भौतिकी के सूत्र

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	$\mathrm{\Delta }t$	$\mathrm{\Delta }t$
विस्थापन	$\mathrm{\Delta }x$	$\mathrm{\Delta }\theta$
द्रव्यमान संरक्षण	$\mathrm{\Delta }m=0$	$\mathrm{\Delta }m=0$
ऊर्जा संरक्षण	$\mathrm{\Delta }E=0$	$\mathrm{\Delta }E=0$
संवेग संरक्षण	$\mathrm{\Delta }P=0$	$\mathrm{\Delta }L=0$
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	$\mathrm{\Delta }U=-W$	$\mathrm{\Delta }U=-W$
संवेग	$P=mv$	$L=I\omega$ $=||𝐫\times 𝐏||=m||𝐫\times 𝐯||$
बल	$f=dP/dt=ma=-dU/dx$	$\tau =dL/dt=I\alpha$ $=||𝐫\times 𝐟||=m||𝐫\times 𝐚||$
जड़त्व आघूर्ण	$m=\int dm=\mathrm{\Sigma }{m}_i$	$I=\int r^{2}dm=\mathrm{\Sigma }{r}^2{m}_i$
आवेग	$J=\int fdt$	$J=\int \tau dt$
कार्य	$W=\int fdx=𝐝\cdot 𝐟$	$W=\int \tau d\theta$
शक्ति	$P=dW/dt=fv$	$P=dW/dt=\tau \omega$
गतिज ऊर्जा	$K=m{v}^2/2=P^2/2m$	$K=I{w}^2/2=\mathrm{\Sigma }{R}^{2}m$
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	$\mathrm{\Delta }v=at$	$\mathrm{\Delta }\omega =\alpha t$
समय	$\mathrm{\Delta }(v^2)=2a\mathrm{\Delta }x$	$\mathrm{\Delta }({\omega }^2)=2\alpha \mathrm{\Delta }\theta$
त्वरण	$\mathrm{\Delta }x=t\mathrm{\Delta }v/2$	$\mathrm{\Delta }\theta =t\mathrm{\Delta }\omega /2$
प्रा०वेग	$\mathrm{\Delta }x=-a{t}^2/2+v_{2}t$	$\mathrm{\Delta }\theta =-\alpha t^2/2+{\omega }_{2}t$
अंतिमवेग	$\mathrm{\Delta }x=+a{t}^2/2+v_{1}t$	$\mathrm{\Delta }\theta =+\alpha t^2/2+{\omega }_{1}t$

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	$𝐝=𝐢cos\omega t+𝐣sin\omega t$
uniform circular motion tangential speed	$𝐯={𝐝}^{\prime }=-\omega r(𝐢\sin \omega t-𝐣\cos \omega t)$
uniform circular motion tangential component, scalar	$a_t=\alpha r$
uniform circular motion centripetal acceleration	$𝐚={𝐝}^{″}=-{\omega }^2𝐝=-v^2𝐧/r$
uniform circular motion centripetal acceleration scalar	$\alpha =v^2/r$
uniform circular motion centripetal force	$f=-m{v}^2/r$
uniform circular motion revolution time	$T=2\pi r/v$

Elasticity

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

घर्षण (Friction)

normal force	$f_n=𝐟\cdot 𝐧$
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	$f={\mu }_d\rho a{v}^2/2$
terminal velocity	$v_t=\sqrt{2fg/({\mu }_d\rho A)}$
friction creates heat and sound	$\mathrm{\Delta }E=f_{k}d$

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	$F/A=E\mathrm{\Delta }L/L$
shear modulus	$F/A=G\mathrm{\Delta }x/L$
bulk modulus	$F/A=B\mathrm{\Delta }V/V$

अन्य

inertial frames	$x_{PA}=x_{PB}+x_{AB}$
. . .	$v_{PA}=v_{PB}+v_{AB}$
. . .	$a_{PA}=a_{PB}+0$
trajectory	$y=x\tan \theta -g{x}^2/2(V_0\cos \theta {)}^2$
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	$\mathrm{\Delta }{E}_{mec}=0$ when all forces are conservative
thrust	$t=R{v}_{rel}=ma$
ideal rocket equation	$\mathrm{\Delta }v=ln(m_i/m_f)v_{rel}$
parallel axis theorem	$I=I_{com}+m{r}^2$
list of moments of inertia
indeterminate systems

द्रब्यमान केन्द्र एवं संघट्ट (Center of mass and collisions)

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

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 ?	$K=I_{com}{\omega }^2/2+M{v}_{com}^2/2$
rolling down a ramp along axis x	$a_{com,x}=-\frac{g\sin \theta }{1+I_{com}/M{R}^2}$

उष्मागतिकी (Thermodynamics)

Zeroth Law of Thermodynamics	$(A=B)\wedge (B=C)\Rightarrow A=C$ (where "=" denotes systems in thermal equilibrium
First Law of Thermodynamics	$\mathrm{\Delta }{E}_{int}=Q+W$
Second Law of Thermodynamics	$\mathrm{\Delta }S\ge 0$
Third Law of Thermodynamics	$S=S_{structural}+CT$
temperature	$T$
molecules	$N$
degrees of freedom	$f$
heat	$Q$, $\mathrm{\Delta }E$ due to $\mathrm{\Delta }T$ (energy)
thermal mass (extensive property)	$C_{th}=Q/\mathrm{\Delta }T$
specific heat capacity (bulk property)	$c_{th}=Q/\mathrm{\Delta }Tm$
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	$P=Q/t=A(T_H-T_C)/\mathrm{\Sigma }(R_i)$
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	$P=\sigma ϵA{T}_{sys}^4$
thermal absorption	$P=\sigma ϵA{T}_{env}^4$
adiabatic	$\mathrm{\Delta }Q=0$
ideal gas law	$PV=kTN$
work, constant temperature	$W=kTNln(V_f/V_i)$
work due to gas expansion	$W=\underset{i}{\overset{f}{\int }}pdV$
. . . adiabatic	$\mathrm{\Delta }{E}_{int}=W$
. . . constant volume	$\mathrm{\Delta }{E}_{int}=Q$
. . . free expansion	$\mathrm{\Delta }{E}_{int}=0$
. . . closed cycle	$Q+W=0$
work, constant volume	$W=0$
work, constant pressure	$W=p\mathrm{\Delta }V$
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	$P(v)=4\pi (m/(2\pi kT){)}^{3/2}{V}^2{e}^{-(m{v}^2/(2kT))}$
molecular specific heat at a constant volume	$C_V=Q/(N\mathrm{\Delta }T)$
?	$\mathrm{\Delta }{E}_{int}=N{C}_V\mathrm{\Delta }T$
molecular specific heat at a constant pressure	$C_p=Q/(N\mathrm{\Delta }T)$
?	$W=p\mathrm{\Delta }V=Nk\mathrm{\Delta }T$
?	$k=C_p-C_V$
adiabatic expansion	$p{V}^{\gamma }=constant$
adiabatic expansion	$T{V}^{\gamma -1}=constant$
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	$\mathrm{\Delta }S=\underset{i}{\overset{f}{\int }}(1/T)dQ\approx Q/T_{avg}$
entropy change	$\mathrm{\Delta }S=kNln(V_f/V_i)+N{C}_{V}ln(T_f/T_i)$
entropic force	$f=-TdS/dx$
engine efficiency	$ϵ=|W|/|Q_H|$
Carnot engine efficiency	${ϵ}_c=(|Q_H|-|Q_L|)/|Q_H|=(T_H-T_L)/T_H$
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	$\varphi$
phase	$(\omega t+\varphi )$
damping force	$f_d=-bv$
phase	$ky-\omega t$
wavenumber	$k$
phase constant	$\varphi$
linear density	$\mu$
harmonic number	$n$
harmonic series	$f=v/\lambda =nv/(2L)$
wavelength	$\lambda =k/(2\pi )$
bulk modulus	$B=\mathrm{\Delta }p/(\mathrm{\Delta }V/V)$
path length difference	$\mathrm{\Delta }L$
resonance	${\omega }_d=\omega$
phase difference	$\varphi =2\pi \mathrm{\Delta }L/\lambda$
fully constructive interference	$\mathrm{\Delta }L/\lambda =n$
fully destructive interference	$\mathrm{\Delta }L/\lambda =n+0.5$
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	$s(t)=[2s_m\cos {\omega }^{\prime }t]\cos \omega t$
beat frequency	$f_{beat}=f_1-f_2$
Doppler effect	$f^{\prime }=f(v+-v_D)/(v+-v_S)$
sonic boom angle	$\sin \theta =v/v_s$
average wave power	$P_{avg}=\mu v{\omega }^2{x}_m^2/2$
pressure amplitude	$\mathrm{\Delta }{p}_m=(v\rho \omega )x_m$
wave equation	$\frac{\partial y}{\partial x^2}=\frac{1}{v^2}\frac{{\partial }^{2}y}{\partial t^2}$
wave superposition	$x^{\prime }(y,t)=x_1(y,t)+x_2(y,t)$
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	${\omega }^{\prime }=\sqrt{(k/m)-(b^2/4m^2)}$
wave displacement	$x(t)=x_m\cos (\omega t+\varphi )$
wave displacement when damped	$x(t)=x_m\cos ({\omega }^{\prime }t+\varphi )(e^{-bt/2m})$
wave velocity	$v(t)=x_m\sin (\omega t+\varphi )(-\omega )$
wave acceleration	$a(t)=x_m\cos (\omega t+\varphi )(-{\omega }^2)$
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	$x^{\prime }(y,t)=x_m\sin (ky-\omega t+\varphi /2)(2\cos \varphi /2)$
standing wave	$x^{\prime }(y,t)=\cos (\omega t)(2y\sin ky)$
sound displacement function	$x(y,t)=x_m\cos (ky-\omega t)$
sound pressure-variation function	$\mathrm{\Delta }p(y,t)=\sin (ky-\omega t)\mathrm{\Delta }{p}_m$
potential harmonic energy	$E_U(t)=k{x}^2/2=k{x}_m^2{\cos }^2(\omega t+\varphi )/2$
kinetic harmonic energy	$E_K(t)=k{x}^2/2=k{x}_m^2{\sin }^2(\omega t+\varphi )/2$
total harmonic energy	$E(t)=k{x}_m^2/2=E_U+E_K$
damped mechanical energy	$E_{mec}(t)=k{e}^{-bt/m}{x}_m^2/2$

गुरुत्वाकर्षण (Gravitation)

gravitational constant	$G$ (force)(distance/mass)^2
gravitational force	$f_G=G{m}_1{m}_2/r^2$
superposition applies	$𝐅=\mathrm{\Sigma }{𝐅}_i=\int d𝐅$
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	$U=-G{m}_1{m}_2/r\approx m{a}_{g}y$
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	$E=-G{m}_1{m}_2/a2$
standard gravity	$a_g=G{m}_{Earth}/r_{Earth}^2\approx 9.81m/s^2$
weight, points toward the center of gravity	$f_g=-f_n=mg$
path independence	$W_{ab,1}=W_{ab,2}=\cdots$
Einstein field equations	$R_{\mu \nu }-\frac{1}{2}{g}_{\mu \nu }R+g_{\mu \nu }\mathrm{\Lambda }=\frac{8\pi G}{c^4}{T}_{\mu \nu }$

तरलगतिकी (Fluid dynamics)

density	$\rho =\mathrm{\Delta }m/\mathrm{\Delta }V$
pressure	$p=\mathrm{\Delta }F/\mathrm{\Delta }A$
pressure difference	$\mathrm{\Delta }p=\rho g\mathrm{\Delta }y$
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	$weigh{t}_{app}=weight-F_b$
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	$𝐅=q(𝐄+𝐯\times 𝐁)$
Gauss' law	$\oint 𝐄\cdot d𝐀={\mathrm{\Phi }}_E=q_{enc}/{ϵ}_0$
Gauss' law for magnetic fields	$\oint 𝐁\cdot d𝐀={\mathrm{\Phi }}_B=0$
Faraday's law of induction	$\oint 𝐄\cdot d𝐬=-d{\mathrm{\Phi }}_B/dt=-ℰ$
Ampere-maxwell law	$\oint 𝐁\cdot d𝐬={\mu }_0(i_{enc}+i_{d,enc})$
elementary charge	$e$
electric charge	$q=ne$
conservation of charge	$\mathrm{\Delta }q=0$
linear charge density	$\lambda =q/l^1$
surface charge density	$\sigma =q/l^2$
volume charge density	$\rho =q/l^3$
electric constant	${ϵ}_0$ (time)^2(charge)^2/(mass)(volume)
magnetic constant	${\mu }_0$ (force)(time)^2/(charge)^2
Coulomb's law	$F=q_1{q}_2/(4\pi {ϵ}_0)r^2$
electric field	$𝐄=𝐅/q$
electric field lines	end at a negative charge
Gaussian surface	$𝐀$
flux notation implies a normal unit vector	$\cdot d𝐀\to \cdot 𝐧d𝐀$
electric flux	${\mathrm{\Phi }}_E=\oint 𝐄\cdot d𝐀$
magnetic flux	${\mathrm{\Phi }}_B=\int 𝐁\cdot d𝐀$
magnetic flux given assumptions	${\mathrm{\Phi }}_B=BA$
dielectric constant	$\kappa \ge 1$
dielectric	${ϵ}_0\to {ϵ}_0\kappa$
Gauss' law with dialectric	$q_{enc}={ϵ}_0\oint \kappa 𝐄\cdot d𝐀$
Biot-Savart law	$𝐁=\int \frac{{\mu }_0}{4\pi }\frac{(id𝐬)\times 𝐫}{r^3},$
Lenz's law	induced current always opposes its cause
inductance (with respect to time)	$L=-ℰ/q^{″}$
inductance from coils	$L=N{\mathrm{\Phi }}_B/i$
inductance of a solenoid	$L/l={\mu }_0{n}^{2}A$
displacement current	$i_d={ϵ}_{0}d{\mathrm{\Phi }}_E/dt$
displacement vector	$𝐝$
electric dipole moment	$𝐩=q𝐝$
electric dipole torque	$\mathit{\tau }=𝐩\times 𝐄$
electric dipole potential energy	$U=-𝐩\cdot 𝐄$
magnetic dipole moment of a coil, magnitude only	$\mu =iNA$
magnetic dipole moment torque	$\mathit{\tau }=\mathit{\mu }\times 𝐁$
magnetic dipole moment potential energy	$U=-\mathit{\mu }\cdot 𝐁$
electric field accelerating a charged mass	$a=qE/m$
electric field of a charged point	$E=q/{ϵ}_{0}4\pi r^2\hat{r}$
electric field of a dipole moment	$E=p/{ϵ}_{0}2\pi z^3$
electric field of a charged line	$E=\lambda /{ϵ}_{0}2\pi r$
electric field of a charged ring	$E=qz/{ϵ}_{0}4\pi (z^2+R^2{)}^{3/2}$
electric field of a charged conducting surface	$E=\sigma /{ϵ}_0$
electric field of a charged non-conducting surface	$E=\sigma /{ϵ}_{0}2$
electric field of a charged disk	$E=\sigma (1-z)/{ϵ}_{0}2\sqrt{z^2+R^2}$
electric field outside spherical shell r>=R	$E=q/{ϵ}_{0}4\pi r^2$
electric field inside spherical shell r<R	$E=0$
electric field of uniform charge r<=R	$E=qr/{ϵ}_{0}4\pi R^3$
electric field energy density	$u={ϵ}_0{E}^2/2$
electric potential versus electric potential energy	(energy)/(charge) versus (energy)
electric potential energy	$U=-W_{\mathrm{\infty }}$
electric potential	$V=-W_{\mathrm{\infty }}/q=U/q$
electric potential difference	$\mathrm{\Delta }V=-W/q=\mathrm{\Delta }U/q$
electric potential from electric field	$\mathrm{\Delta }V=-\underset{i}{\overset{f}{\int }}𝐄\cdot d𝐬$
electric field from electric potential	$\mathrm{\nabla }V=-𝐄$
electric potential of a charged point	$V=q/{ϵ}_{0}4\pi r$
electric potential of a set of charged points	$V=\mathrm{\Sigma }{V}_i=(1/{ϵ}_{0}4\pi )\mathrm{\Sigma }{q}_i/r_i$
electric potential of a dipole	$V=p\cos \theta /{ϵ}_{0}4\pi r^2$
electric potential of continuous charge	$V=\int dV=(1/{ϵ}_{0}4\pi )\int dq/r$
electric potential energy of a pair of charged points	$V{q}_2=U=W=q_1{q}_2/{ϵ}_{0}4\pi r$
capacitance	$C=q/V$ (charge)^2/(energy)
capacitance of parallel plates	$C={ϵ}_{0}A/d$
capacitance of a cylinder	$C={ϵ}_{0}2\pi L/\ln (b/a)$
capacitance of a sphere	$C={ϵ}_{0}4\pi ba/(b-a)$
capacitance of an isolated sphere	$C={ϵ}_{0}4\pi R$
capacitors in parallel	$C_{eq}^{+1}=\mathrm{\Sigma }{C}_i^{+1}$
capacitors in series	$C_{eq}^{-1}=\mathrm{\Sigma }{C}_i^{-1}$
capacitor potential energy	$U=q^2/C2=C{V}^2/2$
current	$i=dq/dt$
drift speed	${𝐯}_d$
current density	$𝐉=ne{𝐯}_d/m^3$
current density magnitude	$J=i/A$
current density to get current	$i=\int JdA$
resistance	$R=V/i$
resistivity	$\rho =𝐄/𝐉$
resistivity temperature coefficient	$\alpha$
resistivity across temperature	$\rho -{\rho }_0={\rho }_0\alpha (T-T_0)$
resistivity and resistance	$RA=\rho L$
electrical conductivity	$\sigma =1/\rho =𝐉/𝐄$
resistor power dissipation	$P=i^{2}R=V^2/R$
internal resistance	$i=ℰ/(R+r)$
resistors in series	$R_{eq}^{+1}=\mathrm{\Sigma }{R}_i^{+1}$
resistors in parallel	$R_{eq}^{-1}=\mathrm{\Sigma }{R}_i^{-1}$
Kirchoff's current law	$i_{in}=i_{out}$
Ohm's law	$V=iR$
emf	$ℰ=dW/dq=iR$
emf rules	loop, resistance, emf
electrical power	$P=iV$
emf power	$P_{emf}=iℰ$
electric potential difference across a real battery	$p=ℰ-iR$
magnetic field force on a moving charge	${𝐅}_B=q𝐯\times 𝐁$
magnetic field force on a current	${𝐅}_B=i𝐋\times 𝐁$
Hall effect	$n=Bi/Vle$
circulating charged particle	$|q|vB=m{v}^2/r$
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	$B={\mu }_{0}i\varphi /4\pi R$
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	$𝐁={\mu }_0\mathit{\mu }/2\pi z^3$
self induction of emf	${ℰ}_L=-Ldi/dt$
magnetic energy	$U_B=L{i}^2/2$
magnetic energy density	$u_B=B^2/2{\mu }_0$
mutual induction	${ℰ}_1=-Md{i}_2/dt,{ℰ}_2=-Md{i}_1/dt$
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	$R{q}^{\prime }+C^{-1}q=ℰ$
RC circuit capacitive time constant	$\tau =RC$
RC circuit charging a capacitor	$q=Cℰ(1-e^{-t/RC})$
RL circuit ODE with respect to time	$L{i}^{″}+R{i}^{\prime }=ℰ$
RL circuit time constant	${\tau }_L=L/R$
RL circuit rise of current	$i=ℰ/R(1-e^{-t/{\tau }_L})$
RL circuit decay of current	$i=ℰe^{-t/{\tau }_L}/R=i_0{e}^{-t/{\tau }_L}$
LC circuit ODE with respect to time	$L{q}^{″}+C^{-1}q=ℰ$
LC circuit	$\omega =1/\sqrt{LC}$
LC circuit charge	$q=Qcos(\omega t+\varphi )$
LC circuit current	$i=-\omega Qsin(\omega t+\varphi )$
LC circuit electrical potential energy	$U_E=q^2/2C=Q^{2}co{s}^2(\omega t+\varphi )/2C$
LC circuit magnetic potential energy	$U_B=Q^{2}si{n}^2(\omega t+\varphi )/2C$
RLC circuit ODE with respect to time	$L{q}^{″}+R{q}^{\prime }+C^{-1}q=ℰ$
RLC circuit charge	$q=Qe{T}^{-Rt/2L}cos({\omega }^{\prime }t+\varphi )$
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	$tan\varphi =X_L-X_C/R$
electromagnetic resonance	${\omega }_d=\omega =1/\sqrt{LC}$
AC current	$I_{rms}=I/\sqrt{2}$
AC voltage	$V_{rms}=V/\sqrt{2}$
AC emf	${ℰ}_{rms}={ℰ}_m/\sqrt{2}$
AC power	$P_{avg}=ℰI_{rms}cos\varphi$

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	$c=1/\sqrt{{\mu }_0{ϵ}_0}=E/B$
Poynting vector	$𝐒={\mu }_0^{-1}𝐄\times 𝐁$
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)	$\mathrm{\Delta }p=\mathrm{\Delta }U/c$
radiation momentum with total reflection (elastic)	$\mathrm{\Delta }p=2\mathrm{\Delta }U/c$
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	$I=I_{0}co{s}^2\theta$
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	${\theta }_c=si{n}^{-1}{n}_2/n_1$
angle of total polarisation	${\theta }_B=ta{n}^{-1}{n}_2/n_1$
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	$\mathrm{\Delta }L=dsin\theta$
double slit minima	$dsin\theta =(N+1/2)\lambda$
double slit maxima	$dsin\theta =N\lambda$
double-slit interference intensity	$I=4I_{0}co{s}^2(\pi dsin\theta /\lambda )$
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	$I(\theta )=I_0(co{s}^2\mathrm{B})(sin\alpha /\alpha {)}^2$
. . .	$\alpha =\pi asin\theta /\lambda$
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	$\mathrm{\Delta }{\theta }_{hw}=\lambda /Ndcos\theta$
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	$t^{\prime }=\gamma (t-xv/c^2)$
. . .	$x^{\prime }=\gamma (x-vt)$
. . .	$y^{\prime }=y$
. . .	$z^{\prime }=z$
time dilation	$\mathrm{\Delta }t=\gamma \mathrm{\Delta }{t}_0$
length contraction	$L=L_0/\gamma$
relativistic Doppler effect	$f=f_0\sqrt{1-(v/c)/1+(v/c)}$
Doppler shift	$v=|\mathrm{\Delta }\lambda |c/{\lambda }_0$
momentum	$𝐩=\gamma m𝐯$
rest energy	$E_0=m{c}^2$
total energy	$E=E_0+K=m{c}^2+K=\gamma m{c}^2=\sqrt{(pc{)}^2+(m{c}^2{)}^2}$
Energy Removed	$Q=-\mathrm{\Delta }m{c}^2$
kinetic energy	$K=E-m{c}^2=\gamma m{c}^2-m{c}^2=m{c}^2(\gamma -1)$

कण भौतिकी (Particle Physics)

standard model	see 4x4 chart of particles
Planck's constant	$h$, in energy/frequency
Reduced Planck's constant	$\hslash =h/2\pi$, in energy/frequency
Planck–Einstein equation	$E=hf$
threshold frequency	$f_0$
work function	$\mathrm{\Phi }=h{f}_0$
photoelectric kinetic energy	$K_{max}=hf-\mathrm{\Phi }$
photon momentum	$p=hf/c=h/\lambda$
de Broglie wavelength	$\lambda =h/p$
Schrodinger's equation	$i\hslash \frac{\partial }{\partial t}\mathrm{\Psi }(𝐫,t)=\hat{H}\mathrm{\Psi }(𝐫,t)$
Schrodinger's equation one dimensional motion	$d^2\psi /d{x}^2+8{\pi }^{2}m[E-U(x)]\psi /h^2=0$
Schrodinger's equation free particle	$d^2\psi /d{x}^2+k^2\psi =0$
Heisenberg's uncertainty principle	$\mathrm{\Delta }x\cdot \mathrm{\Delta }{p}_x\ge \hslash$
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	$E_n=-m{e}^4/8{ϵ}_0^2{h}^2{n}^2=13.61eV/n^2$, 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	${\mathit{\mu }}_{orb}=-e𝐋/2m$
orbital magnetic dipole moment components	${\mathit{\mu }}_{orb,z}=-m_{ℒ}{\mu }_B$
spin magnetic dipole moment	${\mathit{\mu }}_{𝐬}=-e𝐒/m=gq𝐒/2m$
orbital magnetic dipole moment	${\mathit{\mu }}_{orb}=-e{𝐋}_{orb}/2m$
spin magnetic dipole moment potential	$U=-{\mathit{\mu }}_s\cdot {𝐁}_{ext}=-{\mu }_{s,z}{B}_{ext}$
orbital magnetic dipole moment potential	$U=-{\mathit{\mu }}_{orb}\cdot {𝐁}_{ext}=-{\mu }_{orb,z}{B}_{ext}$
Bohr magneton	${\mu }_B=e\hslash /2m$
angular momentum components	$L_z=mℒ\hslash$
spin angular momentum magnitude	$S=\hslash \sqrt{s(s+1)}$
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	$\mathrm{\Delta }=M-A$
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	$\begin{array}{cc}{ℒ}_{\mathrm{Q}\mathrm{C}\mathrm{D}}& ={\overline{\psi }}_i(i{\gamma }^{\mu }(D_{\mu }{)}_{ij}-m{\delta }_{ij}){\psi }_j-\frac{1}{4}{G}_{\mu \nu }^a{G}_a^{\mu \nu }\\ & ={\overline{\psi }}_i(i{\gamma }^{\mu }{\partial }_{\mu }-m){\psi }_i-g{G}_{\mu }^a{\overline{\psi }}_i{\gamma }^{\mu }{T}_{ij}^a{\psi }_j-\frac{1}{4}{G}_{\mu \nu }^a{G}_a^{\mu \nu }\\ \end{array}$
Noether's theorem	${\displaystyle }$
Electroweak interaction	:${ℒ}_{EW}={ℒ}_g+{ℒ}_f+{ℒ}_h+{ℒ}_y.$ ${ℒ}_g=-\frac{1}{4}{W}_a^{\mu \nu }{W}_{\mu \nu }^a-\frac{1}{4}{B}^{\mu \nu }{B}_{\mu \nu }$ ${ℒ}_f={\bar{Q}}_{i}iD/Q_i+{\bar{u}}_i^{c}iD/u_i^c+{\bar{d}}_i^{c}iD/d_i^c+{\bar{L}}_{i}iD/L_i+{\bar{e}}_i^{c}iD/e_i^c$ ${ℒ}_h=|D_{\mu }h{|}^2-\lambda {(|h{|}^2-\frac{v^2}{2})}^2$ ${ℒ}_y=-y_{uij}{ϵ}^{ab}{h}_b^{†}{\bar{Q}}_{ia}{u}_j^c-y_{dij}h{\bar{Q}}_i{d}_j^c-y_{eij}h{\bar{L}}_i{e}_j^c+h.c.$
Quantum electrodynamics	:$ℒ=\overline{\psi }(i{\gamma }^{\mu }{D}_{\mu }-m)\psi -\frac{1}{4}{F}_{\mu \nu }{F}^{\mu \nu },$

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.

इन्हें भी देखें

भौतिकी में प्रायः प्रयुक्त चर

सन्दर्भ

• साँचा:Cite book
• साँचा:Cite book

बाहरी कड़ियाँ

• Physics formulae at xs4all.nl