Lists of physics equations

This page seeks to provide a list of elementary physics formulae commonly appearing in high-school and college introductory physics courses.

Where possible this list has avoided using any specific system of units.

SI Prefixes

SI prefixes

Prefix		Base 10	Decimal	Adoption
Name	Symbol
quetta	Q	1030	1000000000000000000000000000000	2022
ronna	R	1027	1000000000000000000000000000
yotta	Y	1024	1000000000000000000000000	1991
zetta	Z	1021	1000000000000000000000
exa	E	1018	1000000000000000000	1975
peta	P	1015	1000000000000000
tera	T	1012	1000000000000	1960
giga	G	109	1000000000
mega	M	106	1000000	1873
kilo	k	103	1000	1795
hecto	h	102	100
deca	da	101	10
—	—	100	1	—
deci	d	10−1	0.1	1795
centi	c	10−2	0.01
milli	m	10−3	0.001
micro	μ	10−6	0.000001	1873
nano	n	10−9	0.000000001	1960
pico	p	10−12	0.000000000001
femto	f	10−15	0.000000000000001	1964
atto	a	10−18	0.000000000000000001
zepto	z	10−21	0.000000000000000000001	1991
yocto	y	10−24	0.000000000000000000000001
ronto	r	10−27	0.000000000000000000000000001	2022
quecto	q	10−30	0.000000000000000000000000000001
Notes

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	$m=\int dm=\Sigma m_{i}$	$I=\int r^{2}dm=\Sigma r^{2}m_{i}$
impulse	$J=\int fdt$	$J=\int \tau dt$
work	$W=\int fdx=\mathbf {d} \cdot \mathbf {f}$	$W=\int \tau d\theta$
power	$P=dW/dt=fv$	$P=dW/dt=\tau \omega$
kinetic energy	$K=mv^{2}/2=P^{2}/2m$	$K=I\ w^{2}/2=\Sigma R^{2}m$
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.

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	$\Delta x=-at^{2}/2+v_{2}t$	$\Delta \theta =-\alpha t^{2}/2+\omega _{2}t$
final velocity	$\Delta x=+at^{2}/2+v_{1}t$	$\Delta \theta =+\alpha t^{2}/2+\omega _{1}t$

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	$W=-k\Delta (x^{2})/2$

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	$\Delta E=f_{k}d$

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	$F/A=B\Delta V/V$

Other

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

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	$v_{1f}={(m_{1}-m_{2}) \over (m_{1}+m_{2})}v_{1i}$
. . .	$v_{2f}={(2m_{1}) \over (m_{1}+m_{2})}v_{1i}$

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	$a_{com,x}=-{\frac {g\sin \theta }{1+I_{com}/MR^{2}}}$

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	$K_{C}=|Q_{L}|/(|Q_{H}|-|Q_{L}|)=T_{L}/(T_{H}-T_{L})$

Waves

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	$E_{mec}(t)=ke^{-bt/m}x_{m}^{2}/2$

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	$R_{\mu \nu }-{1 \over 2}g_{\mu \nu }\,R+g_{\mu \nu }\Lambda ={8\pi G \over c^{4}}T_{\mu \nu }$

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

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	$P_{avg}={\mathcal {E}}I_{rms}cos\phi$

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	$d=N\lambda /2sin\theta$

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	$K=E-mc^{2}=\gamma mc^{2}-mc^{2}=mc^{2}(\gamma -1)$

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	:${\mathcal {L}}={\bar {\psi }}(i\gamma ^{\mu }D_{\mu }-m)\psi -{\frac {1}{4}}F_{\mu \nu }F^{\mu \nu }\;,$

The Postulates of 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.

See also

Variables commonly used in physics

References

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

External links

• Physics formulae at xs4all.nl