction — Key Numericals

Motional EMF (general)ε = ∫(v × B)·dl (for arbitrary conductor shape)

Rod rotating about endε = ½BωL²

Rectangular loop entering fieldε = BLv (constant while entering/exiting B region)

Energy methodPower dissipated = ε²/R = B²L²v²/R = work done per second by external agent

Mutual Inductance (coils)M = μ₀N₁N₂A/l (coils on same core, l = length)

LC circuit energyE_total = ½LI² + ½q²/C = const (oscillates between L and C)

⚛️

Modern Physics

CLASS 12

Dual Nature of Radiation & Matter

Photon EnergyE = hν = hc/λ (h = 6.626 × 10⁻³⁴ J·s)

Photoelectric EffectEinstein: KE_max = hν − φ

Work Function (φ)Minimum energy to eject electron from surface

Threshold Frequencyν₀ = φ/h (below this, no photoelectrons)

Stopping PotentialeV₀ = KE_max = hν − φ

de Broglie Wavelengthλ = h/p = h/(mv) (matter waves)

de Broglie (accelerated e⁻)λ = h/√(2meV) = 1.226/√V nm (V in volts)

Photoelectric Effect — Observations

Observation	Classical Prediction	Quantum (Einstein)
Below ν₀, no emission	Should work at any ν	Correct: needs hν ≥ φ
KE depends on ν, not intensity	KE ∝ intensity	Correct: KE = hν − φ
Instantaneous emission	Time delay expected	Correct: single photon interaction
More intensity = more electrons	Same	Correct: more photons = more e⁻

💡

Wave-particle duality: Light behaves as wave (interference, diffraction) AND particle (photoelectric, Compton effect). Similarly, matter (electrons) shows wave properties (electron diffraction by Davisson-Germer experiment).

Davisson-Germer Experiment

SetupElectron beam directed at nickel crystal, detector measures scattered intensity

ResultPeaks in scattered electrons at specific angles → confirmed de Broglie hypothesis

Formula usednλ = d sin φ (Bragg condition for electron waves)

Calculated λMatched de Broglie λ = h/√(2meV) with experimental value

SignificanceFirst direct evidence of wave nature of matter

Bohr Model — Limitations

Limitation	Details
Multi-electron atoms	Only works for hydrogen-like (single electron) systems
Fine structure	Cannot explain splitting of spectral lines in magnetic field (Zeeman effect)
Intensities	Cannot predict relative intensities of spectral lines
Quantum mechanics	Replaced by quantum mechanical model (Schrödinger equation, orbitals)

Nuclear Physics — Additional Concepts

Nuclear ForceStrong, short-range (~1-3 fm), charge independent, attractive, saturates

Magic Numbers2, 8, 20, 28, 50, 82, 126 — nuclei with these N or Z are more stable

Nuclear Fission triggerU-235 + slow neutron → unstable compound → fission + 2-3 neutrons + energy

Controlled FissionNuclear reactor: fuel (U-235), moderator (heavy water/graphite), control rods (Cd/B), coolant

Breeder ReactorProduces more fuel than it consumes (U-238 → Pu-239)

Nuclear WasteHigh-level waste: long-lived isotopes; stored in deep geological repositories

Radioactive datingt = (1/λ)ln(N₀/N) — Carbon-14 dating: t_(1/2) = 5730 years

Atoms — Bohr Model

Postulate 1Electron moves in circular orbits (centripetal = Coulomb force)

Postulate 2Angular momentum quantized: L = mvr = nh/(2π)

Postulate 3Radiation emitted when electron jumps: hν = E_i − E_f

Bohr Model — Key Results (Hydrogen)

Radius of n-th orbitr_n = n²a₀ (a₀ = 0.529 Å = Bohr radius)

Velocityv_n = cα/n (α = 1/137 = fine structure constant)

EnergyEₙ = −13.6/n² eV (n = 1, 2, 3...)

Ground state (n=1)E₁ = −13.6 eV, r₁ = 0.529 Å

Ionization Energy13.6 eV (energy to remove e⁻ from n=1)

Hydrogen Spectral Series

Series	Transition	Region	Wavelength Formula
Lyman	n → 1	Ultraviolet	1/λ = R(1 − 1/n²), n = 2,3...
Balmer	n → 2	Visible	1/λ = R(1/4 − 1/n²), n = 3,4...
Paschen	n → 3	Infrared	1/λ = R(1/9 − 1/n²), n = 4,5...
Brackett	n → 4	Infrared	1/λ = R(1/16 − 1/n²), n = 5,6...
Pfund	n → 5	Infrared	1/λ = R(1/25 − 1/n²), n = 6,7...

Rydberg ConstantR = 1.097 × 10⁷ m⁻¹

Nuclei — Composition & Properties

IsotopesSame Z (protons), different A (mass number). Ex: ¹H, ²H, ³H

IsobarsSame A, different Z. Ex: ¹⁴₆C, ¹⁴₇N

IsotonesSame N (neutrons), different Z. Ex: ¹⁴₆C, ¹⁵₇N

Size of NucleusR = R₀A^(1/3), R₀ = 1.2 fm (1 fm = 10⁻¹⁵ m)

Nuclear Densityρ = mass/volume ≈ 2.3 × 10¹⁷ kg/m³ (constant!)

Mass Defect & Binding Energy

Mass DefectΔm = [Zm_p + Nm_n] − M_nucleus (always positive)

Binding EnergyBE = Δmc² = [Zm_p + Nm_n − M]c²

BE per nucleonBE/A = Δmc²/A (stability indicator)

Peak stabilityFe-56 has highest BE/A ≈ 8.8 MeV/nucleon

1 amu energy1 amu × c² = 931.5 MeV

Nuclear Fission & Fusion

Process	Reaction	Energy Release	Conditions
Fission	Heavy nucleus splits	~200 MeV per fission	Slow neutrons (U-235)
Fusion	Light nuclei combine	~26.7 MeV per D-T	Very high T (~10⁷ K)

Chain ReactionEach fission produces neutrons that cause more fission

Critical MassMinimum mass needed to sustain chain reaction

Nuclear ReactorUses controlled fission; moderator (slow neutrons), control rods (absorb neutrons)

D-T Reaction²H + ³H → ⁴He + n + 17.6 MeV

Solar energy sourceProton-proton chain: 4p → ⁴He + 2e⁺ + 2ν + 26.7 MeV

Radioactivity

Decay LawN = N₀e^(−λt) (exponential decay)

Half-lifet_(1/2) = ln(2)/λ = 0.693/λ

Mean lifeτ = 1/λ = t_(1/2)/ln(2) = 1.44 × t_(1/2)

ActivityA = −dN/dt = λN (Becquerel, Bq = 1 decay/s)

α Decay²³⁸₉₂U → ²³⁴₉₀Th + ⁴₂He (Z→Z−2, A→A−4)

β⁻ Decayn → p + e⁻ + ν̄ (Z→Z+1, A→A)

β⁺ Decayp → n + e⁺ + ν (Z→Z−1, A→A)

γ DecayExcited nucleus → ground state + γ photon (no change in Z or A)

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Radioactivity: Radioactivity is a nuclear phenomenon (unaffected by temperature, pressure, chemical bonding). α particles: He nucleus (high ionizing, low penetration). β particles: fast electrons (moderate). γ rays: EM waves (low ionizing, high penetration).

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Electronics

CLASS 12

Semiconductors — Basics

Type	Conductivity	Example	Charge Carriers
Conductor	High (10⁴⁻⁷ S/m)	Cu, Al	Free electrons
Insulator	Very low (~10⁻¹⁰ S/m)	Glass, Rubber	Negligible
Intrinsic SC	Moderate	Pure Si, Ge	e⁻ and holes (equal)
Extrinsic SC	Controlled	Doped Si, Ge	Majority carriers

Energy Gap (Si)E_g = 1.1 eV

Energy Gap (Ge)E_g = 0.7 eV

Temperature EffectConductivity increases with temperature (opposite of metals)

Extrinsic Semiconductors

Type	Dopant	Majority	Minority	Energy Level
n-type	Pentavalent (P, As, Sb)	Electrons	Holes	Donor level (just below conduction band)
p-type	Trivalent (B, Al, Ga)	Holes	Electrons	Acceptor level (just above valence band)

n = n_e = n_h × e^(−E_g/2kT)Intrinsic carrier concentration relation

n_e × n_h = n_i²Law of mass action (always holds)

p-n Junction Diode

Formationp-type and n-type semiconductors joined → depletion region

Depletion regionRegion devoid of free carriers, has built-in potential (~0.7V for Si)

Forward Biasp connected to +ve → depletion width ↓ → current flows easily

Reverse Biasp connected to −ve → depletion width ↑ → very small current (leakage)

Forward V-II = I₀(e^(eV/kT) − 1) (Shockley equation)

Knee VoltageSi: ~0.7V, Ge: ~0.3V (forward bias threshold)

Diode Applications

Device	Function	Details
Half-wave Rectifier	AC → pulsating DC	One diode, efficiency ≈ 40.6%
Full-wave Rectifier	AC → smoother DC	Two (or 4 bridge) diodes, efficiency ≈ 81.2%
LED	Light emission	Forward biased diode, direct bandgap semiconductors
Photodiode	Light detection	Reverse biased, current ∝ light intensity
Solar Cell	Light → electricity	p-n junction, no external bias needed
Zener Diode	Voltage regulation	Operates in reverse breakdown, V_Z constant

Zener Diode — Voltage Regulator

BreakdownReverse breakdown voltage V_Z is nearly constant

RegulationV_L = V_Z (constant), regardless of load changes

Series ResistanceR_S = (V_in − V_Z)/(I_Z + I_L) (design formula)

Zener CurrentI_Z = (V_in − V_Z)/R_S − I_L

Bipolar Junction Transistor (BJT)

	n-p-n	p-n-p
Emitter	n-type (heavily doped)	p-type (heavily doped)
Base	p-type (thin, lightly doped)	n-type (thin, lightly doped)
Collector	n-type (moderately doped)	p-type (moderately doped)
Current	Electrons flow E → C	Holes flow E → C
Biasing	EB: forward, CB: reverse	EB: forward, CB: reverse

Transistor — Current Relations

Current relationI_E = I_B + I_C

Current gain (β)β = I_C/I_B (typically 20-300)

Current gain (α)α = I_C/I_E (typically 0.95-0.99)

Relationβ = α/(1−α), α = β/(1+β)

CE Amplifier gainA_V = −β(R_C/R_B) = −g_m × R_C

Transistor as Amplifier (CE)

Input SignalApplied to base-emitter (forward biased)

OutputTaken from collector-emitter through R_C

Voltage GainA_V = ΔV_C/ΔV_B = −β × R_C/R_in

Power GainA_P = β² × R_C/R_in = A_V × A_I

Phase reversal180° phase shift between input and output

Logic Gates

Gate	Symbol	Boolean	Output
AND	D-shape	Y = A·B	HIGH only if both HIGH
OR	Curved D	Y = A + B	HIGH if any input HIGH
NOT	Triangle + circle	Y = Ā	Inverts input
NAND	AND + circle	Y = ¯(A·B)	NOT of AND
NOR	OR + circle	Y = ¯(A+B)	NOT of OR
XOR	OR with extra curve	Y = A⊕B = AḂ + ĀB	HIGH if inputs different
XNOR	XOR + circle	Y = A⊙B = A·B + Ā·Ḃ	HIGH if inputs same

Logic Gates — Truth Tables

A	B	AND	OR	NAND	NOR	XOR
0	0	0	0	1	1	0
0	1	0	1	1	0	1
1	0	0	1	1	0	1
1	1	1	1	0	0	0

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Universal gates: NAND and NOR are called universal gates because any other logic gate (AND, OR, NOT) can be constructed using only NAND gates or only NOR gates. This is fundamental in digital circuit design.

Universal Gate Constructions

Gate to Build	Using NAND Only	Using NOR Only
NOT	NAND(A, A)	NOR(A, A)
AND	NAND(NAND(A,B), NAND(A,B))	NOR(NOR(A,A), NOR(B,B))
OR	NAND(NAND(A,A), NAND(B,B))	NOR(NOR(A,B), NOR(A,B))
XOR	Complex (5 NAND gates)	Complex (5 NOR gates)

Transistor Configurations

Configuration	Input	Output	Current Gain	Voltage Gain	Power Gain
Common Base (CB)	E-B	C-B	α = I_C/I_E (< 1)	High	Medium
Common Emitter (CE)	B-E	C-E	β = I_C/I_B (high)	High	Highest
Common Collector (CC)	B-C	E-C	1 + β	Unity (< 1)	Medium

CE: Phase reversal180° phase shift between input and output — most widely used

CC: Emitter FollowerOutput follows input (no inversion), high input impedance, low output impedance

CB: Current gain < 1Used for high frequency applications, no phase reversal

Feedback in Amplifiers

Positive FeedbackOutput fed back in-phase → increases gain, can cause oscillation

Negative FeedbackOutput fed back out-of-phase → decreases gain but improves stability, bandwidth, reduces distortion

Gain with feedbackA_f = A/(1 + βA) where β = feedback fraction

Benefits of -ve feedbackStabilized gain, increased bandwidth, reduced noise, improved linearity

Number Systems in Digital Electronics

System	Base	Digits	Example (decimal 10)
Binary	2	0, 1	1010
Octal	8	0-7	12
Decimal	10	0-9	10
Hexadecimal	16	0-9, A-F	A

Binary → Decimal1010 = 1×2³ + 0×2² + 1×2¹ + 0×2⁰ = 8 + 0 + 2 + 0 = 10

1 byte= 8 bits, range: 0 to 255 (unsigned) or −128 to 127 (signed)

Communication Systems

TransmitterConverts message signal into modulated form for transmission

ChannelMedium through which signal travels (wire, optical fiber, free space)

ReceiverReceives and demodulates the signal to recover original message

Bandwidth of signalRange of frequencies contained in the signal

Modulation

Type	What varies	Bandwidth	Quality
AM (Amplitude)	Amplitude of carrier	~10 kHz	Low quality, affected by noise
FM (Frequency)	Frequency of carrier	~200 kHz	High quality, noise immune

AM — Modulation Indexm_a = A_m/A_c (0 ≤ m_a ≤ 1 for undistorted)

AM — Sidebandsf_c ± f_m (two sideband frequencies)

Need for modulation1) Antenna size ∝ λ, 2) Avoid mixing, 3) Power radiated ∝ 1/λ²

EM Wave Communication Bands

Band	Frequency	Propagation	Use
Ground Wave	Low to Medium (≤ 2 MHz)	Follows Earth surface	AM radio
Sky Wave	High (2-30 MHz)	Reflected by ionosphere	Short wave radio
Space Wave	VHF/UHF (> 30 MHz)	Line of sight + reflection	TV, FM, satellite

Max range (space wave)d = √(2Rh) — R = Earth radius, h = antenna height

Satellite CommunicationUses geostationary satellite (h ≈ 36,000 km, T = 24h)

Bandwidth for TV~6 MHz per channel

Bandwidth for voice~3.1 kHz (telephone)

Propagation of EM Waves

Type	Frequency Range	How it Propagates	Applications
Ground Wave	Up to 2 MHz	Follows Earth curvature, attenuated by ground absorption	AM radio broadcasting, naval communication
Sky Wave	2-30 MHz	Reflected by ionosphere (E, F layers), can skip across globe	SW radio, amateur radio, long-distance comm
Space Wave	30 MHz+	Line of sight; can be reflected by troposphere	TV, FM, satellite, mobile, radar

Ionosphere layersD (60-90 km, absorbs MF), E (90-120 km, reflects HF), F (150-400 km, reflects HF/FX)

Skip distanceMinimum distance from transmitter where sky wave returns to Earth

FadingVariation in signal strength due to interference of multiple sky wave paths

Satellite & Fiber Optic Communication

Geostationary satelliteOrbits west to east in equatorial plane, T = 24h, appears fixed from Earth

Coverage angle≈ 120° from geostationary satellite → 3 satellites cover entire Earth

Uplink/DownlinkUplink: higher freq (to overcome noise), Downlink: lower freq

Optical FiberTotal internal reflection guides light; low loss (0.2 dB/km), high bandwidth

Step index fiberCore (high n) + cladding (low n) → light reflects at boundary

Graded index fiberRefractive index gradually decreases → light curves smoothly

Acceptance anglesin θ_a = √(n₁² − n₂²) — max angle for light entering fiber

Numerical ApertureNA = sin θ_a = √(n₁² − n₂²) — light gathering ability

ModesSingle-mode (laser, long distance), Multi-mode (LED, short distance)

Internet & Mobile Communication

InternetNetwork of networks using TCP/IP protocol stack

Mobile (cellular)Area divided into cells, each with base station; handoff between cells

1GAnalog voice (AMPS), ~1980s

2GDigital voice (GSM, CDMA), SMS, ~1990s

3GMobile internet, video calling, ~2001

4G LTEHigh-speed internet, streaming, ~2010

5GUltra-fast, low latency, IoT support, ~2020

Frequency reuseSame frequency used in non-adjacent cells to increase capacity

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Formula Sheet

QUICK REFERENCE

Mechanics Formulas

v = u + at

s = ut + ½at²

v² = u² + 2as

Projectile RangeR = u² sin 2θ / g

Max Height (projectile)H = u² sin²θ / (2g)

Time of FlightT = 2u sin θ / g

Centripetal ForceF = mv²/r = mω²r

F = maNewton's Second Law

f_k = μ_k NKinetic Friction

W = F·d cos θWork done

KE = ½mv²Kinetic Energy

PE (spring)U = ½kx²

PE (gravity)U = mgh

P = F·vPower

p = mvMomentum

J = FΔt = ΔpImpulse

τ = r × FTorque

L = IωAngular Momentum

τ = IαRotational analog of F = ma

I_cm + Md²Parallel Axis Theorem

F = GMm/r²Newton's Gravitation

g = GM/R²Acceleration due to gravity

vₑ = √(2gR)Escape Velocity

vₒ = √(gR)Orbital Velocity

T² ∝ R³Kepler's Third Law

Y = stress/strainYoung's Modulus

P + ½ρv² + ρghBernoulli's Equation

v = √(2gh)Torricelli's theorem

v_t = 2r²(ρ−σ)g/(9η)Terminal Velocity

T = 2T cos θ/(ρgr)Capillary Rise

Thermodynamics Formulas

ΔL = L₀ α ΔTLinear thermal expansion

ΔV = V₀ γ ΔTVolume thermal expansion

Q = mcΔTHeat (specific heat)

Q = mLHeat (latent heat)

PV = nRTIdeal Gas Law

v_rms = √(3RT/M)RMS Speed

KE per molecule½mv² = (3/2)kT

Cp − Cv = RMayer's Relation

γ = Cp/CvAdiabatic index

PV^γ = constAdiabatic Process

ΔU = Q − WFirst Law of Thermodynamics

W = nRT ln(V₂/V₁)Isothermal Work

η = 1 − T₂/T₁Carnot Efficiency

P = eσAT⁴Stefan-Boltzmann Law

λ_max T = bWien's Displacement Law

Oscillations & Waves Formulas

F = −kxSHM restoring force

x = A sin(ωt + φ)SHM displacement

T = 2π√(m/k)SHM period (spring)

T = 2π√(L/g)Simple Pendulum

E = ½kA²Total SHM Energy

v = fλ = ω/kWave Speed

v_string = √(T/μ)Wave on string

v_sound = √(γP/ρ)Sound in gas

y = 2A sin(kx) cos(ωt)Standing Wave

f_n = nv/(2L)String harmonics

f_n = nv/(2L)Open pipe harmonics

f_n = nv/(4L) (n=1,3,5...)Closed pipe harmonics

f_beat = |f₁ − f₂|Beat Frequency

f' = f(v ± v₀)/(v ∓ vₛ)Doppler Effect

I ∝ 1/λ⁴Rayleigh Scattering

Optics Formulas

1/v + 1/u = 1/fMirror Formula

n₁ sin i = n₂ sin rSnell's Law

sin θ_c = n₂/n₁Critical Angle (TIR)

1/f = (n−1)(1/R₁ − 1/R₂)Lens Maker's Equation

P = 1/fLens Power (Dioptre)

μ = sin((A+δ_m)/2)/sin(A/2)Prism Formula

y_n = nλD/dYDSE Bright Fringe

β = λD/dFringe Width

a sin θ = nλSingle Slit Diffraction (minima)

tan i_B = n₂/n₁Brewster's Angle

I = I₀ cos²θMalus's Law

M = fₒ/fₑTelescope Magnification

Electromagnetism Formulas

F = kq₁q₂/r²Coulomb's Law

E = kQ/r²Electric Field (point charge)

V = kQ/rElectric Potential

∮E·dA = Q/ε₀Gauss's Law

C = ε₀A/dParallel Plate Capacitor

U = ½CV²Capacitor Energy

V = IROhm's Law

R = ρl/AResistance

P = VI = I²R = V²/RElectrical Power

dB = (μ₀/4π)(Idl × r̂)/r²Biot-Savart Law

B = μ₀I/(2πr)B (long wire)

B = μ₀nIB (solenoid)

F = qv × BLorentz Force

F = BIl sin θForce on wire

τ = m × B = NIAB sin θTorque on loop

∮B·dl = μ₀I_encAmpere's Law

ε = −dΦ/dtFaraday's Law

ε = −L dI/dtSelf Inductance

U = ½LI²Inductor Energy

V = V₀ sin(ωt)AC Voltage

Z = √(R² + (X_L−X_C)²)Impedance

f₀ = 1/(2π√(LC))Resonant Frequency

c = 1/√(μ₀ε₀)Speed of Light

V_s/V_p = N_s/N_pTransformer

Modern Physics Formulas

E = hν = hc/λPhoton Energy

KE = hν − φPhotoelectric Effect

λ = h/(mv)de Broglie Wavelength

λ = 1.226/√V nmde Broglie (electron, V in volts)

Eₙ = −13.6/n² eVBohr Energy Levels

rₙ = 0.529n² ÅBohr Orbit Radii

1/λ = R(1/n_f² − 1/n_i²)Hydrogen Spectrum

R = R₀A^(1/3)Nuclear Radius

BE = Δmc²Binding Energy

1 amu = 931.5 MeVMass-Energy equivalence

N = N₀e^(−λt)Radioactive Decay Law

t_(1/2) = 0.693/λHalf-life

τ = 1/λMean Life

A = λNActivity

Electronics Formulas

n_e × n_h = n_i²Mass Action Law

I_E = I_B + I_CTransistor Current

β = I_C/I_BCurrent Gain (CE)

α = I_C/I_ECurrent Gain (CB)

β = α/(1−α)α-β Relation

m_a = A_m/A_cAM Modulation Index

d = √(2Rh)Max range (space wave)

Important Constants

Constant	Symbol	Value
Speed of light	c	3 × 10⁸ m/s
Gravitational constant	G	6.674 × 10⁻¹¹ N·m²/kg²
Planck constant	h	6.626 × 10⁻³⁴ J·s
Electron charge	e	1.602 × 10⁻¹⁹ C
Electron mass	mₑ	9.109 × 10⁻³¹ kg
Proton mass	mₚ	1.673 × 10⁻²⁷ kg
Neutron mass	mₙ	1.675 × 10⁻²⁷ kg
Avogadro number	N_A	6.022 × 10²³ /mol
Boltzmann constant	k_B	1.381 × 10⁻²³ J/K
Universal gas constant	R	8.314 J/(mol·K)
Permittivity of vacuum	ε₀	8.854 × 10⁻¹² C²/(N·m²)
Permeability of vacuum	μ₀	4π × 10⁻⁷ T·m/A
Stefan-Boltzmann constant	σ	5.670 × 10⁻⁸ W/(m²·K⁴)
Rydberg constant	R_∞	1.097 × 10⁷ m⁻¹
Bohr radius	a₀	0.529 × 10⁻¹⁰ m
Atomic mass unit	u	1.661 × 10⁻²⁷ kg = 931.5 MeV/c²
Acceleration due to gravity	g	9.8 m/s²
Rydberg energy	hcR_∞	13.6 eV
Fine structure constant	α	1/137

⚠️

Memorization strategy: Focus on understanding derivations rather than rote memorization. Most formulas can be derived from a few fundamental ones: F = ma, F = GMm/r², F = kq₁q₂/r², E = hν, and the conservation laws (energy, momentum, angular momentum, charge).

Key Derivations Summary

Derivation	Starting Point	Result	Chapter
v² = u² + 2as	a = dv/dt = v(dv/dx)	Eliminates time from kinematics	Motion
T = 2π√(L/g)	Restoring torque τ = −mgL sin θ ≈ −mgLθ	SHM analogy	Oscillations
Escape velocity	KE + PE = 0 at ∞: ½mv² − GMm/r = 0	vₑ = √(2gR)	Gravitation
Kepler's 3rd	Gravity provides centripetal: GMm/r² = mv²/r	T² ∝ R³	Gravitation
v_rms = √(3RT/M)	PV = Nm·v²_rms/3 and PV = nRT	Molecular speed	KTG
Bohr radius	Coulomb force = centripetal, L = nħ	rₙ = n²a₀ = 0.529n² Å	Atoms
Eₙ = −13.6/n² eV	KE + PE = ½kZe²/r − kZe²/r	Energy levels	Atoms
F = qvB sin θ	Lorentz force from Biot-Savart	Magnetic force	Magnetism
ε = −dΦ/dt	Energy conservation + Lenz law	Faraday law	EMI
β = 1 − T₂/T₁	W = Q₁ − Q₂, T ∝ Q for Carnot	Carnot efficiency	Thermo
eV₀ = hν − φ	Energy conservation (photon → electron KE)	Photoelectric effect	Modern Physics
h/√(2meV)	de Broglie: λ = h/p, KE = eV	Electron wavelength	Modern Physics
I = (k/4πε₀)(q₁q₂/r²)	Coulomb's law from Gauss	k = 9×10⁹	Electrostatics

Common Mistakes to Avoid

Mistake	Correction	Topic
Using g = 10 instead of 9.8	Unless specified, use g = 9.8 m/s²	Mechanics
Forgetting vector nature of F, v, a	Always account for direction and use vector math	Motion, Forces
Confusing mass and weight	Mass (kg) is invariant; Weight = mg changes with g	Gravitation
Wrong sign convention in mirror/lens	Use New Cartesian: u is always −ve	Optics
Forgetting ε₀ in Gauss law	∮E·dA = Q/ε₀, NOT just Q	Electrostatics
Using V instead of V₀ in AC	V₀ = peak, V_rms = V₀/√2 for power calculations	AC Circuits
Adding KE of recoil nucleus	Total KE = KE_particle + KE_nucleus, use momentum conservation	Nuclear Physics
Confusing half-life and mean life	t_(1/2) = 0.693/λ, τ = 1/λ; τ = 1.44 × t_(1/2)	Radioactivity
Using f for both force and frequency	Be explicit: F for force (N), f for frequency (Hz)	General
Forgetting that Cp − Cv = R only for ideal gas	Real gases deviate, especially at high P and low T	Thermodynamics
Mixing up α, β, γ expansions	α = linear, β ≈ 2α (area), γ ≈ 3α (volume)	Thermal Properties
Wrong formula for closed pipe	Closed pipe has only ODD harmonics: fₙ = n×v/4L	Waves

Exam Strategy — Time Management

Board ExamsAttempt all questions. Show all steps — marks for steps even if answer is wrong. Draw neat diagrams.

JEE Main60 questions, 180 min → 3 min per question. Attempt easy ones first. Skip and revisit tough ones.

NEET45 Physics Qs, 180 min total (3 subjects). ~1 min per Physics Q. Focus on formula-based Qs.

High weightage topicsMechanics (30%), Electrodynamics (25%), Modern Physics (15%), Optics (10%), Thermodynamics (10%), Waves (10%)

Graph-based QsAlways check slope, intercept, area. Practice v-t, F-x, P-V, I-V graphs.

Units checkVerify final answer units. If units don't match, you made an error.

Dimensional analysis trickUse dimensional analysis to eliminate wrong options in MCQs.

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