Physics Comprehensive Formula Sheet - JEE/NEET Essential Formulas

📋 Introduction

This comprehensive physics formula sheet contains all essential formulas, equations, and constants needed for JEE Advanced and NEET preparation. Formulas are organized by topic and include units, conditions, and applications.

⚛️ Mechanics

Kinematics

Linear Motion:
v = u + at
s = ut + ½at²
v² = u² + 2as
s = vt - ½at²
s = [(u + v)/2]t

Average velocity: v_avg = Δs/Δt
Average acceleration: a_avg = Δv/Δt

Projectile Motion:
Time of flight: T = 2u sin θ/g
Maximum height: H = u² sin²θ/2g
Horizontal range: R = u² sin 2θ/g
Equation of trajectory: y = x tan θ - (gx²/2u²cos²θ)

Position at time t:
x = u cos θ · t
y = u sin θ · t - ½gt²
v_x = u cos θ
v_y = u sin θ - gt

Relative Motion:
v_AB = v_A - v_B
v_AC = v_AB + v_BC

Circular Motion:
Angular velocity: ω = v/r = 2π/T = 2πf
Angular acceleration: α = Δω/Δt
Linear velocity: v = ωr
Tangential acceleration: a_t = αr
Centripetal acceleration: a_c = v²/r = ω²r
Total acceleration: a = √(a_t² + a_c²)

Period: T = 2πr/v = 2π/ω
Frequency: f = 1/T = ω/2π

Newton’s Laws of Motion

Newton's Laws:
First Law: F = 0 (equilibrium)
Second Law: F = ma
Third Law: F_AB = -F_BA

Force Analysis:
ΣF = ma (vector form)
ΣF_x = ma_x, ΣF_y = ma_y

Friction:
Static friction: f_s ≤ μ_s N
Maximum static friction: f_max = μ_s N
Kinetic friction: f_k = μ_k N
Angle of repose: tan θ = μ_s

Tension in strings:
Same string: T₁ = T₂
Pulley system: T distribution based on masses
Inclined plane: Components mg sin θ, mg cos θ

Connected Bodies:
For two masses connected by string:
a = (m₂g - m₁g sin θ)/(m₁ + m₂)
T = m₁(g sin θ + a) = m₂(g - a)

Circular Dynamics:
Centripetal force: F_c = mv²/r = mω²r
Banking of roads: tan θ = v²/(rg)
Conical pendulum: T cos θ = mg, T sin θ = mv²/r

Work, Energy and Power

Work Done:
Constant force: W = F·d·cosθ
Variable force: W = ∫F·ds
Work by gravity: W_g = mgh
Work by spring: W_s = ½k(x₂² - x₁²)

Power:
Instantaneous: P = dW/dt = F·v
Average: P_avg = W/t
Rotational: P = τ·ω

Energy:
Kinetic Energy (translational): KE = ½mv²
Kinetic Energy (rotational): KE = ½Iω²
Potential Energy (gravitational): PE = mgh
Potential Energy (elastic): PE = ½kx²
Mechanical Energy: E = KE + PE

Work-Energy Theorem:
W_net = ΔKE = KE_final - KE_initial

Conservation of Energy:
ΔKE + ΔPE = 0 (conservative forces only)
W_nc = ΔKE + ΔPE (with non-conservative forces)

Power Efficiency:
η = Output Power/Input Power

Rotational Motion

Angular Kinematics:
ω = ω₀ + αt
θ = ω₀t + ½αt²
ω² = ω₀² + 2αθ
θ = ωt - ½αt²

Linear-Angular Relations:
v = rω
a_t = rα
a_c = ω²r

Moment of Inertia:
Point mass: I = mr²
Continuous body: I = ∫r² dm
Parallel axis theorem: I = I_cm + Md²
Perpendicular axis theorem: I_z = I_x + I_y (lamina)

Common Moments of Inertia:
Rod (about center): I = ML²/12
Rod (about end): I = ML²/3
Ring: I = MR²
Disk: I = MR²/2
Sphere: I = 2MR²/5
Hollow sphere: I = 2MR²/3

Torque and Angular Momentum:
τ = r × F = Iα
Angular momentum: L = r × p = Iω
τ = dL/dt
Conservation: L = constant (if τ = 0)

Rolling Motion:
Pure rolling: v_cm = ωR
KE_total = ½mv² + ½Iω²
KE_trans = ½mv²
KE_rot = ½Iω²
Acceleration down incline: a = g sin θ/(1 + I/(MR²))

Gravitation

Newton's Law of Gravitation:
F = G(m₁m₂/r²)
G = 6.67 × 10⁻¹¹ N·m²/kg²

Gravitational Field:
Field strength: g = F/m = GM/r²
Gravitational potential: V = -GM/r
Field and potential: g = -dV/dr

Orbital Motion:
Orbital velocity: v = √(GM/r)
Orbital period: T = 2π√(r³/GM)
Angular velocity: ω = √(GM/r³)
Centripetal force: F_c = mv²/r = GMm/r²

Escape Velocity:
v_e = √(2GM/r) = √(2gr) (for Earth's surface)
v_e = 11.2 km/s (Earth)

Kepler's Laws:
First law: Planets move in ellipses with Sun at focus
Second law: dA/dt = L/(2m) = constant (equal areas in equal times)
Third law: T² ∝ a³, T² = (4π²/GM)a³

Variation of g:
With height: g(h) = g₀/(1 + h/R)²
With depth: g(d) = g₀(1 - d/R)
At Earth's surface: g = 9.8 m/s²

Satellite Motion:
Geostationary orbit: r = [GMT²/(4π²)]^(1/3)
Geostationary period: T = 24 hours
Binding energy: E = -GMm/(2r)

🔥 Thermodynamics

Heat and Temperature

Temperature Scales:
Celsius to Fahrenheit: F = (9/5)C + 32
Fahrenheit to Celsius: C = (5/9)(F - 32)
Celsius to Kelvin: K = C + 273.15
Kelvin to Celsius: C = K - 273.15

Heat Transfer:
Conduction: Q = kAΔT·t/d
Convection: Q = hAΔT·t
Radiation: Q = εσAT⁴t (Stefan-Boltzmann)
Newton's law of cooling: dQ/dt = -k(T - T₀)

Specific Heat:
Q = mcΔT
Specific heat capacity: c = Q/(mΔT)
Molar specific heat: C = Q/(nΔT)
Latent heat: Q = mL

Calorimetry:
Principle of calorimetry: Heat lost = Heat gained
Q₁ + Q₂ + ... + Q_n = 0
Water equivalent: W = mc

Laws of Thermodynamics

First Law of Thermodynamics:
ΔU = Q - W
For cyclic process: ΔU = 0, Q = W
Work done by gas: W = ∫PdV

Work in Different Processes:
Isobaric: W = PΔV
Isochoric: W = 0
Isothermal: W = nRT ln(V₂/V₁)
Adiabatic: W = (P₁V₁ - P₂V₂)/(γ - 1)

Internal Energy:
For ideal gas: U = nC_vT
ΔU = nC_vΔT

Second Law of Thermodynamics:
Clausius statement: Heat cannot flow from cold to hot spontaneously
Kelvin-Planck statement: No 100% efficient heat engine possible

Entropy:
ΔS = ∫dQ_rev/T
For reversible process: ΔS = Q_rev/T
For irreversible process: ΔS > Q/T

Heat Engine:
Efficiency: η = W/Q_H = 1 - Q_C/Q_H
Carnot efficiency: η = 1 - T_C/T_H
Coefficient of performance (refrigerator): COP = Q_C/W

Thermodynamic Processes:
Ideal Gas Equation: PV = nRT
Van der Waals Equation: (P + a/V²)(V - b) = RT
Adiabatic process: PV^γ = constant
Isothermal process: PV = constant

Kinetic Theory of Gases

Kinetic Theory Assumptions:
- Large number of molecules
- Molecules in random motion
- No intermolecular forces (ideal gas)
- Elastic collisions
- Volume of molecules negligible

Pressure of Gas:
P = (1/3)ρv_rms² = (2/3)(KE)/V
P = (1/3)nmv²_avg

Root Mean Square Speed:
v_rms = √(3kT/m) = √(3RT/M)
Average speed: v_avg = √(8kT/πm)
Most probable speed: v_p = √(2kT/m)

Relation between speeds:
v_p : v_avg : v_rms = √2 : √(8/π) : √3

Kinetic Energy:
Average KE per molecule: (3/2)kT
Total KE: (3/2)nRT
KE per mole: (3/2)RT

Degrees of Freedom:
f = Translational + Rotational + Vibrational
Energy per molecule per degree of freedom: (1/2)kT
Energy per mole per degree of freedom: (1/2)RT

Specific Heats:
C_p - C_v = R (Mayer's relation)
γ = C_p/C_v
Monatomic: C_v = (3/2)R, C_p = (5/2)R, γ = 5/3
Diatomic: C_v = (5/2)R, C_p = (7/2)R, γ = 7/5

⚡ Electromagnetism

Electrostatics

Coulomb's Law:
F = k(q₁q₂/r²) = (1/4πε₀)(q₁q₂/r²)
k = 9 × 10⁹ N·m²/C²
ε₀ = 8.854 × 10⁻¹² C²/N·m²

Electric Field:
E = F/q = k(Q/r²)
Electric field due to point charge: E = kQ/r²
Electric field lines: Start from positive, end at negative

Electric Flux:
Φ = ∮E·dA
Gauss's Law: Φ = Q_enc/ε₀
Electric flux through Gaussian surface: Φ = ∮E·dA = Q_enc/ε₀

Electric Potential:
V = W/q = kQ/r
Potential difference: V = W/q
Electric potential energy: U = qV
Potential due to multiple charges: V = kΣ(q_i/r_i)

Equipotential Surfaces:
V = constant
E ⊥ equipotential surface
Work done moving charge: W = qΔV

Capacitors:
Capacitance: C = Q/V
Parallel plate capacitor: C = ε₀A/d
Capacitor with dielectric: C = κε₀A/d
Energy stored: U = ½CV² = ½QV = Q²/2C

Combinations:
Series: 1/C_eq = 1/C₁ + 1/C₂ + ...
Parallel: C_eq = C₁ + C₂ + ...

Current Electricity

Electric Current:
I = Q/t = dQ/dt
Current density: J = I/A = nqv_d
Drift velocity: v_d = I/(nAq)

Ohm's Law:
V = IR
Resistance: R = ρL/A
Resistivity: ρ = RA/L
Conductivity: σ = 1/ρ

Power and Energy:
Power: P = VI = I²R = V²/R
Energy: E = Pt = VIt
Heating effect: H = I²Rt (Joule's law)

Kirchhoff's Laws:
Junction rule: ΣI_in = ΣI_out
Loop rule: ΣV = 0

Wheatstone Bridge:
Balance condition: R₁/R₂ = R₃/R₄
Galvanometer current: I_g = 0 when balanced

Potentiometer:
Comparison: E₁/E₂ = l₁/l₂
Measurement: V = IR (unknown R = kl/I)

Electrochemical Cells:
EMF of cell: E_cell = E_cathode - E_anode
Nernst equation: E = E° - (RT/nF)ln(Q)
Standard conditions: 25°C, 1 atm, 1 M

Magnetism

Magnetic Field:
Biot-Savart Law: dB = (μ₀/4π)(Idlsinθ/r²)
Ampere's Law: ∮B·dl = μ₀I_enc
Magnetic field due to straight wire: B = μ₀I/(2πr)
Magnetic field at center of loop: B = μ₀I/(2r)

Force on Moving Charge:
Lorentz force: F = q(v × B)
Force on current-carrying conductor: F = ILB sinθ
Torque on current loop: τ = NIAB sinθ

Magnetic Materials:
Magnetic field intensity: H = B/μ₀ - M
Magnetization: M = χ_m H
Magnetic susceptibility: χ_m = μ_r - 1
Relative permeability: μ_r = 1 + χ_m

Types of magnetism:
Diamagnetic: χ_m < 0, weak repulsion
Paramagnetic: χ_m > 0, weak attraction
Ferromagnetic: χ_m >> 0, strong attraction

Magnetic Force:
Between parallel wires: F/L = μ₀I₁I₂/(2πd)
Circular motion in magnetic field: r = mv/(qB)
Helical motion: pitch = v_parallel × T = 2πmv_parallel/(qB)

Electromagnetic Induction

Faraday's Laws:
First law: Induced EMF = -dΦ/dt
Second law: ε = -N(dΦ/dt)
Magnetic flux: Φ = ∫B·dA = BA cosθ

Lenz's Law:
Induced current opposes change in flux
Direction: Using right-hand rule

Self Induction:
Self-inductance: L = NΦ/I = μ₀N²A/l (solenoid)
Induced EMF: ε = -L(dI/dt)
Energy stored: U = ½LI²

Mutual Induction:
Mutual inductance: M = N₂Φ₂₁/I₁
Induced EMF: ε₂ = -M(dI₁/dt)

Transformers:
Turns ratio: V₂/V₁ = N₂/N₁
Current ratio: I₂/I₁ = N₁/N₂
Efficiency: η = P_out/P_in = (V₂I₂)/(V₁I₁)

AC Generator:
Induced EMF: ε = NABω sin(ωt)
Maximum EMF: ε_max = NABω
Frequency: f = ω/(2π)

Induced Electric Field:
From changing magnetic field: ∮E·dl = -dΦ_B/dt
Displacement current: I_d = ε₀(dΦ_E/dt)

🔬 Optics

Ray Optics

Reflection:
Law of reflection: θ_i = θ_r
Image formation by mirrors:
Concave mirror: 1/f = 1/u + 1/v
Convex mirror: 1/f = 1/u + 1/v
Magnification: m = -v/u = h_i/h_o

Refraction:
Snell's law: n₁sin i = n₂sin r
Refractive index: n = c/v = sin i/sin r
Critical angle: sin θ_c = n₂/n₁
Total internal reflection: θ_i > θ_c

Lenses:
Lens maker's formula: 1/f = (n-1)(1/R₁ - 1/R₂)
Lens formula: 1/f = 1/v - 1/u
Magnification: m = v/u = h_i/h_o
Power of lens: P = 1/f (in diopters)

Lens Combinations:
Two lenses in contact: 1/F = 1/f₁ + 1/f₂
Two lenses separated: 1/F = 1/f₁ + 1/f₂ - d/(f₁f₂)

Optical Instruments:
Simple microscope: M = D/f
Compound microscope: M = (D/f_e) × (v_o/f_o)
Astronomical telescope: M = f_o/f_e

Wave Optics

Interference:
Young's double slit: d sin θ = nλ
Fringe width: β = λD/d
Path difference: Δ = d sin θ
Constructive: Δ = nλ, Destructive: Δ = (2n+1)λ/2

Coherent sources: Same frequency, constant phase difference
Intensity distribution: I = I₁ + I₂ + 2√(I₁I₂)cos δ

Diffraction:
Single slit: a sin θ = nλ (minima)
Central maximum width: 2λD/a
Diffraction grating: d sin θ = nλ
Resolving power: R = nN

Polarization:
Brewster's angle: tan i_p = n₂/n₁
Malus's law: I = I₀cos²θ
Polaroids: I = I₀cos²θ

Huygens' Principle:
Each point on wavefront acts as source of secondary wavelets
New wavefront is envelope of secondary wavelets

⚛️ Modern Physics

Atomic Structure

Bohr Model:
Radius: r_n = n²(h²ε₀)/(πmZe²) = n²a₀/Z
Energy: E_n = -(13.6Z²)/n² eV
Frequency: ν = (E_i - E_f)/h
Wavelength: 1/λ = R∞Z²(1/n_f² - 1/n_i²)

Rydberg formula: 1/λ = R∞(1/n₁² - 1/n₂²)
Rydberg constant: R∞ = 1.097 × 10⁷ m⁻¹

Hydrogen spectrum:
Lyman: n_f = 1 (UV region)
Balmer: n_f = 2 (Visible region)
Paschen: n_f = 3 (IR region)
Brackett: n_f = 4 (IR region)

de Broglie wavelength:
λ = h/p = h/(mv)
λ = h/√(2mE) (for particles)
λ = h/√(2meV) (for electrons with V volts)

Heisenberg Uncertainty Principle:
Δx·Δp ≥ h/(4π)
ΔE·Δt ≥ h/(4π)

Nuclear Physics

Radioactive Decay:
Decay constant: λ = 0.693/T½
Half-life: T½ = 0.693/λ
Number of nuclei: N = N₀e^(-λt)
Activity: A = λN = A₀e^(-λt)

Nuclear Reactions:
Mass-energy equivalence: E = mc²
Binding energy: B = (Zm_p + Nm_n - M)c²
Q-value: Q = [M_initial - M_final]c²

Alpha decay: ᵐXₙ → ᵐ⁻⁴Yₙ₋₂ + ⁴He₂
Beta decay: ᵐXₙ → ᵐYₙ₊₁ + β⁻ + ν̄
Gamma decay: ᵐXₙ* → ᵐXₙ + γ

Nuclear Stability:
Magic numbers: 2, 8, 20, 28, 50, 82, 126
Liquid drop model: Semi-empirical mass formula

Semiconductor Devices

Semiconductor Physics:
Conductivity: σ = q(nμₙ + pμ_p)
Intrinsic carrier concentration: n_i = √(N_cN_v)e^(-E_g/2kT)

Diode Equation:
I = I₀(e^(qV/kT) - 1)
Forward bias: V > 0, I ≈ I₀e^(qV/kT)
Reverse bias: V < 0, I ≈ -I₀

Transistor:
Current gain (common emitter): β = I_c/I_b
Current gain (common base): α = I_c/I_e
Relationship: β = α/(1 - α)

Logic Gates:
NOT: Y = Ā
AND: Y = A·B
OR: Y = A + B
NAND: Y = (A·B)̄
NOR: Y = (A + B)̄
XOR: Y = A⊕B

🔬 Experimental Physics

Measurements and Errors

Significant Figures:
All certain digits + first uncertain digit
Operations: Follow least precise measurement

Errors:
Absolute error: Δx = |x_measured - x_true|
Relative error: (Δx/x) × 100%
Percentage error: (Δx/x) × 100%

Propagation of Errors:
For addition/subtraction: Δ(A ± B) = ΔA + ΔB
For multiplication/division: Δ(AB)/AB = ΔA/A + ΔB/B
For powers: Δ(A^n)/A^n = n(ΔA/A)

Vernier Calipers:
Least count: 1 MSD - 1 VSD
Zero error: Positive/negative
Reading: MSR + (VSD × LC)

Screw Gauge:
Least count: Pitch/Number of divisions
Zero error: Positive/negative
Reading: MSR + (HSR × LC)

📊 Physical Constants

Fundamental Constants

Speed of light: c = 3 × 10⁸ m/s
Planck's constant: h = 6.626 × 10⁻³⁴ J·s
Electron charge: e = 1.602 × 10⁻¹⁹ C
Electron mass: m_e = 9.109 × 10⁻³¹ kg
Proton mass: m_p = 1.673 × 10⁻²⁷ kg
Neutron mass: m_n = 1.675 × 10⁻²⁷ kg
Avogadro's number: N_A = 6.022 × 10²³ mol⁻¹
Gas constant: R = 8.314 J/(mol·K)
Boltzmann constant: k = 1.381 × 10⁻²³ J/K
Gravitational constant: G = 6.674 × 10⁻¹¹ N·m²/kg²
Permittivity of free space: ε₀ = 8.854 × 10⁻¹² C²/N·m²
Permeability of free space: μ₀ = 4π × 10⁻⁷ T·m/A
Rydberg constant: R∞ = 1.097 × 10⁷ m⁻¹

🎯 Usage Tips

Memory Techniques:

1. Group related formulas by topic
2. Understand derivations for better retention
3. Practice dimensional analysis to verify formulas
4. Create visual associations for complex relationships
5. Use real-world examples to understand applications

Problem-Solving Strategy:

1. Identify the concept and relevant formula
2. Check units and conditions for formula validity
3. Substitute values carefully with proper units
4. Solve step-by-step showing all calculations
5. Verify answer using alternative methods if possible

Use this comprehensive physics formula sheet as your quick reference guide for JEE/NEET preparation! Regular practice with these formulas will significantly enhance your problem-solving speed and accuracy. 🎯