Complete Physics Formula Sheet - Class 12
A comprehensive pedagogical guide for quick revision.
Electric Charges and Fields
Properties of Charge
Where \( K = \frac{1}{4\pi\epsilon_0} = 9 \times 10^9 \, \text{N}\cdot\text{m}^2/\text{C}^2 \)
(i) Quantisation: \( q = ne \) (where \( n=0,1,2... \), \( e = 1.6 \times 10^{-19} C \))
(ii) Additivity: \( q_{net} = \sum q \)
(iii) Conservation: Total charge of an isolated system is constant.
Coulomb's Law
(ii) Additivity: \( q_{net} = \sum q \)
(iii) Conservation: Total charge of an isolated system is constant.
\( F = K \frac{q_1 q_2}{r^2} \)
Where \( K = \frac{1}{4\pi\epsilon_0} = 9 \times 10^9 \, \text{N}\cdot\text{m}^2/\text{C}^2 \)
Vector form: \( \vec{F}_{12} = K \frac{q_1 q_2}{r_{12}^2} \hat{r}_{12} \) where \( \hat{r}_{12} = \frac{\vec{r}_{12}}{r_{12}} \)
Principle of Superposition
\( \vec{F}_1 = \vec{F}_{12} + \vec{F}_{13} + ... + \vec{F}_{1n} \)
\( \epsilon_0 = 8.854 \times 10^{-12} \, \text{C}^2/\text{N}\cdot\text{m}^2 \)
Electric Field
\( \epsilon_0 = 8.854 \times 10^{-12} \, \text{C}^2/\text{N}\cdot\text{m}^2 \)
\( \vec{E} = \frac{\vec{F}}{q_0} \) or \( \vec{E} = K \frac{q}{r^2} \)
Vector form: \( \vec{E} = \frac{1}{4\pi\epsilon_0} \frac{q}{r^2} \hat{r} \)
Relative Permittivity
Vector form: \( \vec{E} = \frac{1}{4\pi\epsilon_0} \frac{q}{r^2} \hat{r} \)
\( \epsilon_r = \frac{\epsilon}{\epsilon_0} \) (Permittivity in medium / Permittivity in free space)
HSC Physics Board Papers with Solution
- Physics - March 2025 - English Medium View Answer Key
- Physics - March 2025 - Marathi Medium View Answer Key
- Physics - March 2025 - Hindi Medium View Answer Key
- Physics - March 2024 - English Medium View Answer Key Answer Key
- Physics - March 2024 - Marathi Medium View Answer Key
- Physics - March 2024 - Hindi Medium View Answer Key
- Physics - March 2023 - English Medium View Answer Key
- Physics - July 2023 - English Medium View Answer Key
- Physics - March 2022 - English Medium View Answer Key
- Physics - July 2022 - English Medium View Answer Key
- Physics - July 2021 - English Medium View Answer Key
- Physics - February 2020 - English Medium View Answer Key
- Physics - February 2019 - English Medium View Answer Key
- Physics - February 2018 - English Medium View Answer Key
- Physics - March 2013 View
- Physics - October 2013 View Answer Key
- Physics - March 2014 View Answer Key
- Physics - October 2014 View Answer Key
- Physics - March 2015 View Answer Key
- Physics - October 2015 View Answer Key
- Physics - March 2016 View Answer Key
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- Physics - March 2017 View Answer Key
- Physics - July 2017 View Answer Key
Charge Distributions & Gauss's Law
Charge Densities
(i) Linear: \( \lambda = dq/dl \)
(ii) Surface: \( \sigma = dq/dA \)
(iii) Volume: \( \rho = dq/dV \)
Electric Dipole
(ii) Surface: \( \sigma = dq/dA \)
(iii) Volume: \( \rho = dq/dV \)
Dipole Moment: \( \vec{p} = q(2\vec{a}) \) (direction \( -q \) to \( +q \))
(i) At axial position: \( \vec{E}_{axial} = \frac{1}{4\pi\epsilon_0} \frac{2\vec{p}}{r^3} \)
(ii) At equatorial position: \( \vec{E}_{equa} = \frac{-1}{4\pi\epsilon_0} \frac{\vec{p}}{r^3} \)
Torque & Flux
(i) At axial position: \( \vec{E}_{axial} = \frac{1}{4\pi\epsilon_0} \frac{2\vec{p}}{r^3} \)
(ii) At equatorial position: \( \vec{E}_{equa} = \frac{-1}{4\pi\epsilon_0} \frac{\vec{p}}{r^3} \)
Torque: \( \vec{\tau} = \vec{p} \times \vec{E} \) or \( \tau = pE \sin\theta \)
Electric Flux: \( \phi_E = \oint \vec{E} \cdot d\vec{A} = \oint E dA \cos\theta \)
Gauss Theorem & Applications
Electric Flux: \( \phi_E = \oint \vec{E} \cdot d\vec{A} = \oint E dA \cos\theta \)
\( \phi_E = \frac{q}{\epsilon_0} = \oint \vec{E} \cdot d\vec{A} \)
(i) Infinitely long wire: \( E = \frac{\lambda}{2\pi\epsilon_0 r} \)
(ii) Infinite plane sheet: \( E = \frac{\sigma}{2\epsilon_0} \)
(i) Infinitely long wire: \( E = \frac{\lambda}{2\pi\epsilon_0 r} \)
(ii) Infinite plane sheet: \( E = \frac{\sigma}{2\epsilon_0} \)
Applications of Gauss's Law (Spheres)
(iii) Uniformly charged thin spherical shell
(a) Outside (\(r > R\)): \( E_{out} = \frac{\sigma R^2}{\epsilon_0 r^2} \)
(b) At surface (\(r = R\)): \( E_{surf} = \frac{\sigma}{\epsilon_0} \)
(c) Internal point (\(r < R\)): \( E_{in} = 0 \)
(iv) Solid non-conducting sphere
(b) At surface (\(r = R\)): \( E_{surf} = \frac{\sigma}{\epsilon_0} \)
(c) Internal point (\(r < R\)): \( E_{in} = 0 \)
(a) Outside (\(r > R\)): \( E_{out} = \frac{\rho R^3}{3\epsilon_0 r^2} \)
(b) At surface (\(r = R\)): \( E_{surf} = \frac{\rho R}{3\epsilon_0} \)
(c) Internal Point (\(r < R\)): \( E_{in} = \frac{\rho r}{3\epsilon_0} \)
(b) At surface (\(r = R\)): \( E_{surf} = \frac{\rho R}{3\epsilon_0} \)
(c) Internal Point (\(r < R\)): \( E_{in} = \frac{\rho r}{3\epsilon_0} \)
Electrostatic Potential & Capacitance
Potential
\( V = \frac{W}{q_0} \); Work \( W = q \times \Delta V \)
Relation to Field: \( V = - \int_{\infty}^r \vec{E} \cdot d\vec{r} \)
Due to point charge: \( V = \frac{1}{4\pi\epsilon_0} \frac{q}{r} \)
Potential due to Dipole
Relation to Field: \( V = - \int_{\infty}^r \vec{E} \cdot d\vec{r} \)
Due to point charge: \( V = \frac{1}{4\pi\epsilon_0} \frac{q}{r} \)
(a) End on: \( V = \frac{1}{4\pi\epsilon_0} \frac{p}{r^2} \)
(b) Equatorial: \( V = 0 \)
(c) Any point: \( V = \frac{1}{4\pi\epsilon_0} \frac{p \cos\theta}{r^2} \)
Work & Energy (Dipole)
(b) Equatorial: \( V = 0 \)
(c) Any point: \( V = \frac{1}{4\pi\epsilon_0} \frac{p \cos\theta}{r^2} \)
Work to rotate: \( W = pE(\cos\theta_0 - \cos\theta) \)
Potential Energy: \( U_{\theta} = -pE \cos\theta = -\vec{p} \cdot \vec{E} \)
Capacitance
Potential Energy: \( U_{\theta} = -pE \cos\theta = -\vec{p} \cdot \vec{E} \)
\( C = \frac{q}{V} \)
Isolated Spherical Conductor: \( C = 4\pi\epsilon_0 K a \)
Parallel Plate Capacitor: \( C = \frac{K \epsilon_0 A}{d} \) (For air \( K=1 \), \( C_0 = \frac{\epsilon_0 A}{d} \))
Force between plates: \( F = \frac{1}{2} qE \)
Energy of Charged Conductor
Isolated Spherical Conductor: \( C = 4\pi\epsilon_0 K a \)
Parallel Plate Capacitor: \( C = \frac{K \epsilon_0 A}{d} \) (For air \( K=1 \), \( C_0 = \frac{\epsilon_0 A}{d} \))
Force between plates: \( F = \frac{1}{2} qE \)
\( U = \frac{1}{2} qV \) or \( U = \frac{1}{2} \frac{q^2}{C} \) or \( U = \frac{1}{2} CV^2 \)
Capacitance (Continued)
Formulas
Energy Density: \( u = \frac{U}{Ad} = \frac{1}{2} \epsilon_0 E^2 \)
With Dielectric Slab: \( C = \frac{\epsilon_0 A}{(d-t) + \frac{t}{K}} \)
Series Combination: \( \frac{1}{C} = \frac{1}{C_1} + \frac{1}{C_2} + \frac{1}{C_3} \)
Parallel Combination: \( C = C_1 + C_2 + C_3 \)
With Dielectric Slab: \( C = \frac{\epsilon_0 A}{(d-t) + \frac{t}{K}} \)
Series Combination: \( \frac{1}{C} = \frac{1}{C_1} + \frac{1}{C_2} + \frac{1}{C_3} \)
Parallel Combination: \( C = C_1 + C_2 + C_3 \)
Current Electricity
Current: \( I = \frac{q}{t} \) or \( I(t) = \lim_{\Delta t \to 0} \frac{\Delta Q}{\Delta t} \)
Ohm's Law: \( V = RI \)
Resistivity: \( \rho = \frac{RA}{l} \) or \( \rho = \frac{m}{ne^2\tau} \)
Drift Velocity: \( v_d = \frac{eE\tau}{m} \) or \( v_d = \left( \frac{eV}{ml} \right) \tau \)
Current Density: \( j = i/A \) or \( \vec{j} = n e \vec{v}_d \)
Mobility: \( \mu = \frac{v_d}{E} \)
Vector form of Ohm's Law: \( \vec{j} = \sigma \vec{E} \)
Resistance & Temperature
Ohm's Law: \( V = RI \)
Resistivity: \( \rho = \frac{RA}{l} \) or \( \rho = \frac{m}{ne^2\tau} \)
Drift Velocity: \( v_d = \frac{eE\tau}{m} \) or \( v_d = \left( \frac{eV}{ml} \right) \tau \)
Current Density: \( j = i/A \) or \( \vec{j} = n e \vec{v}_d \)
Mobility: \( \mu = \frac{v_d}{E} \)
Vector form of Ohm's Law: \( \vec{j} = \sigma \vec{E} \)
\( R_t = R_0(1 + \alpha t) \) where \(\alpha\) is temp coefficient.
Combination
(A) Series: \( R = R_1 + R_2 + R_3 \)
(B) Parallel: \( \frac{1}{R} = \frac{1}{R_1} + \frac{1}{R_2} + \frac{1}{R_3} \)
Power & Energy
(B) Parallel: \( \frac{1}{R} = \frac{1}{R_1} + \frac{1}{R_2} + \frac{1}{R_3} \)
Power \( P = i^2 R = V^2/R \)
Heat \( H = i^2 R t \)
Heat \( H = i^2 R t \)
Current Electricity (Circuits)
Terminal Voltage: \( V = E - ir \)
Internal Resistance: \( r = R \left[ \frac{E}{V} - 1 \right] \)
Kirchhoff's Laws: 1. \( \sum i = 0 \) (Junction rule)
2. \( \sum iR = \sum E \) (Loop rule)
Wheatstone Bridge: \( \frac{P}{Q} = \frac{R}{S} \)
Meter Bridge: \( S = R \left[ \frac{100-l}{l} \right] \)
Potentiometer: \( r = R \left( \frac{l_1}{l_2} - 1 \right) \)
Internal Resistance: \( r = R \left[ \frac{E}{V} - 1 \right] \)
Kirchhoff's Laws: 1. \( \sum i = 0 \) (Junction rule)
2. \( \sum iR = \sum E \) (Loop rule)
Wheatstone Bridge: \( \frac{P}{Q} = \frac{R}{S} \)
Meter Bridge: \( S = R \left[ \frac{100-l}{l} \right] \)
Potentiometer: \( r = R \left( \frac{l_1}{l_2} - 1 \right) \)
Moving Charges and Magnetism
Laws & Fields
Biot Savart Law: \( d\vec{B} = \frac{\mu_0}{4\pi} \frac{i (d\vec{l} \times \vec{r})}{r^3} \)
Relation: \( c = \frac{1}{\sqrt{\mu_0 \epsilon_0}} \)
Circular Coil (Axis): \( B = \frac{\mu_0 N I a^2}{2(a^2 + x^2)^{3/2}} \)
Ampere's Circuital Law: \( \oint \vec{B} \cdot d\vec{l} = \mu_0 i \)
Infinite Straight Wire: \( B = \frac{\mu_0 i}{2\pi r} \)
Solenoid: \( B = \mu_0 n i \); Toroid: \( B = \mu_0 n i \)
Lorentz Force
Relation: \( c = \frac{1}{\sqrt{\mu_0 \epsilon_0}} \)
Circular Coil (Axis): \( B = \frac{\mu_0 N I a^2}{2(a^2 + x^2)^{3/2}} \)
Ampere's Circuital Law: \( \oint \vec{B} \cdot d\vec{l} = \mu_0 i \)
Infinite Straight Wire: \( B = \frac{\mu_0 i}{2\pi r} \)
Solenoid: \( B = \mu_0 n i \); Toroid: \( B = \mu_0 n i \)
\( \vec{F} = q(\vec{v} \times \vec{B}) \)
Cyclotron: \( r = \frac{mv}{qB} \), \( T = \frac{2\pi m}{qB} \), \( K_{max} = \frac{q^2 B^2 R^2}{2m} \)
Cyclotron: \( r = \frac{mv}{qB} \), \( T = \frac{2\pi m}{qB} \), \( K_{max} = \frac{q^2 B^2 R^2}{2m} \)
Forces & Instruments
Force on Conductor: \( F = i B L \sin\theta \)
Field at center of loop: \( B = \frac{\mu_0 i}{2a} \) (or \( \frac{\mu_0 N i}{2a} \))
Straight conductor (finite): \( B = \frac{\mu_0 i}{4\pi r}(\sin\phi_1 + \sin\phi_2) \)
Torque on Bar Magnet: \( \vec{\tau} = \vec{M} \times \vec{B} = MB \sin\theta \)
Potential Energy: \( U = -MB \cos\theta \)
Moving Coil Galvanometer: Deflection \( \phi = \left( \frac{NAB}{K} \right) I \)
Bohr Magneton: \( M_{min} = \frac{eh}{4\pi m_e} \)
Field at center of loop: \( B = \frac{\mu_0 i}{2a} \) (or \( \frac{\mu_0 N i}{2a} \))
Straight conductor (finite): \( B = \frac{\mu_0 i}{4\pi r}(\sin\phi_1 + \sin\phi_2) \)
Torque on Bar Magnet: \( \vec{\tau} = \vec{M} \times \vec{B} = MB \sin\theta \)
Potential Energy: \( U = -MB \cos\theta \)
Moving Coil Galvanometer: Deflection \( \phi = \left( \frac{NAB}{K} \right) I \)
Bohr Magneton: \( M_{min} = \frac{eh}{4\pi m_e} \)
Magnetism and Matter
Bar Magnet (Solenoid equiv): \( B = \frac{\mu_0 2M}{4\pi r^3} \)
Earth's Field: \( B_E = \sqrt{B_H^2 + B_V^2} \), \( \theta = \tan^{-1} \frac{B_V}{B_H} \)
Magnetisation: \( \vec{I} = \frac{\vec{M}}{V} \)
Magnetic Intensity: \( \vec{H} = \frac{\vec{B}}{\mu_0} - \vec{I} \)
Relative Permeability: \( \mu_r = 1 + \chi_m \)
Curie's Law: \( I = C \left( \frac{H}{T} \right) \)
Gauss Law for Magnetism: \( \oint \vec{B} \cdot d\vec{A} = 0 \)
Earth's Field: \( B_E = \sqrt{B_H^2 + B_V^2} \), \( \theta = \tan^{-1} \frac{B_V}{B_H} \)
Magnetisation: \( \vec{I} = \frac{\vec{M}}{V} \)
Magnetic Intensity: \( \vec{H} = \frac{\vec{B}}{\mu_0} - \vec{I} \)
Relative Permeability: \( \mu_r = 1 + \chi_m \)
Curie's Law: \( I = C \left( \frac{H}{T} \right) \)
Gauss Law for Magnetism: \( \oint \vec{B} \cdot d\vec{A} = 0 \)
Electromagnetic Induction
Magnetic Flux: \( \phi_B = \vec{B} \cdot \vec{A} = BA \cos\theta \)
Induced EMF: \( e = - \frac{d\phi_B}{dt} \) (for N turns \( e = -N \frac{d\phi}{dt} \))
Induced Current: \( i = \frac{e}{R} \)
Motional EMF: \( e = Bvl \)
Self Inductance: \( L = \frac{N \phi_B}{i} \); Solenoid \( L = \frac{\mu_0 N^2 A}{l} \)
Energy stored: \( U = \frac{1}{2} L i_0^2 \)
Mutual Inductance: \( M = \frac{\mu_0 N_1 N_2 A}{l} \)
Induced EMF: \( e = - \frac{d\phi_B}{dt} \) (for N turns \( e = -N \frac{d\phi}{dt} \))
Induced Current: \( i = \frac{e}{R} \)
Motional EMF: \( e = Bvl \)
Self Inductance: \( L = \frac{N \phi_B}{i} \); Solenoid \( L = \frac{\mu_0 N^2 A}{l} \)
Energy stored: \( U = \frac{1}{2} L i_0^2 \)
Mutual Inductance: \( M = \frac{\mu_0 N_1 N_2 A}{l} \)
Alternating Current
Mean Value: \( i_m = \frac{2}{\pi} i_0 = 0.637 i_0 \)
RMS Value: \( i_{rms} = \frac{i_0}{\sqrt{2}} = 0.707 i_0 \)
Reactance: \( X_L = \omega L \), \( X_C = \frac{1}{\omega C} \)
Impedance (LCR): \( Z = \sqrt{R^2 + (X_L - X_C)^2} \)
Resonance Frequency: \( f = \frac{1}{2\pi \sqrt{LC}} \)
Power Factor: \( \cos\phi = \frac{R}{Z} \)
Transformer: \( \frac{V_s}{V_p} = \frac{N_s}{N_p} = r \)
RMS Value: \( i_{rms} = \frac{i_0}{\sqrt{2}} = 0.707 i_0 \)
Reactance: \( X_L = \omega L \), \( X_C = \frac{1}{\omega C} \)
Impedance (LCR): \( Z = \sqrt{R^2 + (X_L - X_C)^2} \)
Resonance Frequency: \( f = \frac{1}{2\pi \sqrt{LC}} \)
Power Factor: \( \cos\phi = \frac{R}{Z} \)
Transformer: \( \frac{V_s}{V_p} = \frac{N_s}{N_p} = r \)
Electromagnetic Waves
Displacement Current: \( i_d = \epsilon_0 \frac{d\phi_E}{dt} \)
Maxwell's Equations: 1. \( \oint \vec{E} \cdot d\vec{A} = q/\epsilon_0 \)
2. \( \oint \vec{B} \cdot d\vec{A} = 0 \)
3. \( \oint \vec{E} \cdot d\vec{l} = -d\phi_B/dt \)
4. \( \oint \vec{B} \cdot d\vec{l} = \mu_0(i + i_d) \)
Relation: \( c = \frac{1}{\sqrt{\mu_0 \epsilon_0}} \), \( \frac{E}{B} = c \)
Maxwell's Equations: 1. \( \oint \vec{E} \cdot d\vec{A} = q/\epsilon_0 \)
2. \( \oint \vec{B} \cdot d\vec{A} = 0 \)
3. \( \oint \vec{E} \cdot d\vec{l} = -d\phi_B/dt \)
4. \( \oint \vec{B} \cdot d\vec{l} = \mu_0(i + i_d) \)
Relation: \( c = \frac{1}{\sqrt{\mu_0 \epsilon_0}} \), \( \frac{E}{B} = c \)
Ray Optics
Mirror Equation: \( \frac{1}{v} + \frac{1}{u} = \frac{1}{f} \), \( f = R/2 \)
Snell's Law: \( \frac{\sin i}{\sin r} = {}_1n_2 = \frac{n_2}{n_1} \)
Lens Formula: \( \frac{1}{v} - \frac{1}{u} = \frac{1}{f} \)
Power: \( P = 1/f \)
Prism: \( n = \frac{\sin((A+\delta_m)/2)}{\sin(A/2)} \)
Refraction at Spherical Surface: \( \frac{n_2}{v} - \frac{n_1}{u} = \frac{n_2 - n_1}{R} \)
Snell's Law: \( \frac{\sin i}{\sin r} = {}_1n_2 = \frac{n_2}{n_1} \)
Lens Formula: \( \frac{1}{v} - \frac{1}{u} = \frac{1}{f} \)
Power: \( P = 1/f \)
Prism: \( n = \frac{\sin((A+\delta_m)/2)}{\sin(A/2)} \)
Refraction at Spherical Surface: \( \frac{n_2}{v} - \frac{n_1}{u} = \frac{n_2 - n_1}{R} \)
Wave Optics
Interference: \( I = I_1 + I_2 + 2\sqrt{I_1 I_2} \cos\phi \)
Bright Fringes: \( x = m \frac{D\lambda}{d} \)
Fringe Width: \( W = \frac{D\lambda}{d} \)
Diffraction (Single slit): \( e \sin\theta = \pm m\lambda \) (minima)
Brewster's Law: \( \mu = \tan i_B \)
Bright Fringes: \( x = m \frac{D\lambda}{d} \)
Fringe Width: \( W = \frac{D\lambda}{d} \)
Diffraction (Single slit): \( e \sin\theta = \pm m\lambda \) (minima)
Brewster's Law: \( \mu = \tan i_B \)
Dual Nature of Radiation
Einstein's Photoelectric Eq: \( K_{max} = h\nu - \phi_0 = h(\nu - \nu_0) \)
Energy: \( E = h\nu = hc/\lambda \)
De Broglie Wavelength: \( \lambda = \frac{h}{p} = \frac{h}{mv} = \frac{h}{\sqrt{2mK}} \)
Heisenberg's Uncertainty: \( \Delta x \cdot \Delta p \approx \hbar \)
Energy: \( E = h\nu = hc/\lambda \)
De Broglie Wavelength: \( \lambda = \frac{h}{p} = \frac{h}{mv} = \frac{h}{\sqrt{2mK}} \)
Heisenberg's Uncertainty: \( \Delta x \cdot \Delta p \approx \hbar \)
Atoms
Bohr Quantization: \( mvr = \frac{nh}{2\pi} \)
Radius: \( r_n \propto n^2 \)
Energy: \( E_n = - \frac{13.6}{n^2} \, \text{eV} \)
Rydberg Formula: \( \frac{1}{\lambda} = R \left[ \frac{1}{n_1^2} - \frac{1}{n_2^2} \right] \)
Radius: \( r_n \propto n^2 \)
Energy: \( E_n = - \frac{13.6}{n^2} \, \text{eV} \)
Rydberg Formula: \( \frac{1}{\lambda} = R \left[ \frac{1}{n_1^2} - \frac{1}{n_2^2} \right] \)
Nuclei
Size: \( R = R_0 A^{1/3} \)
Mass Energy: \( E = mc^2 \)
Radioactive Decay: \( N = N_0 e^{-\lambda t} \)
Half Life: \( T_{1/2} = \frac{0.6931}{\lambda} \)
Mass Energy: \( E = mc^2 \)
Radioactive Decay: \( N = N_0 e^{-\lambda t} \)
Half Life: \( T_{1/2} = \frac{0.6931}{\lambda} \)
Semiconductors
\( n_e n_h = n_i^2 \)
Gains: \( \alpha = \frac{i_c}{i_e} \), \( \beta = \frac{i_c}{i_b} \)
Relation: \( \alpha = \frac{\beta}{1+\beta} \)
Logic Gates (Summary)
Gains: \( \alpha = \frac{i_c}{i_e} \), \( \beta = \frac{i_c}{i_b} \)
Relation: \( \alpha = \frac{\beta}{1+\beta} \)
| Gate | Boolean Expression |
|---|---|
| OR | \( Y = A + B \) |
| AND | \( Y = A \cdot B \) |
| NOT | \( Y = \bar{A} \) |
| NAND | \( Y = \overline{A \cdot B} \) |
| NOR | \( Y = \overline{A + B} \) |
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