भौतिकी के सूत्र

आधारभूत यांत्रिकी (Fundamentals of Mechanics)

Foundational equations in translation and rotation.

Quantity	Translation	Rotation
समय	<math>t</math>	<math>t</math>
स्थिति	<math>x </math>	<math>\theta </math> in radians
द्रव्यमान	<math>m</math>	<math>m</math>
समयान्तर	<math>\Delta t</math>	<math>\Delta t</math>
विस्थापन	<math>\Delta x</math>	<math>\Delta \theta</math>
द्रव्यमान संरक्षण	<math>\Delta m = 0 </math>	<math>\Delta m = 0 </math>
ऊर्जा संरक्षण	<math>\Delta E = 0 </math>	<math>\Delta E = 0 </math>
संवेग संरक्षण	<math>\Delta P = 0 </math>	<math>\Delta L = 0 </math>
वेग	<math> v = dx/dt </math>	<math>\omega = d\theta/dt </math>
त्वरण	<math> a = dv/dt </math>	<math>\alpha = d\omega/dt </math>
झटका	<math>j = da/dt </math>	<math>j = d\alpha/dt </math>
स्थितिज ऊर्जा परिवर्तन	<math>\Delta U = -W</math>	<math>\Delta U = -W</math>
संवेग	<math>P = mv </math>	<math>L = I\omega </math> <math> =	\mathbf{r} \times \mathbf{P}	= m	\mathbf{r} \times \mathbf{v}	</math>
बल	<math>f = dP/dt = ma = -dU/dx </math>	<math>\tau = dL/dt = I\alpha </math> <math> =	\mathbf{r} \times \mathbf{f}	=m	\mathbf{r} \times \mathbf{a}	</math>
जड़त्व आघूर्ण	<math>m = \int dm = \Sigma m_i</math>	<math>I = \int r^2 dm = \Sigma r^2m_i</math>
आवेग	<math>J=\int f dt</math>	<math>J=\int \tau dt</math>
कार्य	<math>W = \int f dx = \mathbf{d} \cdot \mathbf{f}</math>	<math>W = \int \tau d\theta </math>
शक्ति	<math> P = dW/dt = fv </math>	<math> P = dW/dt = \tau\omega</math>
गतिज ऊर्जा	<math>K = mv^2/2 = P^2/2m </math>	<math>K = I \ w^2 / 2 = \Sigma R^2m</math>
न्यूटन का तीसरा नियम	<math> f_{ab} = - f_{ba} </math>	<math>\tau_{ab} = -\tau_{ba} </math>

Every conservative force has a potential energy. By following two principles one can consistently assign a non-relative value to U:

• Wherever the force is zero, its potential energy is defined to be zero as well.
• Whenever the force does work, potential energy is lost.

स्थिर त्वरण (Constant acceleration)

Equations in translation and rotation, assuming constant acceleration.

भौतिक राशि	रेखीय गति	घुर्णन गति
विस्थापन	<math>\Delta v = at</math>	<math>\Delta \omega = \alpha t</math>
समय	<math>\Delta(v^2) = 2a\Delta x</math>	<math>\Delta(\omega^2) = 2\alpha\Delta \theta</math>
त्वरण	<math>\Delta x = t\Delta v/2</math>	<math>\Delta \theta = t\Delta \omega/2</math>
प्रा०वेग	<math>\Delta x = -at^2/2 + v_2t</math>	<math>\Delta \theta = -\alpha t^2/2 + \omega_2t</math>
अंतिमवेग	<math>\Delta x = +at^2/2 + v_1t</math>	<math>\Delta \theta = +\alpha t^2/2 + \omega_1t</math>

एकसमान वृत्तीय गति (Uniform circular motion)

uniform circular motion angular to linear displacement	<math>x = \theta r</math>
uniform circular motion angular to linear speed	<math>v = \theta \omega</math>
uniform circular motion angular to linear acceleration normal component	<math>a_r = \omega^2r</math>
uniform circular motion	<math>\mathbf{d} = \mathbf{i}cos\omega t + \mathbf{j}sin\omega t</math>
uniform circular motion tangential speed	<math>\mathbf{v} = \mathbf{d}' = -\omega r (\mathbf{i}\sin\omega t - \mathbf{j}\cos\omega t)</math>
uniform circular motion tangential component, scalar	<math>a_t = \alpha r</math>
uniform circular motion centripetal acceleration	<math>\mathbf{a} = \mathbf{d} = -\omega^2\mathbf{d} = -v^2\mathbf{n}/r</math>
uniform circular motion centripetal acceleration scalar	<math>\alpha=v^2/r</math>
uniform circular motion centripetal force	<math>f = -mv^2/r</math>
uniform circular motion revolution time	<math>T=2\pi r/v</math>

Elasticity

elastic force, lies parallel to spring	<math>f = -kd</math>
elastic potential energy	<math>U=kx^2/2</math>
elastic work, positive when relaxes	<math>W = -k\Delta(x^2)/2</math>

घर्षण (Friction)

normal force	<math>f_n = \mathbf{f}\cdot\mathbf{n}</math>
static friction maximum, lies tangent to the surface	<math>f=\mu_sf_n</math>
kinetic friction, lies tangent to the surface	<math>f=\mu_kf_n</math>
drag force, tangent to the path	<math>f =\mu_d\rho a v^2/2</math>
terminal velocity	<math>v_t=\sqrt{2fg/(\mu_d\rho A)}</math>
friction creates heat and sound	<math>\Delta E = f_kd</math>

प्रतिबाधा एवं विकृत्ति (Stress and strain)

stress	<math></math>
strain	<math></math>
modulus of elasticity	<math>\lambda = {stress}/{strain}</math>
yield strength	<math></math>
ultimate strength	<math></math>
Young's modulus	<math>F/A = E\Delta L/L</math>
shear modulus	<math>F/A = G\Delta x/L</math>
bulk modulus	<math>F/A = B\Delta V/V</math>

अन्य

inertial frames	<math>x_{PA} = x_{PB} + x_{AB}</math>
. . .	<math>v_{PA} = v_{PB} + v_{AB}</math>
. . .	<math>a_{PA} = a_{PB} + 0</math>
trajectory	<math>y=x\tan\theta-gx^2/2(V_0\cos\theta)^2</math>
flight distance	<math>v_0^2\sin{2\theta}/g</math>
tension, lies within the cord	<math>f_t = f</math>
mechanical energy	<math> E_{mec}=K + U</math>
mechanical energy is conserved	<math> \Delta E_{mec} = 0</math> when all forces are conservative
thrust	<math>t = Rv_{rel}=ma</math>
ideal rocket equation	<math>\Delta v = ln(m_i/m_f)v_{rel}</math>
parallel axis theorem	<math>I = I_{com} + mr^2</math>
list of moments of inertia
indeterminate systems

द्रब्यमान केन्द्र एवं संघट्ट (Center of mass and collisions)

center of mass COM	<math>\mathbf{r}_{com}=M^{-1}\Sigma m_i \mathbf{r}_i</math>
. . .	<math>x_{com}=M^{-1}\int x dm, \cdots</math>
for constant density:	<math>x_{com}=V^{-1}\int x dV, \cdots</math>
COM is in all planes of symmetry	<math></math>
elastic collision	<math>\Delta E_k = 0</math>
inelastic collision	<math>\Delta E_k = </math>maximum
conservation of momentum in a two body collision	<math>\mathbf{P}_{1i}+\mathbf{P}_{2i}=\mathbf{P}_{1f}+\mathbf{P}_{2f} </math>
system COM remains inert	<math>\mathbf{v}_{com}={(\mathbf{P}_{1i}+\mathbf{P}_{2i})\over(M_1+M_2)} = const</math>
elastic collision, 1D, M2 stationary	<math>v_{1f}={(m_1 - m_2)\over(m_1 + m_2)}v_{1i}</math>
. . .	<math>v_{2f}={(2m_1)\over(m_1 + m_2)}v_{1i}</math>

चिकने तल पर लुढ़कना (Smooth rolling)

rolling distance	<math>x_{arc}=R\theta</math>
rolling distance ?	<math>x_{com}=R\alpha</math>
rolling velocity	<math>v_{com}=R\omega</math>
rolling ?	<math>K = I_{com}\omega^2/2 + Mv^2_{com}/2</math>
rolling down a ramp along axis x	<math>a_{com,x}=-\frac{g\sin\theta}{1+I_{com}/MR^2}</math>

उष्मागतिकी (Thermodynamics)

Zeroth Law of Thermodynamics	<math>(A = B) \land (B=C) \Rightarrow A=C</math> (where "=" denotes systems in thermal equilibrium
First Law of Thermodynamics	<math>\Delta E_{int} = Q + W</math>
Second Law of Thermodynamics	<math>\Delta S \ge 0</math>
Third Law of Thermodynamics	<math>S = S_{structural} + CT</math>
temperature	<math>T</math>
molecules	<math>N</math>
degrees of freedom	<math>f</math>
heat	<math>Q</math>, <math>\Delta E</math> due to <math>\Delta T</math> (energy)
thermal mass (extensive property)	<math>C_{th} = Q/\Delta T</math>
specific heat capacity (bulk property)	<math>c_{th} = Q/\Delta Tm</math>
enthalpy of vaporization	<math>L_v = Q/m</math>
enthalpy of fusion	<math>L_f = Q/m</math>
thermal conductivity	<math>\kappa</math>
thermal resistance	<math>R=L/ \kappa</math>
thermal conduction rate	<math>P = Q/t = A(T_H - T_C)/R</math>
thermal conduction rate through a composite slab	<math>P = Q/t = A(T_H - T_C)/\Sigma(R_i)</math>
linear coefficient of thermal expansion	<math> dL/dt = \alpha L</math>
volume coefficient of thermal expansion	<math>dV/dt = 3 \alpha V </math>
Boltzmann constant	<math>k</math> (energy)/(temperature)
Stefan-Boltzmann constant	<math>\sigma</math> (power)/(area)(temp)^4
thermal radiation	<math>P = \sigma \epsilon A T ^4_{sys}</math>
thermal absorption	<math>P = \sigma \epsilon A T ^4_{env}</math>
adiabatic	<math>\Delta Q = 0 </math>
ideal gas law	<math>PV = kTN</math>
work, constant temperature	<math>W=kTNln(V_f/V_i)</math>
work due to gas expansion	<math>W = \int_{i}^{f}pdV</math>
. . . adiabatic	<math>\Delta E_{int} = W</math>
. . . constant volume	<math>\Delta E_{int} = Q</math>
. . . free expansion	<math>\Delta E_{int} = 0</math>
. . . closed cycle	<math>Q + W = 0</math>
work, constant volume	<math>W=0</math>
work, constant pressure	<math>W=p\Delta V</math>
translational energy	<math>E_{k,avg} = kTf/2</math>
internal energy	<math>E_{int} = NkTf/2</math>
mean speed	<math>v_{avg}= \sqrt{(kT/m)(8/\pi)}</math>
mode speed	<math>v_{prb} = \sqrt{(kT/m)2}</math>
root mean square speed	<math>v_{rms} = \sqrt{(kT/m)3}</math>
mean free path	<math>\lambda = 1/(\sqrt{2} \pi d^2 N / V)</math>?
Maxwell–Boltzmann distribution	<math>P(v)=4\pi(m/(2\pi kT))^{3/2}V^2e^{-(mv^2/(2kT))}</math>
molecular specific heat at a constant volume	<math>C_V = Q/(N\Delta T)</math>
?	<math>\Delta E_{int} = NC_V \Delta T</math>
molecular specific heat at a constant pressure	<math>C_p = Q/(N\Delta T)</math>
?	<math>W = p \Delta V = Nk \Delta T</math>
?	<math>k = C_p - C_V</math>
adiabatic expansion	<math>pV^{\gamma} = constant</math>
adiabatic expansion	<math>TV^{\gamma - 1} = constant</math>
multiplicity of configurations	<math>W = N!/n_1!n_2!</math>
microstate in one half of the box	<math>n_1, n_2</math>
Boltzmann's entropy equation	<math>S = klnW</math>
irreversibility	<math></math>
entropy	<math>S = - k\sum_i P_i \ln P_i \!</math>
entropy change	<math>\Delta S = \int_i^f(1/T)dQ \approx Q/T_{avg}</math>
entropy change	<math>\Delta S = kNln(V_f/V_i) + NC_Vln(T_f/T_i)</math>
entropic force	<math>f = -TdS/dx</math>
engine efficiency	W|/|Q_H|</math>
Carnot engine efficiency	Q_H|-|Q_L|)/|Q_H| = (T_H-T_L)/T_H</math>
refrigeration performance	Q_L|/|W|</math>
Carnot refrigeration performance	Q_L|/(|Q_H|-|Q_L|) = T_L/(T_H-T_L)</math>

तरंग

torsion constant	<math>\kappa = -\tau / \theta</math>
phasor	<math></math>
node	<math></math>
antinode	<math></math>
period	<math>T</math>
amplitude	<math>x_m</math>
decibel	<math>dB</math>
frequency	<math>f = 1/T = \omega /(2\pi)</math>
angular frequency	<math>\omega = 2\pi f = 2\pi / T</math>
phase angle	<math>\phi</math>
phase	<math>(\omega t + \phi)</math>
damping force	<math>f_d = -bv</math>
phase	<math>ky -\omega t</math>
wavenumber	<math>k</math>
phase constant	<math>\phi</math>
linear density	<math>\mu</math>
harmonic number	<math>n</math>
harmonic series	<math>f = v/\lambda = nv/(2L) </math>
wavelength	<math>\lambda = k/(2\pi)</math>
bulk modulus	<math>B = \Delta p /(\Delta V / V)</math>
path length difference	<math>\Delta L</math>
resonance	<math>\omega_d = \omega</math>
phase difference	<math>\phi = 2 \pi \Delta L / \lambda </math>
fully constructive interference	<math>\Delta L/\lambda = n</math>
fully destructive interference	<math>\Delta L/\lambda = n+0.5</math>
sound intensity	<math>I = P/A = \rho v \omega^2 s^2_m/2</math>
sound power source	<math>P_s</math>
sound intensity over distance	<math>I = P_s/(4\pi r^2)</math>
sound intensity standard reference	<math>I_0</math>
sound level	<math>\Beta = (10 dB)log(I/I_0)</math>
pipe, two open ends	<math>f=v/\lambda = nv/(2L)</math>
pipe, one open end	<math>f = v/\lambda = nv/(4L)</math> for n odd
beats	<math>s(t) = [2s_m\cos\omega ' t ] \cos \omega t</math>
beat frequency	<math>f_{beat} = f_1 - f_2</math>
Doppler effect	<math>f' = f(v+-v_D)/(v+-v_S)</math>
sonic boom angle	<math>\sin \theta = v/v_s</math>
average wave power	<math>P_{avg}=\mu v \omega^2 x_m^2/2</math>
pressure amplitude	<math>\Delta p_m = (v\rho \omega)x_m</math>
wave equation	<math>\frac{\partial y}{\partial x^2} = \frac{1}{v^2} \frac{\partial ^2 y}{\partial t^2}</math>
wave superposition	<math>x'(y,t) = x_1(y,t) + x_2(y,t)</math>
wave speed	<math>v = \omega/k = \lambda/T = \lambda f</math>
speed of sound	<math>v = \sqrt{B/ \rho }</math>
wave speed on a stretched string	<math>v=\sqrt{f_t/\mu}</math>
angular frequency of an angular simple harmonic oscillator	<math>\omega = \sqrt{I/\kappa}</math>
angular frequency of a low amplitude simple pendulum	<math>\omega = \sqrt{L/g}</math>
angular frequency of a low amplitude physical pendulum	<math>\omega = \sqrt{I/mgh}</math>
angular frequency of a linear simple harmonic oscillator	<math>\omega = \sqrt{k/m} </math>
angular frequency of a linear damped harmonic oscillator	<math>\omega ' = \sqrt{(k/m)-(b^2/4m^2)}</math>
wave displacement	<math>x(t)=x_m\cos(\omega t + \phi)</math>
wave displacement when damped	<math>x(t)=x_m\cos(\omega 't+\phi)(e^{-bt/2m})</math>
wave velocity	<math>v(t)=x_m\sin(\omega t + \phi)(- \omega)</math>
wave acceleration	<math>a(t)=x_m\cos(\omega t + \phi)(- \omega^2 )</math>
transverse wave	<math>x(y,t) = x_m\sin(ky-\omega t)</math>
wave traveling backwards	<math>x(y,t) = x_m\sin(ky+\omega t)</math>
resultant wave	<math>x'(y,t) = x_m\sin(ky-\omega t + \phi/2)(2\cos\phi/2)</math>
standing wave	<math>x'(y,t) = \cos(\omega t)(2y\sin ky)</math>
sound displacement function	<math>x(y,t) = x_m\cos(ky-\omega t)</math>
sound pressure-variation function	<math>\Delta p(y,t) = \sin(ky-\omega t)\Delta p_m</math>
potential harmonic energy	<math>E_U(t) = kx^2/2 = kx_m^2\cos^2(\omega t + \phi)/2</math>
kinetic harmonic energy	<math>E_K(t) = kx^2/2 = kx_m^2\sin^2(\omega t + \phi)/2</math>
total harmonic energy	<math>E(t) = kx_m^2/2 = E_U + E_K</math>
damped mechanical energy	<math>E_{mec}(t) = ke^{-bt/m}x^2_m/2</math>

गुरुत्वाकर्षण (Gravitation)

gravitational constant	<math>G</math> (force)(distance/mass)^2
gravitational force	<math>f_G = Gm_1m_2/r^2</math>
superposition applies	<math>\mathbf{F} = \Sigma \mathbf{F}_i = \int d\mathbf{F}</math>
gravitational acceleration	<math>a_g = Gm/r^2</math>
free fall acceleration	<math>a_f = a_g - \omega^2R</math>
shell theorem for gravitation
potential energy from gravity	<math>U = -Gm_1m_2/r \approx ma_gy</math>
escape speed	<math>v = \sqrt{2Gm/r}</math>
Kepler's law 1	planets move in an ellipse, with the star at a focus
Kepler's law 2	<math>A = 0</math>
Kepler's law 3	<math>T^2 = (4\pi^2/Gm)r^3</math>
orbital energy	<math>E = - Gm_1m_2/a2</math>
standard gravity	<math> a_g = Gm_{Earth}/r_{Earth}^2 \approx 9.81m/s^2</math>
weight, points toward the center of gravity	<math>f_g = -f_n = mg </math>
path independence	<math>W_{ab,1}=W_{ab,2}=\cdots</math>
Einstein field equations	<math>R_{\mu \nu} - {1 \over 2}g_{\mu \nu}\,R + g_{\mu \nu} \Lambda = {8 \pi G \over c^4} T_{\mu \nu}</math>

तरलगतिकी (Fluid dynamics)

density	<math>\rho = \Delta m / \Delta V</math>
pressure	<math>p = \Delta F / \Delta A</math>
pressure difference	<math>\Delta p = \rho g\Delta y</math>
pressure at depth	<math>p = p_0 + \rho gh</math>
barometer versus manometer	<math></math>
Pascal's principle	<math></math>
Archimedes' Principle	<math></math>
buoyant force	<math>F_b = m_fg</math>
gravitational force when floating	<math>F_g = F_b</math>
apparent weight	<math>weight_{app} = weight - F_b</math>
ideal fluid	<math></math>
equation of continuity	<math>R_V = Av =</math> constant
Bernoulli's equation	<math>p + \rho v^2/2 + \rho gy =</math> constant

विद्युतचुम्बकत्व (Electromagnetism)

Lorentz force	<math>\mathbf{F} = q (\mathbf{E} + \mathbf{v} \times \mathbf{B})</math>
Gauss' law	<math>\oint\mathbf{E}\cdot d \mathbf{A} = \Phi_E = q_{enc}/\epsilon_0</math>
Gauss' law for magnetic fields	<math>\oint \mathbf{B} \cdot d \mathbf{A} = \Phi_B = 0</math>
Faraday's law of induction	<math>\oint\mathbf{E}\cdot d\mathbf{s} = -d\Phi_B/dt = -\mathcal{E}</math>
Ampere-maxwell law	<math>\oint \mathbf{B} \cdot d\mathbf{s} = \mu_0(i_{enc} + i_{d,enc})</math>
elementary charge	<math>e</math>
electric charge	<math>q = ne</math>
conservation of charge	<math>\Delta q = 0</math>
linear charge density	<math>\lambda = q/l^1</math>
surface charge density	<math>\sigma = q/l^2</math>
volume charge density	<math>\rho = q/l^3</math>
electric constant	<math>\epsilon_0</math> (time)^2(charge)^2/(mass)(volume)
magnetic constant	<math>\mu_0</math> (force)(time)^2/(charge)^2
Coulomb's law	<math>F = q_1q_2/(4\pi\epsilon_0)r^2</math>
electric field	<math>\mathbf{E} =\mathbf{F}/q</math>
electric field lines	end at a negative charge
Gaussian surface	<math>\mathbf{A}</math>
flux notation implies a normal unit vector	<math>\cdot d \mathbf{A} \to \cdot \mathbf{n} d \mathbf{A}</math>
electric flux	<math>\Phi_E = \oint\mathbf{E}\cdot d \mathbf{A}</math>
magnetic flux	<math>\Phi_B = \int \mathbf{B}\cdot d\mathbf{A}</math>
magnetic flux given assumptions	<math>\Phi_B = BA</math>
dielectric constant	<math>\kappa \ge 1</math>
dielectric	<math>\epsilon_0 \to \epsilon_0\kappa</math>
Gauss' law with dialectric	<math>q_{enc} = \epsilon_0 \oint \kappa\mathbf{E}\cdot d \mathbf{A}</math>
Biot-Savart law	<math> \mathbf{B} = \int\frac{\mu_0}{4\pi}\ \frac{(id\mathbf{s}) \times \mathbf{r}}{r^3},</math>
Lenz's law	induced current always opposes its cause
inductance (with respect to time)	<math>L=-\mathcal{E}/q</math>
inductance from coils	<math>L=N\Phi_B/i</math>
inductance of a solenoid	<math>L/l=\mu_0n^2A</math>
displacement current	<math>i_d = \epsilon_0 d\Phi_E/dt</math>
displacement vector	<math>\mathbf{d}</math>
electric dipole moment	<math>\mathbf{p} = q\mathbf{d}</math>
electric dipole torque	<math>\mathbf{\tau}=\mathbf{p}\times\mathbf{E}</math>
electric dipole potential energy	<math>U = -\mathbf{p}\cdot\mathbf{E}</math>
magnetic dipole moment of a coil, magnitude only	<math>\mu=iNA</math>
magnetic dipole moment torque	<math>\mathbf{\tau}=\mathbf{\mu}\times\mathbf{B}</math>
magnetic dipole moment potential energy	<math>U=-\mathbf{\mu}\cdot\mathbf{B}</math>
electric field accelerating a charged mass	<math>a = qE/m</math>
electric field of a charged point	<math>E = q / \epsilon_0 4 \pi r^2 \hat{r} </math>
electric field of a dipole moment	<math>E = p / \epsilon_0 2 \pi z^3 </math>
electric field of a charged line	<math>E = \lambda / \epsilon_0 2\pi r</math>
electric field of a charged ring	<math>E = qz/\epsilon_04\pi(z^2 + R^2)^{3/2}</math>
electric field of a charged conducting surface	<math>E = \sigma / \epsilon_0</math>
electric field of a charged non-conducting surface	<math>E = \sigma /\epsilon_0 2</math>
electric field of a charged disk	<math>E = \sigma (1 - z)/ \epsilon_0 2 \sqrt{z^2 + R^2}</math>
electric field outside spherical shell r>=R	<math>E = q/\epsilon_0 4 \pi r^2</math>
electric field inside spherical shell r<R	<math>E = 0</math>
electric field of uniform charge r<=R	<math>E = qr/\epsilon_0 4 \pi R^3</math>
electric field energy density	<math>u = \epsilon_0 E^2/2</math>
electric potential versus electric potential energy	(energy)/(charge) versus (energy)
electric potential energy	<math>U = - W_{\infty}</math>
electric potential	<math>V = -W_{\infty}/q = U/q</math>
electric potential difference	<math>\Delta V = -W/q = \Delta U/q</math>
electric potential from electric field	<math>\Delta V = -\int_i^f \mathbf{E}\cdot d\mathbf{s}</math>
electric field from electric potential	<math>\nabla V = -\mathbf{E}</math>
electric potential of a charged point	<math>V = q/\epsilon_0 4 \pi r</math>
electric potential of a set of charged points	<math>V = \Sigma V_i = (1/\epsilon_0 4 \pi) \Sigma q_i/r_i</math>
electric potential of a dipole	<math>V = p\cos\theta/\epsilon_0 4 \pi r^2</math>
electric potential of continuous charge	<math>V = \int dV = (1/\epsilon_0 4 \pi)\int dq/r</math>
electric potential energy of a pair of charged points	<math>Vq_2 = U = W = q_1q_2/\epsilon_04\pi r</math>
capacitance	<math>C = q/V</math> (charge)^2/(energy)
capacitance of parallel plates	<math>C = \epsilon_0A/d</math>
capacitance of a cylinder	<math>C = \epsilon_0 2 \pi L/\ln(b/a)</math>
capacitance of a sphere	<math>C = \epsilon_0 4 \pi ba/(b-a)</math>
capacitance of an isolated sphere	<math>C = \epsilon_0 4 \pi R</math>
capacitors in parallel	<math>C_{eq}^{+1} = \Sigma C_i^{+1}</math>
capacitors in series	<math>C_{eq}^{-1} = \Sigma C_i^{-1}</math>
capacitor potential energy	<math>U=q^2/C2 = CV^2/2</math>
current	<math>i = dq/dt</math>
drift speed	<math>\mathbf{v}_d</math>
current density	<math>\mathbf{J} = ne\mathbf{v}_d/m^3</math>
current density magnitude	<math>J = i/A</math>
current density to get current	<math>i = \int JdA</math>
resistance	<math>R = V/i</math>
resistivity	<math>\rho = \mathbf{E}/\mathbf{J}</math>
resistivity temperature coefficient	<math>\alpha</math>
resistivity across temperature	<math>\rho - \rho_0 = \rho_0\alpha(T-T_0)</math>
resistivity and resistance	<math>R A = \rho L</math>
electrical conductivity	<math>\sigma = 1/\rho = \mathbf{J}/\mathbf{E}</math>
resistor power dissipation	<math>P = i^2R = V^2/R</math>
internal resistance	<math>i = \mathcal{E}/(R+r)</math>
resistors in series	<math>R_{eq}^{+1}=\Sigma R_i^{+1}</math>
resistors in parallel	<math>R_{eq}^{-1} =\Sigma R_i^{-1}</math>
Kirchoff's current law	<math>i_{in} = i_{out}</math>
Ohm's law	<math>V=iR</math>
emf	<math>\mathcal{E} = dW/dq = iR</math>
emf rules	loop, resistance, emf
electrical power	<math>P=iV</math>
emf power	<math>P_{emf} = i\mathcal{E}</math>
electric potential difference across a real battery	<math>p = \mathcal{E} - iR</math>
magnetic field force on a moving charge	<math>\mathbf{F}_B = q\mathbf{v}\times\mathbf{B}</math>
magnetic field force on a current	<math>\mathbf{F}_B=i\mathbf{L}\times\mathbf{B}</math>
Hall effect	<math>n = Bi/Vle</math>
circulating charged particle	q|vB=mv^2/r</math>
cyclotron resonance condition	<math>f = f_{osc}</math>
magnetic field of a line	<math>B = \mu_0i/2\pi R</math>
magnetic field of a ray	<math>B=\mu_0i/4\pi R</math>
magnetic field at the center of a circular arc	<math>B=\mu_0i\phi/4\pi R</math>
magnetic field of a solenoid	<math>B=\mu_0in</math>
magnetic field of a toroid	<math>B=\mu_0iN/2\pi r</math>
magnetic field of a current carrying coil	<math>\mathbf{B}=\mu_0\mathbf{\mu}/2\pi z^3</math>
self induction of emf	<math>\mathcal{E}_L = -Ldi/dt</math>
magnetic energy	<math>U_B=Li^2/2</math>
magnetic energy density	<math>u_B=B^2/2\mu_0</math>
mutual induction	<math>\mathcal{E}_1=-Mdi_2/dt,\mathcal{E}_2=-Mdi_1/dt</math>
transformation of voltage	<math>V_s N_p = V_p N_s</math>
transformation of current	<math>I_s N_s = I_p N_p</math>
transformation of reistance	<math>R_{eq} = (Np/Ns)^2R</math>
induced magnetic field inside a circular capacitor	<math>B = (\mu_0i_d/2\pi R^2)r</math>
induced magnetic field outside a circular capacitor	<math>B = \mu_0i_d/2\pi rr</math>
RC circuit ODE with respect to time	<math>Rq' + C^{-1}q=\mathcal{E}</math>
RC circuit capacitive time constant	<math>\tau = RC</math>
RC circuit charging a capacitor	<math>q = C\mathcal{E}(1-e^{-t/RC})</math>
RL circuit ODE with respect to time	<math>Li+Ri'=\mathcal{E}</math>
RL circuit time constant	<math>\tau_L=L/R</math>
RL circuit rise of current	<math>i = \mathcal{E}/R(1-e^{-t/\tau_L})</math>
RL circuit decay of current	<math>i=\mathcal{E}e^{-t/\tau_L}/R=i_0e^{-t/\tau_L}</math>
LC circuit ODE with respect to time	<math>Lq+C^{-1}q = \mathcal{E}</math>
LC circuit	<math>\omega = 1/\sqrt{LC}</math>
LC circuit charge	<math>q = Qcos(\omega t + \phi)</math>
LC circuit current	<math>i=-\omega Q sin(\omega t + \phi)</math>
LC circuit electrical potential energy	<math>U_E=q^2/2C=Q^2cos^2(\omega t + \phi)/2C</math>
LC circuit magnetic potential energy	<math>U_B=Q^2sin^2(\omega t + \phi)/2C</math>
RLC circuit ODE with respect to time	<math>Lq + Rq' +C^{-1}q = \mathcal{E} </math>
RLC circuit charge	<math>q = QeT^{-Rt/2L}cos(\omega't+\phi)</math>
resistive load	<math>V_R=I_RR</math>
capacitive load	<math>V_C = I_C X_C</math>
inductive load	<math>V_L = I_L X_L</math>
resistive reactance	<math>X_R = ?</math>
capacitive reactance	<math>X_C = 1/\omega_d C</math>
inductive reactance	<math>X_L = \omega_d L</math>
phase constant	<math>tan\phi=X_L - X_C /R</math>
electromagnetic resonance	<math>\omega_d = \omega = 1/\sqrt{LC}</math>
AC current	<math>I_{rms}=I/\sqrt{2}</math>
AC voltage	<math>V_{rms}=V/\sqrt{2}</math>
AC emf	<math>\mathcal{E}_{rms}=\mathcal{E}_m/\sqrt{2}</math>
AC power	<math>P_{avg}=\mathcal{E}I_{rms}cos\phi</math>

प्रकाश (Light)

electric light component	<math>E = E_m sin(kx-\omega t)</math>
magnetic light component	<math>B = B_m sin(kx-\omega t)</math>
speed of light	<math>c = 1/\sqrt{\mu_0\epsilon_0} = E/B</math>
Poynting vector	<math>\mathbf{S} = \mu_0^{-1}\mathbf{E}\times\mathbf{B}</math>
Poynting vector magnitude	<math>S = EB/\mu_0 = E^2/c\mu_0</math>
rms electric field of light	<math>E_{rms} = E/\sqrt{2}</math>
light intensity	<math>I = E^2_{rms}/c\mu_0</math>
light intensity at the sphere	<math>I = P_s/4\pi r^2</math>
radiation momentum with total absorption (inelastic)	<math>\Delta p = \Delta U/c</math>
radiation momentum with total reflection (elastic)	<math>\Delta p = 2 \Delta U/c</math>
radiation pressure with total absorption (inelastic)	<math>p_r = I/c</math>
radiation pressure with total reflection (elastic)	<math>p_r = 2I/c</math>
intensity from polarizing unpolarized light	<math>I = I_0/2</math>
intensity from polarizing polarized light	<math>I = I_0cos^2\theta</math>
index of refraction of substance f	<math>n_f = c/v_f</math>
angle of reflection	<math>\theta_1=\theta_2</math>
angle of refraction	<math>n_1sin\theta_1 = n_2sin\theta_2</math>
angle of total reflection	<math>\theta_c = sin^{-1}n_2/n_1</math>
angle of total polarisation	<math>\theta_B = tan^{-1}n_2/n_1</math>
image distance in a plane mirror	<math>d_i = -d_o</math>
image distance in a spherical mirror	<math>n_1/d_o + n_2/d_i = (n_2 - n_1)/r</math>
spherical mirror focal length	<math>f =r/2</math>
spherical mirror	<math>1/d_o + 1/d_i = 1/f</math>
lateral magnification m and h negative when upside down	<math>m=h_i/h_o = -d_i/d_o</math>
lens focal length	<math>1/f = 1/d_o +1/d_i</math>
lens focal length from refraction indexes	<math>1/f = (n_{lens}/n_{med}-1)(1/r_1 - 1/r_2)</math>
path length difference	<math>\Delta L = d sin\theta</math>
double slit minima	<math>d sin\theta = (N + 1/2)\lambda</math>
double slit maxima	<math>d sin\theta = N\lambda</math>
double-slit interference intensity	<math>I = 4I_0cos^2(\pi d sin\theta / \lambda)</math>
thin film in air minima	<math>(N + 0/2)\lambda/n_2</math>
thin film in air maxima	<math>2L = (N + 1/2)\lambda/n_2</math>
single-slit minima	<math>a sin \theta = N\lambda</math>
single-slit intensity	<math>I(\theta)=I_0(sin\alpha/\alpha)^2</math>
double slit intensity	<math>I(\theta) = I_0(cos^2\Beta)(sin\alpha/\alpha)^2</math>
. . .	<math>\alpha = \pi a sin\theta/\lambda</math>
circular aperture first minimum	<math>sin\theta = 1.22\lambda/d</math>
Rayleigh's criterion	<math>\theta_R = 1.22\lambda/d</math>
diffraction grating maxima lines	<math>dsin\theta = N\lambda</math>
diffraction grating half-width	<math>\Delta\theta_{hw} = \lambda/Ndcos\theta</math>
diffraction grating dispersion	<math>D=N/d cos\theta</math>
diffraction grating resolving power	<math>R=Nn</math>
diffraction grating lattice distance	<math>d = N\lambda/2sin\theta</math>

विशिष्ट आपेक्षिकता (Special Relativity)

Lorentz factor	<math>\gamma = 1/\sqrt{1-(v/c)^2}</math>
Lorentz transformation	<math>t' = \gamma(t-xv/c^2)</math>
. . .	<math>x'=\gamma(x-vt)</math>
. . .	<math>y' = y</math>
. . .	<math>z' = z</math>
time dilation	<math>\Delta t = \gamma \Delta t_0</math>
length contraction	<math>L = L_0/\gamma</math>
relativistic Doppler effect	<math>f=f_0\sqrt{1-(v/c)/1+(v/c)}</math>
Doppler shift	\Delta\lambda|c/\lambda_0</math>
momentum	<math>\mathbf{p}=\gamma m\mathbf{v}</math>
rest energy	<math>E_0 = mc^2</math>
total energy	<math>E = E_0 + K = mc^2 + K = \gamma mc^2 = \sqrt{(pc)^2 + (mc^2)^2}</math>
Energy Removed	<math>Q = -\Delta mc^2</math>
kinetic energy	<math>K = E - mc^2 = \gamma mc^2 - mc^2 = mc^2(\gamma -1)</math>

कण भौतिकी (Particle Physics)

standard model	see 4x4 chart of particles
Planck's constant	<math>h</math>, in energy/frequency
Reduced Planck's constant	<math>\hbar = h/2\pi</math>, in energy/frequency
Planck–Einstein equation	<math>E = hf</math>
threshold frequency	<math>f_0</math>
work function	<math>\Phi = hf_0</math>
photoelectric kinetic energy	<math>K_{max} = hf - \Phi</math>
photon momentum	<math>p = hf/c = h/\lambda</math>
de Broglie wavelength	<math>\lambda = h/p</math>
Schrodinger's equation	<math>i\hbar\frac{\partial}{\partial t} \Psi(\mathbf{r},\,t) = \hat H \Psi(\mathbf{r},t)</math>
Schrodinger's equation one dimensional motion	<math>d^2\psi/dx^2 + 8\pi^2m[E-U(x)]\psi/h^2 = 0</math>
Schrodinger's equation free particle	<math>d^2\psi/dx^2 + k^2\psi = 0</math>
Heisenberg's uncertainty principle	<math>\Delta x \cdot \Delta p_x \ge \hbar </math>
infinite potential well	<math>E_n = (hn/2L)^2/2m</math>
wavefunction of a trapped electron	<math>\psi_n(x) = A sin(n\pi x/L)</math>, for positive int n
wavefunction probability density	<math>p(x) = \psi^2_n(x)dx</math>
normalization	<math>\int \psi^2_n(x)dx = 1</math>
hydrogen atom orbital energy	<math>E_n = -me^4/8\epsilon_0^2h^2n^2 = 13.61eV/n^2</math>, for positive int n
hydrogen atom spectrum	<math>1/\lambda = R(1/n^2_{low} - 1/n^2_{high})</math>
hydrogen atom radial probability density	<math>P(r) = 4r^2/a^3e^{2r/a}</math>
spin projection quantum number	<math>m_s \in \{-1/2,+1/2\}</math>
orbital magnetic dipole moment	<math>\mathbf{\mu}_{orb} = -e\mathbf{L}/2m</math>
orbital magnetic dipole moment components	<math>\mathbf{\mu}_{orb,z} = -m_\mathcal{L}\mu_B</math>
spin magnetic dipole moment	<math>\mathbf{\mu_s} = -e\mathbf{S}/m = gq\mathbf{S}/2m</math>
orbital magnetic dipole moment	<math>\mathbf{\mu}_{orb}=-e\mathbf{L}_{orb}/2m</math>
spin magnetic dipole moment potential	<math>U = -\mathbf{\mu}_s\cdot\mathbf{B}_{ext} = -\mu_{s,z}B_{ext}</math>
orbital magnetic dipole moment potential	<math>U = -\mathbf{\mu}_{orb}\cdot\mathbf{B}_{ext} = -\mu_{orb,z}B_{ext}</math>
Bohr magneton	<math>\mu_B = e\hbar/2m</math>
angular momentum components	<math>L_z = m\mathcal{L}\hbar</math>
spin angular momentum magnitude	<math>S = \hbar\sqrt{s(s+1)}</math>
cutoff wavelength	<math>\lambda_{min} = hc/K_0</math>
density of states	<math>N(E) = 8\sqrt{2}\pi m^{3/2}E^{1/2}/h^3</math>
occupancy probability	<math>P(E) = 1/(e^{(E-E_F)/kT}+1)</math>
Fermi energy	<math>E_F = (3/16\sqrt{2}\pi)^{2/3}h^2n^{2/3}m</math>
mass number	<math>A = Z+N</math>
nuclear radius	<math>r=r_0A^{1/3}, r_0 \approx 1.2fm</math>
mass excess	<math>\Delta = M - A</math>
radioactive decay	<math>N = N_0e^{-\lambda t}</math>
Hubble constant	<math>H = 71.0km/s</math>
Hubble's law	<math>v=Hr</math>
conservation of lepton number	<math></math>
conservation of baryon number	<math></math>
conservation of strangeness	<math></math>
eightfold way	<math></math>
weak force	<math></math>
strong force	<math> \begin{align} \mathcal{L}_\mathrm{QCD} & = \bar{\psi}_i\left(i \gamma^\mu (D_\mu)_{ij} - m\, \delta_{ij}\right) \psi_j - \frac{1}{4}G^a_{\mu \nu} G^{\mu \nu}_a \\ & = \bar{\psi}_i (i \gamma^\mu \partial_\mu - m)\psi_i - g G^a_\mu \bar{\psi}_i \gamma^\mu T^a_{ij} \psi_j - \frac{1}{4}G^a_{\mu \nu} G^{\mu \nu}_a \, \\ \end{align} </math>
Noether's theorem	<math></math>
Electroweak interaction	:<math>\mathcal{L}_{EW} = \mathcal{L}_g + \mathcal{L}_f + \mathcal{L}_h + \mathcal{L}_y.</math> <math>\mathcal{L}_g = -\frac{1}{4}W_a^{\mu\nu}W_{\mu\nu}^a - \frac{1}{4}B^{\mu\nu}B_{\mu\nu}</math> <math>\mathcal{L}_f = \overline{Q}_i iD\!\!\!\!/\; Q_i+ \overline{u}_i^c iD\!\!\!\!/\; u^c_i+ \overline{d}_i^c iD\!\!\!\!/\; d^c_i+ \overline{L}_i iD\!\!\!\!/\; L_i+ \overline{e}^c_i iD\!\!\!\!/\; e^c_i </math> <math>\mathcal{L}_h = |D_\mu h|^2 - \lambda \left(|h|^2 - \frac{v^2}{2}\right)^2</math> <math>\mathcal{L}_y = - y_{u\, ij} \epsilon^{ab} \,h_b^\dagger\, \overline{Q}_{ia} u_j^c - y_{d\, ij}\, h\, \overline{Q}_i d^c_j - y_{e\,ij} \,h\, \overline{L}_i e^c_j + h.c.</math>
Quantum electrodynamics	:<math>\mathcal{L}=\bar\psi(i\gamma^\mu D_\mu-m)\psi -\frac{1}{4}F_{\mu\nu}F^{\mu\nu}\;,</math>

क्वांटम यांत्रिकी (Quantum Mechanics)

Postulate 1: State of a system	A system is completely specified at any one time by a Hilbert space vector.
Postulate 2: Observables of a system	A measurable quantity corresponds to an operator with eigenvectors spanning the space.
Postulate 3: Observation of a system	Measuring a system applies the observable's operator to the system and the system collapses into the observed eigenvector.
Postulate 4: Probabilistic result of measurement	The probability of observing an eigenvector is derived from the square of its wavefunction.
Postulate 5: Time evolution of a system	The way the wavefunction evolves over time is determined by Shrodinger's equation.

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