भौतिकी के सूत्र

एसआई उपसर्ग (प्रीफिक्स)

SI उपसर्ग

उपसर्ग		10 पर घात के रूप में	दशमलव संख्या के रूप में	शब्दों में		स्वीकरण वर्ष
नाम	चिह्न			भारतीय नाम	यूरोपीय नाम
योट्टा (yotta)	Y	1024	1000000000000000000000000	दस जल्द	septillion	1991
ज़ेट्टा (zetta)	Z	1021	1000000000000000000000	अंक	sextillion	1991
एक्सा (exa)	E	1018	1000000000000000000	महापरार्ध	quintillion	1975
पेटा (peta)	P	1015	1000000000000000	पद्म	quadrillion	1975
टेरा (tera)	T	1012	1000000000000	दस खरब/शंकु	trillion	1960
गिगा (giga)	G	109	1000000000	अरब/शंख	billion	1960
मेगा (mega)	M	106	1000000	दस लक्ष	million	1873
किलो (kilo)	k	103	1000	सहस्र/हजार	thousand	1795
हेक्टो (hecto)	h	102	100	शत/सौ	hundred	1795
डेका (deca)	da	101	10	दस	ten	1795
		100	1	एक	one	–
डेसी (deci)	d	10−1	0.1	दसवाँ	tenth	1795
सेन्टी (centi)	c	10−2	0.01	सौंवा	hundredth	1795
मिली (milli)	m	10−3	0.001	हजारवाँ	thousandth	1795
माइक्रो (micro)	μ	10−6	0.000001	दस-लाखवाँ	millionth	1873
नैनो (nano)	n	10−9	0.000000001	अरबवाँ	billionth	1960
पिको (pico)	p	10−12	0.000000000001	दस-खरबवाँ	trillionth	1960
फेम्टो (femto)	f	10−15	0.000000000000001	पद्मवाँ	quadrillionth	1964
आट्टो (atto)	a	10−18	0.000000000000000001	महापरार्धवाँ	quintillionth	1964
जेप्टो (zepto)	z	10−21	0.000000000000000000001	महाउपाधवाँ	sextillionth	1991
योक्टो (yocto)	y	10−24	0.000000000000000000000001	माधवाँ	septillionth	1991

आधारभूत यांत्रिकी (Fundamentals of Mechanics)

Foundational equations in translation and rotation.

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

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.

भौतिक राशि	रेखीय गति	घुर्णन गति
विस्थापन	$\Delta v=at$	$\Delta \omega =\alpha t$
समय	$\Delta (v^{2})=2a\Delta x$	$\Delta (\omega ^{2})=2\alpha \Delta \theta$
त्वरण	$\Delta x=t\Delta v/2$	$\Delta \theta =t\Delta \omega /2$
प्रा०वेग	$\Delta x=-at^{2}/2+v_{2}t$	$\Delta \theta =-\alpha t^{2}/2+\omega _{2}t$
अंतिमवेग	$\Delta x=+at^{2}/2+v_{1}t$	$\Delta \theta =+\alpha t^{2}/2+\omega _{1}t$

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	$a_{r}=\omega ^{2}r$
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	$f=\mu _{s}f_{n}$
kinetic friction, lies tangent to the surface	$f=\mu _{k}f_{n}$
drag force, tangent to the path	$f=\mu _{d}\rho av^{2}/2$
terminal velocity	$v_{t}={\sqrt {2fg/(\mu _{d}\rho A)}}$
friction creates heat and sound	$\Delta E=f_{k}d$

प्रतिबाधा एवं विकृत्ति (Stress and strain)

stress	${\displaystyle }$
strain	${\displaystyle }$
modulus of elasticity	$\lambda ={stress}/{strain}$
yield strength	${\displaystyle }$
ultimate strength	${\displaystyle }$
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	$\mathbf {r} _{com}=M^{-1}\Sigma m_{i}\mathbf {r} _{i}$
. . .	$x_{com}=M^{-1}\int xdm,\cdots$
for constant density:	$x_{com}=V^{-1}\int xdV,\cdots$
COM is in all planes of symmetry	${\displaystyle }$
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 ?	$K=I_{com}\omega ^{2}/2+Mv_{com}^{2}/2$
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	$(A=B)\land (B=C)\Rightarrow A=C$ (where "=" denotes systems in thermal equilibrium
First Law of Thermodynamics	$\Delta E_{int}=Q+W$
Second Law of Thermodynamics	$\Delta S\geq 0$
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	$P=\sigma \epsilon AT_{sys}^{4}$
thermal absorption	$P=\sigma \epsilon AT_{env}^{4}$
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	$P(v)=4\pi (m/(2\pi kT))^{3/2}V^{2}e^{-(mv^{2}/(2kT))}$
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	${\displaystyle }$
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	$\Delta S=kNln(V_{f}/V_{i})+NC_{V}ln(T_{f}/T_{i})$
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	${\displaystyle }$
node	${\displaystyle }$
antinode	${\displaystyle }$
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	$I=P/A=\rho v\omega ^{2}s_{m}^{2}/2$
sound power source	$P_{s}$
sound intensity over distance	$I=P_{s}/(4\pi r^{2})$
sound intensity standard reference	$I_{0}$
sound level	$\mathrm {B} =(10dB)log(I/I_{0})$
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	$E_{mec}(t)=ke^{-bt/m}x_{m}^{2}/2$

गुरुत्वाकर्षण (Gravitation)

gravitational constant	$G$ (force)(distance/mass)^2
gravitational force	$f_{G}=Gm_{1}m_{2}/r^{2}$
superposition applies	$\mathbf {F} =\Sigma \mathbf {F} _{i}=\int d\mathbf {F}$
gravitational acceleration	$a_{g}=Gm/r^{2}$
free fall acceleration	$a_{f}=a_{g}-\omega ^{2}R$
shell theorem for gravitation
potential energy from gravity	$U=-Gm_{1}m_{2}/r\approx ma_{g}y$
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	$E=-Gm_{1}m_{2}/a2$
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	${\displaystyle }$
Pascal's principle	${\displaystyle }$
Archimedes' Principle	${\displaystyle }$
buoyant force	$F_{b}=m_{f}g$
gravitational force when floating	$F_{g}=F_{b}$
apparent weight	$weight_{app}=weight-F_{b}$
ideal fluid	${\displaystyle }$
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	$F=q_{1}q_{2}/(4\pi \epsilon _{0})r^{2}$
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	$\kappa \geq 1$
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	$L/l=\mu _{0}n^{2}A$
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	$E=qz/\epsilon _{0}4\pi (z^{2}+R^{2})^{3/2}$
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	$Vq_{2}=U=W=q_{1}q_{2}/\epsilon _{0}4\pi r$
capacitance	$C=q/V$ (charge)^2/(energy)
capacitance of parallel plates	$C=\epsilon _{0}A/d$
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	$RA=\rho L$
electrical conductivity	$\sigma =1/\rho =\mathbf {J} /\mathbf {E}$
resistor power dissipation	$P=i^{2}R=V^{2}/R$
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	$B=\mu _{0}i/2\pi R$
magnetic field of a ray	$B=\mu _{0}i/4\pi R$
magnetic field at the center of a circular arc	$B=\mu _{0}i\phi /4\pi R$
magnetic field of a solenoid	$B=\mu _{0}in$
magnetic field of a toroid	$B=\mu _{0}iN/2\pi r$
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	$R_{eq}=(Np/Ns)^{2}R$
induced magnetic field inside a circular capacitor	$B=(\mu _{0}i_{d}/2\pi R^{2})r$
induced magnetic field outside a circular capacitor	$B=\mu _{0}i_{d}/2\pi rr$
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	$i={\mathcal {E}}e^{-t/\tau _{L}}/R=i_{0}e^{-t/\tau _{L}}$
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	$i=-\omega Qsin(\omega t+\phi )$
LC circuit electrical potential energy	$U_{E}=q^{2}/2C=Q^{2}cos^{2}(\omega t+\phi )/2C$
LC circuit magnetic potential energy	$U_{B}=Q^{2}sin^{2}(\omega t+\phi )/2C$
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	$V_{R}=I_{R}R$
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	$I=E_{rms}^{2}/c\mu _{0}$
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	$I=I_{0}cos^{2}\theta$
index of refraction of substance f	$n_{f}=c/v_{f}$
angle of reflection	$\theta _{1}=\theta _{2}$
angle of refraction	$n_{1}sin\theta _{1}=n_{2}sin\theta _{2}$
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	$\Delta L=dsin\theta$
double slit minima	$dsin\theta =(N+1/2)\lambda$
double slit maxima	$dsin\theta =N\lambda$
double-slit interference intensity	$I=4I_{0}cos^{2}(\pi dsin\theta /\lambda )$
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	$asin\theta =N\lambda$
single-slit intensity	$I(\theta )=I_{0}(sin\alpha /\alpha )^{2}$
double slit intensity	$I(\theta )=I_{0}(cos^{2}\mathrm {B} )(sin\alpha /\alpha )^{2}$
. . .	$\alpha =\pi asin\theta /\lambda$
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	$D=N/dcos\theta$
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	$d^{2}\psi /dx^{2}+8\pi ^{2}m[E-U(x)]\psi /h^{2}=0$
Schrodinger's equation free particle	$d^{2}\psi /dx^{2}+k^{2}\psi =0$
Heisenberg's uncertainty principle	$\Delta x\cdot \Delta p_{x}\geq \hbar$
infinite potential well	$E_{n}=(hn/2L)^{2}/2m$
wavefunction of a trapped electron	$\psi _{n}(x)=Asin(n\pi x/L)$, for positive int n
wavefunction probability density	$p(x)=\psi _{n}^{2}(x)dx$
normalization	$\int \psi _{n}^{2}(x)dx=1$
hydrogen atom orbital energy	$E_{n}=-me^{4}/8\epsilon _{0}^{2}h^{2}n^{2}=13.61eV/n^{2}$, for positive int n
hydrogen atom spectrum	$1/\lambda =R(1/n_{low}^{2}-1/n_{high}^{2})$
hydrogen atom radial probability density	$P(r)=4r^{2}/a^{3}e^{2r/a}$
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	$E_{F}=(3/16{\sqrt {2}}\pi )^{2/3}h^{2}n^{2/3}m$
mass number	$A=Z+N$
nuclear radius	$r=r_{0}A^{1/3},r_{0}\approx 1.2fm$
mass excess	$\Delta =M-A$
radioactive decay	$N=N_{0}e^{-\lambda t}$
Hubble constant	$H=71.0km/s$
Hubble's law	$v=Hr$
conservation of lepton number	${\displaystyle }$
conservation of baryon number	${\displaystyle }$
conservation of strangeness	${\displaystyle }$
eightfold way	${\displaystyle }$
weak force	${\displaystyle }$
strong force	${\begin{aligned}{\mathcal {L}}_{\mathrm {QCD} }&={\bar {\psi }}_{i}\left(i\gamma ^{\mu }(D_{\mu })_{ij}-m\,\delta _{ij}\right)\psi _{j}-{\frac {1}{4}}G_{\mu \nu }^{a}G_{a}^{\mu \nu }\\&={\bar {\psi }}_{i}(i\gamma ^{\mu }\partial _{\mu }-m)\psi _{i}-gG_{\mu }^{a}{\bar {\psi }}_{i}\gamma ^{\mu }T_{ij}^{a}\psi _{j}-{\frac {1}{4}}G_{\mu \nu }^{a}G_{a}^{\mu \nu }\,\\\end{aligned}}$
Noether's theorem	${\displaystyle }$
Electroweak interaction	:${\mathcal {L}}_{EW}={\mathcal {L}}_{g}+{\mathcal {L}}_{f}+{\mathcal {L}}_{h}+{\mathcal {L}}_{y}.$ ${\mathcal {L}}_{g}=-{\frac {1}{4}}W_{a}^{\mu \nu }W_{\mu \nu }^{a}-{\frac {1}{4}}B^{\mu \nu }B_{\mu \nu }$ ${\mathcal {L}}_{f}={\overline {Q}}_{i}iD\!\!\!\!/\;Q_{i}+{\overline {u}}_{i}^{c}iD\!\!\!\!/\;u_{i}^{c}+{\overline {d}}_{i}^{c}iD\!\!\!\!/\;d_{i}^{c}+{\overline {L}}_{i}iD\!\!\!\!/\;L_{i}+{\overline {e}}_{i}^{c}iD\!\!\!\!/\;e_{i}^{c}$ ${\mathcal {L}}_{h}=|D_{\mu }h|^{2}-\lambda \left(|h|^{2}-{\frac {v^{2}}{2}}\right)^{2}$ ${\mathcal {L}}_{y}=-y_{u\,ij}\epsilon ^{ab}\,h_{b}^{\dagger }\,{\overline {Q}}_{ia}u_{j}^{c}-y_{d\,ij}\,h\,{\overline {Q}}_{i}d_{j}^{c}-y_{e\,ij}\,h\,{\overline {L}}_{i}e_{j}^{c}+h.c.$
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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सन्दर्भ

• 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