PhysicsModern PhysicsJEE Main & Advanced

Modern Physics Formulas

Complete chapter-wise formula sheet for Modern Physics (Physics) covering key equations, parameters, traps, and concept diagrams for JEE Main & Advanced.

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Chapter Directory (24 Chapters)

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Modern Physics535 Qs

Dual Nature of Radiation & Matter

Photoelectric effect, Einstein's equation, work function, cut-off frequency, and de Broglie matter waves.

7 Formulas

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Modern Physics822 Qs

Atoms & Nuclei

Bohr atomic model, energy levels, Rydberg formula, radioactive decay law, half-life, and nuclear binding energy.

8 Formulas

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Modern Physics462 Qs

Semiconductors & Logic Gates

P-N junction diode, Zener breakdown diode, rectifiers, and truth tables of Boolean logic gates (AND, OR, NOT, NAND, NOR).

8 Formulas

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Formulas Reference (Modern Physics)

3 Chapters match

Modern PhysicsHigh Yield (2 Qs in JEE)535 PYQs

Dual Nature of Radiation & Matter

Photoelectric effect, Einstein's equation, work function, cut-off frequency, and de Broglie matter waves.

Chapter

Einstein's Photoelectric Equation

K_{\text{max}} = h\nu - \Phi_0 = h\nu - h\nu_0 = e V_0

Conservation of energy in photon-electron interaction.

Graph of stopping potential V₀ versus frequency ν has slope h/e (universal constant).

De Broglie Matter Wavelength

\lambda = \frac{h}{p} = \frac{h}{m v} = \frac{h}{\sqrt{2mK}}

Wave-particle duality wavelength associated with moving particles of momentum p.

Wavelength of electron accelerated through potential V: λ = 1.227 / √V nm = √(150/V) Å.

Photon Momentum & Energy

E = h\nu = \frac{hc}{\lambda} \approx \frac{12400}{\lambda \text{ (\AA)}} \text{ eV}, \quad p = \frac{E}{c} = \frac{h}{\lambda}

Energy and momentum of light quanta.

Useful constant: hc ≈ 1240 eV·nm ≈ 12400 eV·Å.

Radiation Pressure on Surface

P = \frac{I}{c} \; (\text{Complete Absorption}), \quad P = \frac{2I}{c} \; (\text{Complete Reflection})

Mechanical force per unit area exerted by electromagnetic waves of intensity I.

Force on perfectly reflecting surface of area A is F = 2IA / c. For oblique incidence at angle θ: P = (2I/c)cos²θ.

De Broglie Wavelength of Accelerated Charged Particles

\lambda = \frac{h}{\sqrt{2mqV}} \implies \lambda_e = \frac{12.27}{\sqrt{V}}\,\text{\AA}, \quad \lambda_p = \frac{0.286}{\sqrt{V}}\,\text{\AA}, \quad \lambda_d = \frac{0.202}{\sqrt{V}}\,\text{\AA}, \quad \lambda_\alpha = \frac{0.101}{\sqrt{V}}\,\text{\AA}

Matter wavelengths of fundamental particles accelerated from rest through potential difference V in volts.

For electron: λ = √(150/V) Å. Ratio of wavelengths for same V: λ_p : λ_d : λ_α = 1 : 1/√2 : 1/(2√2).

Thermal De Broglie Wavelength

\lambda = \frac{h}{\sqrt{3m k_B T}} \approx \frac{25.17}{\sqrt{T}}\,\text{\AA} \quad (\text{for thermal neutron})

Wavelength associated with gas molecules or neutrons in thermal equilibrium at temperature T (Kelvin).

Mean translational kinetic energy is K_avg = (3/2)k_B T.

Photon Flux & Saturation Photocurrent

n = \frac{P}{E} = \frac{P\lambda}{hc} \approx 5 \times 10^{24} P\lambda, \quad i_{\text{sat}} = \eta \cdot n \cdot e

Number of photons incident per second for a source of power P, and resulting saturation current with quantum efficiency η.

In JEE questions, quantum efficiency η is typically 0.1% to 1% (i.e. only 1 out of 100 or 1000 incident photons ejects a photoelectron).

Modern PhysicsTop Priority (3 Qs in JEE)822 PYQs

Atoms & Nuclei

Bohr atomic model, energy levels, Rydberg formula, radioactive decay law, half-life, and nuclear binding energy.

Chapter

Bohr Model Radius & Velocity

r_n = 0.529 \frac{n^2}{Z} \text{ \AA}, \quad v_n = 2.18 \times 10^6 \frac{Z}{n} \text{ m/s} = c \alpha \frac{Z}{n}

Quantized orbit radius and electron speed for hydrogen-like species.

Radius scales as n²/Z; velocity scales as Z/n.

Bohr Energy Levels & Energy Partition

E_n = -13.6 \frac{Z^2}{n^2} \text{ eV}, \quad K_n = -E_n, \quad U_n = 2E_n

Quantized energy levels in hydrogenic atoms.

Ionization energy from ground state is +13.6 Z² eV.

Rydberg Formula for Spectral Lines

\frac{1}{\lambda} = R Z^2 \left(\frac{1}{n_1^2} - \frac{1}{n_2^2}\right) \quad (R \approx 1.097 \times 10^7 \text{ m}^{-1})

Wavelengths of emitted photons during transitions between orbits.

Lyman (n₁=1, UV), Balmer (n₁=2, Visible), Paschen (n₁=3, IR).

Law of Radioactive Decay

N(t) = N_0 e^{-\lambda t}, \quad T_{1/2} = \frac{\ln 2}{\lambda} = \frac{0.693}{\lambda}, \quad \tau_{\text{mean}} = \frac{1}{\lambda} = 1.44 T_{1/2}

Statistical exponential decay of radioactive nuclei.

Fraction remaining after n half-lives is (1/2)^n.

Nuclear Radius & Mass Defect (Binding Energy)

R = R_0 A^{1/3} \; (R_0 \approx 1.2 \text{ fm}), \quad \Delta E = \Delta m \times 931.5 \text{ MeV}

Nuclear volume scales directly with mass number A (constant nuclear density ~ 2.3 × 10¹⁷ kg/m³).

Peak binding energy per nucleon occurs near Iron-56 (~8.8 MeV/nucleon).

Moseley's Law & Characteristic X-Rays

\sqrt{\nu} = a(Z - b), \quad \lambda_{\text{min}} = \frac{hc}{eV} = \frac{12400}{V \text{ (volts)}} \text{ \AA}

Frequency of characteristic X-ray lines where b is screening constant (b = 1 for K-alpha lines). Duane-Hunt minimum cutoff wavelength.

Plot of √ν vs atomic number Z is a straight line, which originally ordered the periodic table by atomic number.

Q-Value of Nuclear Reactions & Threshold

Q = [m_{\text{reactants}} - m_{\text{products}}] c^2 = \sum BE_{\text{products}} - \sum BE_{\text{reactants}}

Net energy liberated or absorbed in nuclear transmutation. Exoergic if Q > 0, endoergic if Q < 0.

Threshold kinetic energy for endoergic reaction: K_{th} = |Q| (1 + m_{projectile} / m_{target}).

Activity & Radioactive Equilibrium

A(t) = \lambda N(t) = A_0 e^{-\lambda t}, \quad 1 \text{ Curie (Ci)} = 3.7 \times 10^{10} \text{ Bq}

Decay rate per second. In successive decay A -> B -> C with secular equilibrium: λ_A N_A = λ_B N_B.

Parallel radioactive decay: effective decay constant λ_eff = λ₁ + λ₂; 1/T_{eff} = 1/T₁ + 1/T₂.

Modern PhysicsHigh Yield (2 Qs in JEE)462 PYQs

Semiconductors & Logic Gates

P-N junction diode, Zener breakdown diode, rectifiers, and truth tables of Boolean logic gates (AND, OR, NOT, NAND, NOR).

Chapter

Intrinsic & Extrinsic Carrier Concentrations

n_e \cdot n_h = n_i^2 \quad (\text{Mass Action Law}), \quad \sigma = e(n_e \mu_e + n_h \mu_h)

Product of electron and hole numbers remains constant at thermal equilibrium.

In n-type, n_e ≈ N_d; in p-type, n_h ≈ N_a.

Half-Wave & Full-Wave Rectifier Efficiency

\eta_{\text{half}} = \frac{0.406 R_L}{r_f + R_L} \le 40.6\%, \quad \eta_{\text{full}} = \frac{0.812 R_L}{r_f + R_L} \le 81.2\%

AC to DC power conversion efficiency.

Output ripple frequency of full-wave rectifier is 2f_in; for half-wave it is f_in.

De Morgan's Laws & Logic Operations

\overline{A + B} = \bar{A} \cdot \bar{B}, \quad \overline{A \cdot B} = \bar{A} + \bar{B}

Fundamental duality theorems for simplifying Boolean digital circuits.

NAND: Y = (A · B)' (0 only when both inputs are 1). NOR: Y = (A + B)' (1 only when both inputs are 0).

Zener Diode as Voltage Regulator

V_{\text{out}} = V_Z, \quad I_s = I_Z + I_L, \quad R_s = \frac{V_{\text{in}} - V_Z}{I_Z + I_L}

Maintains constant terminal load voltage V_Z regardless of unregulated DC input voltage fluctuations.

Zener operates in breakdown region in reverse bias. Minimum input voltage must exceed V_Z.

Transistor Current & Voltage Gains

I_E = I_B + I_C, \quad \beta = \frac{I_C}{I_B} = \frac{\alpha}{1 - \alpha}, \quad A_V = \beta \frac{R_{\text{out}}}{R_{\text{in}}}, \quad A_P = \beta^2 \frac{R_{\text{out}}}{R_{\text{in}}}

Common-emitter amplifier relationships where α < 1 (typically 0.95 to 0.99) and β is large (typically 20 to 100).

In common emitter amplifier, output voltage has a 180° (π radians) phase shift relative to input signal.

Dynamic Resistance & Rectifier Ripple Factor

r_d = \frac{\Delta V}{\Delta I}, \quad r = \sqrt{\left(\frac{I_{\text{rms}}}{I_{\text{dc}}}\right)^2 - 1} \implies r_{\text{half}} = 1.21, \quad r_{\text{full}} = 0.482

Differential resistance of P-N junction and effectiveness of rectifier in delivering pure DC voltage.

A smaller ripple factor indicates a purer DC output with less harmonic fluctuation.

Mass Action Law & Semiconductor Conductivity

n_e \cdot n_h = n_i^2, \quad \sigma = e(n_e \mu_e + n_h \mu_h), \quad \sigma_{\text{n-type}} \approx e N_D \mu_e, \quad \sigma_{\text{p-type}} \approx e N_A \mu_h

Carrier concentration product in thermal equilibrium and electrical conductivity dictated by electron (μ_e) and hole (μ_h) mobilities.

Electron mobility is always higher than hole mobility (μ_e > μ_h) in semiconductors.

Optoelectronic Bandgap Wavelength (LED, Photodiode)

\lambda_{\text{threshold}} = \frac{hc}{E_g} \approx \frac{12400}{E_g \text{ (in eV)}} \,\text{\AA}

Cut-off optical wavelength for light absorption in solar cells/photodiodes and photon emission in LEDs.

Photodiodes are operated in reverse bias because fractional change in reverse saturation current upon illumination is substantially higher than in forward bias.