# भौतिकी के सूत्र

यहाँ जाएँ: भ्रमण, खोज

## एसआई उपसर्ग (प्रीफिक्स)

SI उपसर्ग
१०००n १०n उपसर्ग चिन्ह [१] से लागू संख्या दशमलव रूप SI writing style में
१००० १०२४ योट्टा- यो/Y 1991 दस जल्द १ ००० ००० ००० ००० ००० ००० ००० ०००
१००० १०२१ अं- अं/Z 1991 अंक १ ००० ००० ००० ००० ००० ००० ०००
१००० १०१८ एक्जा- ए/E 1975 दस शङ्ख 1 ००० ००० ००० ००० ००० ०००
१००० १०१५ पद्म- प/P 1975 पद्म १ ००० ००० ००० ००० ०००
१००० १०१२ टेरा- टे/T 1960 दस खरब १ ००० ००० ००० ०००
१००० १० अर्ब- अ/G 1960 अरब १ ००० ००० ०००
१००० १० अद- अद/M 1960 अदन्त १ ००० ०००
१००० १० सहस्र- स्र/k 1795 हजार १ ०००
१०००२/३ १० शत- स/h 1795 सौ १००
१०००१/३ १० दश- द/da 1795 दस १०
१०००0 १०0 (none) (none) NA एक
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 ${\displaystyle t}$ ${\displaystyle t}$
position ${\displaystyle x}$ ${\displaystyle \theta }$ in radians
mass ${\displaystyle m}$ ${\displaystyle m}$
duration ${\displaystyle \Delta t}$ ${\displaystyle \Delta t}$
displacement ${\displaystyle \Delta x}$ ${\displaystyle \Delta \theta }$
conservation of mass ${\displaystyle \Delta m=0}$ ${\displaystyle \Delta m=0}$
conservation of energy ${\displaystyle \Delta E=0}$ ${\displaystyle \Delta E=0}$
conservation of momentum ${\displaystyle \Delta P=0}$ ${\displaystyle \Delta L=0}$
velocity ${\displaystyle v=dx/dt}$ ${\displaystyle \omega =d\theta /dt}$
acceleration ${\displaystyle a=dv/dt}$ ${\displaystyle \alpha =d\omega /dt}$
jerk ${\displaystyle j=da/dt}$ ${\displaystyle j=d\alpha /dt}$
potential energy change ${\displaystyle \Delta U=-W}$ ${\displaystyle \Delta U=-W}$
momentum ${\displaystyle P=mv}$ ${\displaystyle L=I\omega }$ ${\displaystyle =||\mathbf {r} \times \mathbf {P} ||=m||\mathbf {r} \times \mathbf {v} ||}$
force ${\displaystyle f=dP/dt=ma=-dU/dx}$ ${\displaystyle \tau =dL/dt=I\alpha }$ ${\displaystyle =||\mathbf {r} \times \mathbf {f} ||=m||\mathbf {r} \times \mathbf {a} ||}$
inertia ${\displaystyle m=\int dm=\Sigma m_{i}}$ ${\displaystyle I=\int r^{2}dm=\Sigma r^{2}m_{i}}$
impulse ${\displaystyle J=\int fdt}$ ${\displaystyle J=\int \tau dt}$
work ${\displaystyle W=\int fdx=\mathbf {d} \cdot \mathbf {f} }$ ${\displaystyle W=\int \tau d\theta }$
power ${\displaystyle P=dW/dt=fv}$ ${\displaystyle P=dW/dt=\tau \omega }$
kinetic energy ${\displaystyle K=mv^{2}/2=P^{2}/2m}$ ${\displaystyle K=I\ w^{2}/2=\Sigma R^{2}m}$
Newton's Third Law ${\displaystyle f_{ab}=-f_{ba}}$ ${\displaystyle \tau _{ab}=-\tau _{ba}}$

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 ${\displaystyle \Delta v=at}$ ${\displaystyle \Delta \omega =\alpha t}$
time ${\displaystyle \Delta (v^{2})=2a\Delta x}$ ${\displaystyle \Delta (\omega ^{2})=2\alpha \Delta \theta }$
acceleration ${\displaystyle \Delta x=t\Delta v/2}$ ${\displaystyle \Delta \theta =t\Delta \omega /2}$
initial velocity ${\displaystyle \Delta x=-at^{2}/2+v_{2}t}$ ${\displaystyle \Delta \theta =-\alpha t^{2}/2+\omega _{2}t}$
final velocity ${\displaystyle \Delta x=+at^{2}/2+v_{1}t}$ ${\displaystyle \Delta \theta =+\alpha t^{2}/2+\omega _{1}t}$

## एकसमान वृत्तीय गति (Uniform circular motion)

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

## Elasticity

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

## घर्षण (Friction)

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

## प्रतिबाधा एवं विकृत्ति (Stress and strain)

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

## अन्य

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

## द्रब्यमान केन्द्र एवं संघट्ट (Center of mass and collisions)

 center of mass COM ${\displaystyle \mathbf {r} _{com}=M^{-1}\Sigma m_{i}\mathbf {r} _{i}}$ . . . ${\displaystyle x_{com}=M^{-1}\int xdm,\cdots }$ for constant density: ${\displaystyle x_{com}=V^{-1}\int xdV,\cdots }$ COM is in all planes of symmetry elastic collision ${\displaystyle \Delta E_{k}=0}$ inelastic collision ${\displaystyle \Delta E_{k}=}$maximum conservation of momentum in a two body collision ${\displaystyle \mathbf {P} _{1i}+\mathbf {P} _{2i}=\mathbf {P} _{1f}+\mathbf {P} _{2f}}$ system COM remains inert ${\displaystyle \mathbf {v} _{com}={(\mathbf {P} _{1i}+\mathbf {P} _{2i}) \over (M_{1}+M_{2})}=const}$ elastic collision, 1D, M2 stationary ${\displaystyle v_{1f}={(m_{1}-m_{2}) \over (m_{1}+m_{2})}v_{1i}}$ . . . ${\displaystyle v_{2f}={(2m_{1}) \over (m_{1}+m_{2})}v_{1i}}$

## चिकने तल पर लुढ़कना (Smooth rolling)

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

## उष्मागतिकी (Thermodynamics)

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

## तरंग

 torsion constant ${\displaystyle \kappa =-\tau /\theta }$ phasor node antinode period ${\displaystyle T}$ amplitude ${\displaystyle x_{m}}$ decibel ${\displaystyle dB}$ frequency ${\displaystyle f=1/T=\omega /(2\pi )}$ angular frequency ${\displaystyle \omega =2\pi f=2\pi /T}$ phase angle ${\displaystyle \phi }$ phase ${\displaystyle (\omega t+\phi )}$ damping force ${\displaystyle f_{d}=-bv}$ phase ${\displaystyle ky-\omega t}$ wavenumber ${\displaystyle k}$ phase constant ${\displaystyle \phi }$ linear density ${\displaystyle \mu }$ harmonic number ${\displaystyle n}$ harmonic series ${\displaystyle f=v/\lambda =nv/(2L)}$ wavelength ${\displaystyle \lambda =k/(2\pi )}$ bulk modulus ${\displaystyle B=\Delta p/(\Delta V/V)}$ path length difference ${\displaystyle \Delta L}$ resonance ${\displaystyle \omega _{d}=\omega }$ phase difference ${\displaystyle \phi =2\pi \Delta L/\lambda }$ fully constructive interference ${\displaystyle \Delta L/\lambda =n}$ fully destructive interference ${\displaystyle \Delta L/\lambda =n+0.5}$ sound intensity ${\displaystyle I=P/A=\rho v\omega ^{2}s_{m}^{2}/2}$ sound power source ${\displaystyle P_{s}}$ sound intensity over distance ${\displaystyle I=P_{s}/(4\pi r^{2})}$ sound intensity standard reference ${\displaystyle I_{0}}$ sound level ${\displaystyle \mathrm {B} =(10dB)log(I/I_{0})}$ pipe, two open ends ${\displaystyle f=v/\lambda =nv/(2L)}$ pipe, one open end ${\displaystyle f=v/\lambda =nv/(4L)}$ for n odd beats ${\displaystyle s(t)=[2s_{m}\cos \omega 't]\cos \omega t}$ beat frequency ${\displaystyle f_{beat}=f_{1}-f_{2}}$ Doppler effect ${\displaystyle f'=f(v+-v_{D})/(v+-v_{S})}$ sonic boom angle ${\displaystyle \sin \theta =v/v_{s}}$ average wave power ${\displaystyle P_{avg}=\mu v\omega ^{2}x_{m}^{2}/2}$ pressure amplitude ${\displaystyle \Delta p_{m}=(v\rho \omega )x_{m}}$ wave equation ${\displaystyle {\frac {\partial y}{\partial x^{2}}}={\frac {1}{v^{2}}}{\frac {\partial ^{2}y}{\partial t^{2}}}}$ wave superposition ${\displaystyle x'(y,t)=x_{1}(y,t)+x_{2}(y,t)}$ wave speed ${\displaystyle v=\omega /k=\lambda /T=\lambda f}$ speed of sound ${\displaystyle v={\sqrt {B/\rho }}}$ wave speed on a stretched string ${\displaystyle v={\sqrt {f_{t}/\mu }}}$ angular frequency of an angular simple harmonic oscillator ${\displaystyle \omega ={\sqrt {I/\kappa }}}$ angular frequency of a low amplitude simple pendulum ${\displaystyle \omega ={\sqrt {L/g}}}$ angular frequency of a low amplitude physical pendulum ${\displaystyle \omega ={\sqrt {I/mgh}}}$ angular frequency of a linear simple harmonic oscillator ${\displaystyle \omega ={\sqrt {k/m}}}$ angular frequency of a linear damped harmonic oscillator ${\displaystyle \omega '={\sqrt {(k/m)-(b^{2}/4m^{2})}}}$ wave displacement ${\displaystyle x(t)=x_{m}\cos(\omega t+\phi )}$ wave displacement when damped ${\displaystyle x(t)=x_{m}\cos(\omega 't+\phi )(e^{-bt/2m})}$ wave velocity ${\displaystyle v(t)=x_{m}\sin(\omega t+\phi )(-\omega )}$ wave acceleration ${\displaystyle a(t)=x_{m}\cos(\omega t+\phi )(-\omega ^{2})}$ transverse wave ${\displaystyle x(y,t)=x_{m}\sin(ky-\omega t)}$ wave traveling backwards ${\displaystyle x(y,t)=x_{m}\sin(ky+\omega t)}$ resultant wave ${\displaystyle x'(y,t)=x_{m}\sin(ky-\omega t+\phi /2)(2\cos \phi /2)}$ standing wave ${\displaystyle x'(y,t)=\cos(\omega t)(2y\sin ky)}$ sound displacement function ${\displaystyle x(y,t)=x_{m}\cos(ky-\omega t)}$ sound pressure-variation function ${\displaystyle \Delta p(y,t)=\sin(ky-\omega t)\Delta p_{m}}$ potential harmonic energy ${\displaystyle E_{U}(t)=kx^{2}/2=kx_{m}^{2}\cos ^{2}(\omega t+\phi )/2}$ kinetic harmonic energy ${\displaystyle E_{K}(t)=kx^{2}/2=kx_{m}^{2}\sin ^{2}(\omega t+\phi )/2}$ total harmonic energy ${\displaystyle E(t)=kx_{m}^{2}/2=E_{U}+E_{K}}$ damped mechanical energy ${\displaystyle E_{mec}(t)=ke^{-bt/m}x_{m}^{2}/2}$

## गुरुत्वाकर्षण (Gravitation)

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

## तरलगतिकी (Fluid dynamics)

 density ${\displaystyle \rho =\Delta m/\Delta V}$ pressure ${\displaystyle p=\Delta F/\Delta A}$ pressure difference ${\displaystyle \Delta p=\rho g\Delta y}$ pressure at depth ${\displaystyle p=p_{0}+\rho gh}$ barometer versus manometer Pascal's principle Archimedes' Principle buoyant force ${\displaystyle F_{b}=m_{f}g}$ gravitational force when floating ${\displaystyle F_{g}=F_{b}}$ apparent weight ${\displaystyle weight_{app}=weight-F_{b}}$ ideal fluid equation of continuity ${\displaystyle R_{V}=Av=}$ constant Bernoulli's equation ${\displaystyle p+\rho v^{2}/2+\rho gy=}$ constant

## विद्युतचुम्बकत्व (Electromagnetism)

 Lorentz force ${\displaystyle \mathbf {F} =q(\mathbf {E} +\mathbf {v} \times \mathbf {B} )}$ Gauss' law ${\displaystyle \oint \mathbf {E} \cdot d\mathbf {A} =\Phi _{E}=q_{enc}/\epsilon _{0}}$ Gauss' law for magnetic fields ${\displaystyle \oint \mathbf {B} \cdot d\mathbf {A} =\Phi _{B}=0}$ Faraday's law of induction ${\displaystyle \oint \mathbf {E} \cdot d\mathbf {s} =-d\Phi _{B}/dt=-{\mathcal {E}}}$ Ampere-maxwell law ${\displaystyle \oint \mathbf {B} \cdot d\mathbf {s} =\mu _{0}(i_{enc}+i_{d,enc})}$ elementary charge ${\displaystyle e}$ electric charge ${\displaystyle q=ne}$ conservation of charge ${\displaystyle \Delta q=0}$ linear charge density ${\displaystyle \lambda =q/l^{1}}$ surface charge density ${\displaystyle \sigma =q/l^{2}}$ volume charge density ${\displaystyle \rho =q/l^{3}}$ electric constant ${\displaystyle \epsilon _{0}}$ (time)^2(charge)^2/(mass)(volume) magnetic constant ${\displaystyle \mu _{0}}$ (force)(time)^2/(charge)^2 Coulomb's law ${\displaystyle F=q_{1}q_{2}/(4\pi \epsilon _{0})r^{2}}$ electric field ${\displaystyle \mathbf {E} =\mathbf {F} /q}$ electric field lines end at a negative charge Gaussian surface ${\displaystyle \mathbf {A} }$ flux notation implies a normal unit vector ${\displaystyle \cdot d\mathbf {A} \to \cdot \mathbf {n} d\mathbf {A} }$ electric flux ${\displaystyle \Phi _{E}=\oint \mathbf {E} \cdot d\mathbf {A} }$ magnetic flux ${\displaystyle \Phi _{B}=\int \mathbf {B} \cdot d\mathbf {A} }$ magnetic flux given assumptions ${\displaystyle \Phi _{B}=BA}$ dielectric constant ${\displaystyle \kappa \geq 1}$ dielectric ${\displaystyle \epsilon _{0}\to \epsilon _{0}\kappa }$ Gauss' law with dialectric ${\displaystyle q_{enc}=\epsilon _{0}\oint \kappa \mathbf {E} \cdot d\mathbf {A} }$ Biot-Savart law ${\displaystyle \mathbf {B} =\int {\frac {\mu _{0}}{4\pi }}\ {\frac {(id\mathbf {s} )\times \mathbf {r} }{r^{3}}},}$ Lenz's law induced current always opposes its cause inductance (with respect to time) ${\displaystyle L=-{\mathcal {E}}/q''}$ inductance from coils ${\displaystyle L=N\Phi _{B}/i}$ inductance of a solenoid ${\displaystyle L/l=\mu _{0}n^{2}A}$ displacement current ${\displaystyle i_{d}=\epsilon _{0}d\Phi _{E}/dt}$ displacement vector ${\displaystyle \mathbf {d} }$ electric dipole moment ${\displaystyle \mathbf {p} =q\mathbf {d} }$ electric dipole torque ${\displaystyle \mathbf {\tau } =\mathbf {p} \times \mathbf {E} }$ electric dipole potential energy ${\displaystyle U=-\mathbf {p} \cdot \mathbf {E} }$ magnetic dipole moment of a coil, magnitude only ${\displaystyle \mu =iNA}$ magnetic dipole moment torque ${\displaystyle \mathbf {\tau } =\mathbf {\mu } \times \mathbf {B} }$ magnetic dipole moment potential energy ${\displaystyle U=-\mathbf {\mu } \cdot \mathbf {B} }$ electric field accelerating a charged mass ${\displaystyle a=qE/m}$ electric field of a charged point ${\displaystyle E=q/\epsilon _{0}4\pi r^{2}{\hat {r}}}$ electric field of a dipole moment ${\displaystyle E=p/\epsilon _{0}2\pi z^{3}}$ electric field of a charged line ${\displaystyle E=\lambda /\epsilon _{0}2\pi r}$ electric field of a charged ring ${\displaystyle E=qz/\epsilon _{0}4\pi (z^{2}+R^{2})^{3/2}}$ electric field of a charged conducting surface ${\displaystyle E=\sigma /\epsilon _{0}}$ electric field of a charged non-conducting surface ${\displaystyle E=\sigma /\epsilon _{0}2}$ electric field of a charged disk ${\displaystyle E=\sigma (1-z)/\epsilon _{0}2{\sqrt {z^{2}+R^{2}}}}$ electric field outside spherical shell r>=R ${\displaystyle E=q/\epsilon _{0}4\pi r^{2}}$ electric field inside spherical shell r

## प्रकाश (Light)

 electric light component ${\displaystyle E=E_{m}sin(kx-\omega t)}$ magnetic light component ${\displaystyle B=B_{m}sin(kx-\omega t)}$ speed of light ${\displaystyle c=1/{\sqrt {\mu _{0}\epsilon _{0}}}=E/B}$ Poynting vector ${\displaystyle \mathbf {S} =\mu _{0}^{-1}\mathbf {E} \times \mathbf {B} }$ Poynting vector magnitude ${\displaystyle S=EB/\mu _{0}=E^{2}/c\mu _{0}}$ rms electric field of light ${\displaystyle E_{rms}=E/{\sqrt {2}}}$ light intensity ${\displaystyle I=E_{rms}^{2}/c\mu _{0}}$ light intensity at the sphere ${\displaystyle I=P_{s}/4\pi r^{2}}$ radiation momentum with total absorption (inelastic) ${\displaystyle \Delta p=\Delta U/c}$ radiation momentum with total reflection (elastic) ${\displaystyle \Delta p=2\Delta U/c}$ radiation pressure with total absorption (inelastic) ${\displaystyle p_{r}=I/c}$ radiation pressure with total reflection (elastic) ${\displaystyle p_{r}=2I/c}$ intensity from polarizing unpolarized light ${\displaystyle I=I_{0}/2}$ intensity from polarizing polarized light ${\displaystyle I=I_{0}cos^{2}\theta }$ index of refraction of substance f ${\displaystyle n_{f}=c/v_{f}}$ angle of reflection ${\displaystyle \theta _{1}=\theta _{2}}$ angle of refraction ${\displaystyle n_{1}sin\theta _{1}=n_{2}sin\theta _{2}}$ angle of total reflection ${\displaystyle \theta _{c}=sin^{-1}n_{2}/n_{1}}$ angle of total polarisation ${\displaystyle \theta _{B}=tan^{-1}n_{2}/n_{1}}$ image distance in a plane mirror ${\displaystyle d_{i}=-d_{o}}$ image distance in a spherical mirror ${\displaystyle n_{1}/d_{o}+n_{2}/d_{i}=(n_{2}-n_{1})/r}$ spherical mirror focal length ${\displaystyle f=r/2}$ spherical mirror ${\displaystyle 1/d_{o}+1/d_{i}=1/f}$ lateral magnification m and h negative when upside down ${\displaystyle m=h_{i}/h_{o}=-d_{i}/d_{o}}$ lens focal length ${\displaystyle 1/f=1/d_{o}+1/d_{i}}$ lens focal length from refraction indexes ${\displaystyle 1/f=(n_{lens}/n_{med}-1)(1/r_{1}-1/r_{2})}$ path length difference ${\displaystyle \Delta L=dsin\theta }$ double slit minima ${\displaystyle dsin\theta =(N+1/2)\lambda }$ double slit maxima ${\displaystyle dsin\theta =N\lambda }$ double-slit interference intensity ${\displaystyle I=4I_{0}cos^{2}(\pi dsin\theta /\lambda )}$ thin film in air minima ${\displaystyle (N+0/2)\lambda /n_{2}}$ thin film in air maxima ${\displaystyle 2L=(N+1/2)\lambda /n_{2}}$ single-slit minima ${\displaystyle asin\theta =N\lambda }$ single-slit intensity ${\displaystyle I(\theta )=I_{0}(sin\alpha /\alpha )^{2}}$ double slit intensity ${\displaystyle I(\theta )=I_{0}(cos^{2}\mathrm {B} )(sin\alpha /\alpha )^{2}}$ . . . ${\displaystyle \alpha =\pi asin\theta /\lambda }$ circular aperture first minimum ${\displaystyle sin\theta =1.22\lambda /d}$ Rayleigh's criterion ${\displaystyle \theta _{R}=1.22\lambda /d}$ diffraction grating maxima lines ${\displaystyle dsin\theta =N\lambda }$ diffraction grating half-width ${\displaystyle \Delta \theta _{hw}=\lambda /Ndcos\theta }$ diffraction grating dispersion ${\displaystyle D=N/dcos\theta }$ diffraction grating resolving power ${\displaystyle R=Nn}$ diffraction grating lattice distance ${\displaystyle d=N\lambda /2sin\theta }$

## विशिष्ट आपेक्षिकता (Special Relativity)

 Lorentz factor ${\displaystyle \gamma =1/{\sqrt {1-(v/c)^{2}}}}$ Lorentz transformation ${\displaystyle t'=\gamma (t-xv/c^{2})}$ . . . ${\displaystyle x'=\gamma (x-vt)}$ . . . ${\displaystyle y'=y}$ . . . ${\displaystyle z'=z}$ time dilation ${\displaystyle \Delta t=\gamma \Delta t_{0}}$ length contraction ${\displaystyle L=L_{0}/\gamma }$ relativistic Doppler effect ${\displaystyle f=f_{0}{\sqrt {1-(v/c)/1+(v/c)}}}$ Doppler shift ${\displaystyle v=|\Delta \lambda |c/\lambda _{0}}$ momentum ${\displaystyle \mathbf {p} =\gamma m\mathbf {v} }$ rest energy ${\displaystyle E_{0}=mc^{2}}$ total energy ${\displaystyle E=E_{0}+K=mc^{2}+K=\gamma mc^{2}={\sqrt {(pc)^{2}+(mc^{2})^{2}}}}$ Energy Removed ${\displaystyle Q=-\Delta mc^{2}}$ kinetic energy ${\displaystyle K=E-mc^{2}=\gamma mc^{2}-mc^{2}=mc^{2}(\gamma -1)}$

## कण भौतिकी (Particle Physics)

 standard model see 4x4 chart of particles Planck's constant ${\displaystyle h}$, in energy/frequency Reduced Planck's constant ${\displaystyle \hbar =h/2\pi }$, in energy/frequency Planck–Einstein equation ${\displaystyle E=hf}$ threshold frequency ${\displaystyle f_{0}}$ work function ${\displaystyle \Phi =hf_{0}}$ photoelectric kinetic energy ${\displaystyle K_{max}=hf-\Phi }$ photon momentum ${\displaystyle p=hf/c=h/\lambda }$ de Broglie wavelength ${\displaystyle \lambda =h/p}$ Schrodinger's equation ${\displaystyle i\hbar {\frac {\partial }{\partial t}}\Psi (\mathbf {r} ,\,t)={\hat {H}}\Psi (\mathbf {r} ,t)}$ Schrodinger's equation one dimensional motion ${\displaystyle d^{2}\psi /dx^{2}+8\pi ^{2}m[E-U(x)]\psi /h^{2}=0}$ Schrodinger's equation free particle ${\displaystyle d^{2}\psi /dx^{2}+k^{2}\psi =0}$ Heisenberg's uncertainty principle ${\displaystyle \Delta x\cdot \Delta p_{x}\geq \hbar }$ infinite potential well ${\displaystyle E_{n}=(hn/2L)^{2}/2m}$ wavefunction of a trapped electron ${\displaystyle \psi _{n}(x)=Asin(n\pi x/L)}$, for positive int n wavefunction probability density ${\displaystyle p(x)=\psi _{n}^{2}(x)dx}$ normalization ${\displaystyle \int \psi _{n}^{2}(x)dx=1}$ hydrogen atom orbital energy ${\displaystyle E_{n}=-me^{4}/8\epsilon _{0}^{2}h^{2}n^{2}=13.61eV/n^{2}}$, for positive int n hydrogen atom spectrum ${\displaystyle 1/\lambda =R(1/n_{low}^{2}-1/n_{high}^{2})}$ hydrogen atom radial probability density ${\displaystyle P(r)=4r^{2}/a^{3}e^{2r/a}}$ spin projection quantum number ${\displaystyle m_{s}\in \{-1/2,+1/2\}}$ orbital magnetic dipole moment ${\displaystyle \mathbf {\mu } _{orb}=-e\mathbf {L} /2m}$ orbital magnetic dipole moment components ${\displaystyle \mathbf {\mu } _{orb,z}=-m_{\mathcal {L}}\mu _{B}}$ spin magnetic dipole moment ${\displaystyle \mathbf {\mu _{s}} =-e\mathbf {S} /m=gq\mathbf {S} /2m}$ orbital magnetic dipole moment ${\displaystyle \mathbf {\mu } _{orb}=-e\mathbf {L} _{orb}/2m}$ spin magnetic dipole moment potential ${\displaystyle U=-\mathbf {\mu } _{s}\cdot \mathbf {B} _{ext}=-\mu _{s,z}B_{ext}}$ orbital magnetic dipole moment potential ${\displaystyle U=-\mathbf {\mu } _{orb}\cdot \mathbf {B} _{ext}=-\mu _{orb,z}B_{ext}}$ Bohr magneton ${\displaystyle \mu _{B}=e\hbar /2m}$ angular momentum components ${\displaystyle L_{z}=m{\mathcal {L}}\hbar }$ spin angular momentum magnitude ${\displaystyle S=\hbar {\sqrt {s(s+1)}}}$ cutoff wavelength ${\displaystyle \lambda _{min}=hc/K_{0}}$ density of states ${\displaystyle N(E)=8{\sqrt {2}}\pi m^{3/2}E^{1/2}/h^{3}}$ occupancy probability ${\displaystyle P(E)=1/(e^{(E-E_{F})/kT}+1)}$ Fermi energy ${\displaystyle E_{F}=(3/16{\sqrt {2}}\pi )^{2/3}h^{2}n^{2/3}m}$ mass number ${\displaystyle A=Z+N}$ nuclear radius ${\displaystyle r=r_{0}A^{1/3},r_{0}\approx 1.2fm}$ mass excess ${\displaystyle \Delta =M-A}$ radioactive decay ${\displaystyle N=N_{0}e^{-\lambda t}}$ Hubble constant ${\displaystyle H=71.0km/s}$ Hubble's law ${\displaystyle v=Hr}$ conservation of lepton number conservation of baryon number conservation of strangeness eightfold way weak force strong force {\displaystyle {\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 Electroweak interaction :${\displaystyle {\mathcal {L}}_{EW}={\mathcal {L}}_{g}+{\mathcal {L}}_{f}+{\mathcal {L}}_{h}+{\mathcal {L}}_{y}.}$ ${\displaystyle {\mathcal {L}}_{g}=-{\frac {1}{4}}W_{a}^{\mu \nu }W_{\mu \nu }^{a}-{\frac {1}{4}}B^{\mu \nu }B_{\mu \nu }}$ ${\displaystyle {\mathcal {L}}_{f}={\overline {Q}}_{i}iD\!\!\!\!/\;Q_{i}+{\overline {u}}_{i}^{c}iD\!\!\!\!/\;u_{i}^{c}+{\overline {d}}_{i}^{c}iD\!\!\!\!/\;d_{i}^{c}+{\overline {L}}_{i}iD\!\!\!\!/\;L_{i}+{\overline {e}}_{i}^{c}iD\!\!\!\!/\;e_{i}^{c}}$ ${\displaystyle {\mathcal {L}}_{h}=|D_{\mu }h|^{2}-\lambda \left(|h|^{2}-{\frac {v^{2}}{2}}\right)^{2}}$ ${\displaystyle {\mathcal {L}}_{y}=-y_{u\,ij}\epsilon ^{ab}\,h_{b}^{\dagger }\,{\overline {Q}}_{ia}u_{j}^{c}-y_{d\,ij}\,h\,{\overline {Q}}_{i}d_{j}^{c}-y_{e\,ij}\,h\,{\overline {L}}_{i}e_{j}^{c}+h.c.}$ Quantum electrodynamics :${\displaystyle {\mathcal {L}}={\bar {\psi }}(i\gamma ^{\mu }D_{\mu }-m)\psi -{\frac {1}{4}}F_{\mu \nu }F^{\mu \nu }\;,}$

## 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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