भौतिकी के सूत्र

एसआई उपसर्ग (प्रीफिक्स)

SI उपसर्ग

१०००n	१०n	उपसर्ग	चिन्ह	[१] से लागू	संख्या	दशमलव रूप SI writing style में
१०००८	१०२४	योट्टा-	यो/Y	1991	दस जल्द	१ ००० ००० ००० ००० ००० ००० ००० ०००
१०००७	१०२१	अं-	अं/Z	1991	अंक	१ ००० ००० ००० ००० ००० ००० ०००
१०००६	१०१८	एक्जा-	ए/E	1975	दस शङ्ख	1 ००० ००० ००० ००० ००० ०००
१०००५	१०१५	पद्म-	प/P	1975	पद्म	१ ००० ००० ००० ००० ०००
१०००४	१०१२	टेरा-	टे/T	1960	दस खरब	१ ००० ००० ००० ०००
१०००३	१०९	अर्ब-	अ/G	1960	अरब	१ ००० ००० ०००
१०००२	१०६	अद-	अद/M	1960	अदन्त	१ ००० ०००
१०००१	१०३	सहस्र-	स्र/k	1795	हजार	१ ०००
१०००२/३	१०२	शत-	स/h	1795	सौ	१००
१०००१/३	१०१	दश-	द/da	1795	दस	१०
१०००0	१०0	(none)	(none)	NA	एक	१
१०००−१/३	१०−१	दशि-	दि/d	1795	Tenth	०.१
१०००−२/३	१०−२	शति-	शि/c	1795	Hundredth	०.०१
१०००−१	१०−३	सहस्रि-	स्रि/m	1795	Thousandth	०.०० १
१०००−२	१०−६	सूक्ष्म-	सू/µ	1960[2]	Millionth	०.००० ०००१
१०००−३	१०−९	अर्बि-	इ/n	1960	Billionth	०.००० ००० ००१
१०००−४	१०−१२	फैम्टो-	फ/p	1960	Trillionth	०.००० ००० ००० ००१
१०००−५	१०−१५	पद्मि-	पि/f	1964	Quadrillionth	०.००० ००० ००० ००० ००१
१०००−६	१०−१८	एट्टो-	a	1964	Quintillionth	०.००० ००० ००० ००० ००० ००१
१०००−७	१०−२१	अंकि-	इं/z	1991	Sextillionth	०.००० ००० ००० ००० ००० ००० ००१
१०००−८	१०−२४	योक्टो-	y	1991	Septillionth	०.००० ००० ००० ००० ००० ००० ००० ००१
Notes: १. 1795 की तिथियों से उपरोक्त उपसर्ग प्रयोग में लाए जारहे हैं, जबसे मीट्रिक प्रणाली प्रयोग में आई थी । अन्य तिथियाँ आवश्यक रूप से प्रथम प्रयोग की नहीं रहीं हैं, बलकि CGPM के समझौते द्वारा मान्यता की तिथि हैं जो कि 1889 में हुआ था । 2. 2. "मैक्रॉन" शब्द को CGPM नें १९४८ में अनुमोदित किया, पर १९६७-६८ में उसे रद्द कर दिया।

आधारभूत यांत्रिकी (Fundamentals of Mechanics)

Foundational equations in translation and rotation.

Quantity	Translation	Rotation
time	$t$	$t$
position	$x$	$\theta$ in radians
mass	$m$	$m$
duration	$\Delta t$	$\Delta t$
displacement	$\Delta x$	$\Delta \theta$
conservation of mass	$\Delta m = 0$	$\Delta m = 0$
conservation of energy	$\Delta E = 0$	$\Delta E = 0$
conservation of momentum	$\Delta P = 0$	$\Delta L = 0$
velocity	$v = dx/dt$	$\omega = d\theta/dt$
acceleration	$a = dv/dt$	$\alpha = d\omega/dt$
jerk	$j = da/dt$	$j = d\alpha/dt$
potential energy change	$\Delta U = -W$	$\Delta U = -W$
momentum	$P = mv$	$L = I\omega$ $=|| \mathbf{r} \times \mathbf{P}|| = m|| \mathbf{r} \times \mathbf{v}||$
force	$f = dP/dt = ma = -dU/dx$	$\tau = dL/dt = I\alpha$ $=|| \mathbf{r} \times \mathbf{f}||=m|| \mathbf{r} \times \mathbf{a}||$
inertia	$m = \int dm = \Sigma m_i$	$I = \int r^2 dm = \Sigma r^2m_i$
impulse	$J=\int f dt$	$J=\int \tau dt$
work	$W = \int f dx = \mathbf{d} \cdot \mathbf{f}$	$W = \int \tau d\theta$
power	$P = dW/dt = fv$	$P = dW/dt = \tau\omega$
kinetic energy	$K = mv^2/2 = P^2/2m$	$K = I \ w^2 / 2 = \Sigma R^2m$
Newton's Third Law	$f_{ab} = - f_{ba}$	$\tau_{ab} = -\tau_{ba}$

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.

Quantity	Translation	Rotation
displacement	$\Delta v = at$	$\Delta \omega = \alpha t$
time	$\Delta(v^2) = 2a\Delta x$	$\Delta(\omega^2) = 2\alpha\Delta \theta$
acceleration	$\Delta x = t\Delta v/2$	$\Delta \theta = t\Delta \omega/2$
initial velocity	$\Delta x = -at^2/2 + v_2t$	$\Delta \theta = -\alpha t^2/2 + \omega_2t$
final velocity	$\Delta x = +at^2/2 + v_1t$	$\Delta \theta = +\alpha t^2/2 + \omega_1t$

एकसमान वृत्तीय गति (Uniform circular motion)

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

Elasticity

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

घर्षण (Friction)

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

प्रतिबाधा एवं विकृत्ति (Stress and strain)

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

अन्य

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

द्रब्यमान केन्द्र एवं संघट्ट (Center of mass and collisions)

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

चिकने तल पर लुढ़कना (Smooth rolling)

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

उष्मागतिकी (Thermodynamics)

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

तरंग

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

गुरुत्वाकर्षण (Gravitation)

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

तरलगतिकी (Fluid dynamics)

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

विद्युतचुम्बकत्व (Electromagnetism)

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

प्रकाश (Light)

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

विशिष्ट आपेक्षिकता (Special Relativity)

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

कण भौतिकी (Particle Physics)

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

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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संदर्भ

• Halliday, David (2007). Fundamentals of Physics,. Chichester: John Wiley & Sons. आई॰ऍस॰बी॰ऍन॰ 9780470044742. 
• Zettili, Nouredine (2009). Quantum Mechanics. New York: Wiley. आई॰ऍस॰बी॰ऍन॰ 0470026782. 

बाहरी कड़ियाँ

• Physics formulae at xs4all.nl