User:Craig Pemberton/Physics

This page is a summary of the equations one might cover in an introductory physics course.

Where possible this list has avoided using any specific system of units. Axioms are in green, definitions are in blue, and theorems are black.

SI Prefixes

significand$\times 1000^{n}$

n	SI prefix	name	etymology
8	Y	yotta-	[y]octo
7	Z	zetta-	septem
6	E	exa-	hexa
5	P	peta-	penta
4	T	tera-	tetra
3	G	giga-	giant
2	M	mega-	great
1	k	kilo-	thousand (Greek)
-1	m	milli-	thousand (Latin)
-2	µ	micro-	small
-3	n	nano-	dwarf
-4	p	pico-	small
-5	f	femto-	(fifteen/3 = 5) (Danish)
-6	a	atto-	(eighteen/3 = 6) (Danish)
-7	z	zepto-	septem
-8	y	yocto-	[y]octo

Translation and rotation

time	$t$	$t$
mass	$m$	$m$
position	$x$	$\theta$
duration	$\Delta t$	$\Delta t$
displacement	$\Delta x$	$\Delta \theta$
conservation of mass	$\Delta m$	$\Delta m$
velocity	$v=dx/dt$	$\omega =d\theta /dt$
acceleration	$a=dv/dt$	$\alpha =d\omega /dt$
jerk	$j=da/dt$	$j=d\alpha /dt$
inertia	${\color {blue}m=\int dm=\Sigma m_{i}}$	${\color {blue}I=\int r^{2}dm=\Sigma r^{2}m_{i}}$
kinetic energy	$E_{k}=mv^{2}/2$	$E_{k}=I\omega ^{2}/2$
momentum	$P=mv$	$L=I\omega$
conservation of mass	$\Delta m=0$	$\Delta m=0$
conservation of energy	$\Delta E=0$	$\Delta E=0$
conservation of momentum	$\Delta P=0$	$\Delta L=0$
force	${\color {blue}f=dP/dt}=ma=-dU/dx$	${\color {blue}\tau =dL/dt}=I\alpha$ $=\mathbf {r} \times \mathbf {f}$
impulse	${\color {blue}J=\int fdt}$	${\color {blue}J=\int \tau dt}$
work	${\color {blue}W=\int fdx}=\mathbf {f} \cdot \mathbf {d}$	${\color {blue}W=\int \tau d\theta }$
power	${\color {blue}P=dW/dt}=fv$	${\color {blue}P=dW/dt}=\tau \omega$
Newton's Third Law	$f_{ab}=-f_{ba}$	$\tau _{ab}=-\tau _{ba}$

Equations for constant acceleration

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

Common forces

spring force, lies parallel to spring	$f_{k}=-kd$
weight, points toward the centre of gravity	$f_{g}=-f_{n}=mg$
tension, lies within the cord	$f_{t}=f$
static friction maxiumum, lies tangent to the surface	$f_{s}=\mu _{s}f_{n}$
kinetic friction, lies tangent to the surface	$f_{k}=\mu _{k}f_{n}$
drag force, tangent to the path	$f_{d}=\mu _{d}\rho av^{2}/2$

Mapping rotation to translation

centripetal force, points to the axis of rotation	$f_{c}=mv^{2}/r$
angular to linear displacement	$x=\theta r$
angular to linear speed	$v=\theta \omega$
angular to linear acceleration tangential component	$a_{t}=\alpha r$
angular to linear acceleration radial component	$a_{r}=\omega ^{2}r$
angular to linear acceleration centripetal acceleration	$\alpha =v^{2}/r$

Other

work done by a spring, positive when relaxes	$W_{s}=-k\Delta (x^{2})/2$
inertial frames	$x_{PA}=x_{PB}+x_{AB}$
	$v_{PA}=v_{PB}+v_{AB}$
	$a_{PA}=a_{PB}+0$
greatest distance	$1/8$ rotations
trajectory	$y=x\tan \theta -gx^{2}/(2V_{0}\cos \theta )^{2}$
flight distance	$v_{0}^{2}\sin {2\theta }/g$
flight time?	$T=2\pi r/v$
terminal velocity	$v_{t}={\sqrt {2fg/(\mu _{d}\rho A)}}$
elastic potential energy	$u(x)=kx^{2}/2$
mechanical energy	${\color {blue}E_{mec}=K+U}$
work by type of energy	$W=\Delta E_{mec}+\Delta E_{th}+\Delta E_{int}$
friction creates heat	$\Delta E_{th}=f_{k}d$
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
?	$\mathbf {P} _{1i}+\mathbf {P} _{2i}=\mathbf {P} _{1f}+\mathbf {P} _{2f}$
system COM remains inert	$\mathbf {v} _{com}={(\mathbf {P} _{1i}+\mathbf {P} _{2i}) \over (M_{1}+M_{2})}=const$
elastic collision, 1D, M2 stationary	$v_{1f}={(m_{1}-m_{2}) \over (m_{1}+m_{2})}v_{1i}$
	$v_{2f}={(2m_{1}) \over (m_{1}+m_{2})}v_{1i}$
first rocket equation,thrust	$t=Rv_{rel}=Ma$
second rocket equation	$-v_{i}+v_{f}=v_{rel}ln(M_{i}/M_{f})$
parallel axis theorem	$I=I_{com}+Mh^{2}$
smooth rolling	$v_{com}=R\omega$
	$x_{arc}=R\theta$
	$x_{com}=R\alpha$
down a ramp axis x	$a_{com,x}=-{\frac {g\sin \theta }{I_{com}/MR^{2}}}$

Thermodynamics

irreversible	${\displaystyle }$
adiabatic	$\Delta Q=0$
molecules	$N$
temperature	$T$
degrees of freedom	$f$
linear expansion	$dL/dt=\alpha L$
volume expansion	$dV/dt=3\alpha V$
heat	$\Delta E_{Q}$ due to $\Delta T$
heat capacity	$C_{th}=\Delta T/Q$
specific heat	$c_{th}=m\Delta T/Q$
heat of vaporization	$L_{v}=Q/m$
heat of fusion	$L_{f}=Q/m$
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	$Q+W=0$
thermal conductivity	$\kappa$
thermal resistance	$R=L/\kappa$
conduction rate	$P_{cond}=A(T_{H}-T_{C})/(L/\kappa )=Q/t$
steady state conduction through a composite slab	$P_{cond}=A(T_{H}-T_{C})/\Sigma (L/\kappa )$
Stefan-Boltzmann constant	$\sigma =5.67*10^{-8}?$
thermal radiation	$P_{rad}=\sigma \epsilon AT_{sys}^{4}$
thermal absorption	$P_{rad}=\sigma \epsilon AT_{env}^{4}$
Boltzmann Constant	$k=1.38\times 10^{-23}J/K$
ideal gas law	$PV=kTN$
work, constant temperature	$W=kTNln(V_{f}/V_{i})$
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's Speed 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	$TV^{\gamma -1}=constant$
four special processes on page 529	${\displaystyle }$
multiplicity of configurations	$W=N!/n_{1}!n_{2}!$
microstate in one half of the box	$n_{1},n_{2}$
Boltzmann's entropy equation	$S=klnW$
entropy	${\color {blue}S=-k\sum _{i}P_{i}\ln P_{i}\!}$
change in entropy	$\Delta S=\int _{i}^{f}(1/T)dQ\approx Q/T_{avg}$
change in entropy	$\Delta S=kNln(V_{f}/V_{i})+NC_{V}ln(T_{f}/T_{i})$
force due to entropy	$f=-Tds/dx$
engine efficiency	$\epsilon =|W|/|Q_{H}|$
Carnot engine efficiency	$\epsilon _{c}=(|Q_{H}|-|Q_{L}|)/|Q_{H}|=(T_{H}-T_{L})/T_{H}$
refrigerator performance	$K=|Q_{L}|/|W|$
Carnot refrigerator performance	$K_{C}=|Q_{L}|/(|Q_{H}|-|Q_{L}|)=T_{L}/(T_{H}-T_{L})$
Zeroth Law of Thermodynamics	temperature is an equivalence relation
First Law of Thermodynamics	$\Delta E_{int}=Q+W$
Second Law of Thermodynamics	$\Delta S\geq 0$
Third Law of Thermodynamics	$S=S_{structural}+CT$

Waves

phasor	${\displaystyle }$
node	${\displaystyle }$
antinode	${\displaystyle }$
period	$T$
amplitude	$x_{m}$
decibel	$dB$
torsion constant	$\kappa =-\tau /\theta$
amplitude	$x_{m}$
frequency	$f=1/T=\omega /(2\pi )$
angular frequency	$\omega =2\pi f=2\pi /T$
phase angle	$\phi$
phase	$(\omega t+\phi )$
damping constant	$b$
damping force	$f_{d}=-bv$
phase	$ky-\omega t$
angular wave number	$k$
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	$\Delta L/\lambda =n+0.5$
intensity	$I=P/A=\rho v\omega ^{2}s_{m}^{2}/2$
source power	$P_{s}$
sound intensity by distance	$I=P_{s}/(4\pi r^{2})$
standard reference intensity	$I_{0}$
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	$f'=f(v+-v_{D})/(v+-v_{S})$
Mach cone 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	$v=\omega /k=\lambda /T=\lambda f$
wave speed on a stretched string	$v={\sqrt {f_{t}/\mu }}$
angular frequency of a linear simple harmonic oscillator	$\omega ={\sqrt {k/m}}$
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	$\omega ={\sqrt {I/mgh}}$
speed of sound	$v={\sqrt {B/\rho }}$
damped angular frequency	$\omega '={\sqrt {(k/m)-(b^{2}/4m^{2})}}$
displacement	$x(t)=\cos(\omega t+\phi )x_{m}$
damped displacement	$x(t)=\cos(\omega 't+\phi )x_{m}(e^{-bt/2m})$
velocity	$v(t)=\sin(\omega t+\phi )x_{m}(-\omega )$
acceleration	$a(t)=\cos(\omega t+\phi )x_{m}(-\omega ^{2})$
transverse sinusoidal wave	$x(y,t)=\sin(ky-\omega t)x_{m}$
wave travelling backwards	$x(y,t)=\sin(ky+\omega t)x_{m}$
resultant wave	$x'(y,t)=\sin(ky-\omega t+\phi /2)x_{m}(2\cos \phi /2)$
standing wave	$x'(y,t)=\cos(\omega t)(2y\sin ky)$
sound displacement function	$x(y,t)=\cos(ky-\omega t)x_{m}$
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

gravity	$f_{G}=Gm_{1}m_{2}/r^{2}$
potential energy	$E_{G}=-Gm_{1}m_{2}/r$
potential energy	$E_{g}=ma_{g}y$
standard gravity	${\color {blue}a_{g}=Gm_{Earth}/r_{Earth}^{2}}\approx 9.81m/s^{2}$
$Wg=mgd\cos \theta$
?	$\Delta k=W_{lift}+W_{g}$
potential work	$\Delta u=-W$
understand path independence	$W_{ab,1}=W_{ab,2}=\cdots$
?	$W_{pull}=-W_{s}$

Electricity

electric constant	$\epsilon _{0}F/m$
electrostatic constant	$k=1/(4\pi \epsilon _{0})m/F$
Coulomb's law	$F=k|q_{1}||q_{2}|/r^{2}$
elementary charge	$e$ in Coloumbs
electric charge	$q=ne$ for integer n
conservation of charge	$\Delta e=0$
test charge	$q_{0}$
electric field	$\mathbf {E} =\mathbf {F} /q_{0}$
electric field of a point charge	$\mathbf {E} q=\mathbf {F}$
electric field lines	end at a negative charge
r hat?	${\hat {r}}$
electric field of a point charge	$\mathbf {E} =\mathbf {F} /q_{0}=(1/(\epsilon _{0}4\pi ))q/r^{2}{\hat {r}}$
electric dipole moment	$\mathbf {p}$
?	$z$
electric field of a dipole moment	$E=(1/(\epsilon _{0}2\pi ))(p/z^{3})$
electric field of a charged ring	$E=qz/(\epsilon _{0}4\pi (z^{2}+R^{2})^{(}3/2)$
surface charge density	$\sigma =Q/A$
electric field of a charged disk	$E=(\sigma /\epsilon _{0}2)(1-z/{\sqrt {z^{2}+R^{2}}})$
torque on a dipole	$\mathbf {\tau } =\mathbf {p} \times \mathbf {E}$
potential energy of a dipole	$U=-\mathbf {p} \cdot \mathbf {E}$
work done on a dipole	$W_{a}=-W=\Delta U$
Gaussian surface	?
flux	?
?	$\mathbf {A}$
electric flux	$\Phi =\oint \mathbf {E} \cdot d\mathbf {A}$
Gauss' law	$q_{enc}=\epsilon _{0}\Phi$
Gauss' law	$q_{enc}=\epsilon _{0}\oint \mathbf {E} \cdot d\mathbf {A}$
volume charge density	$\rho =Q/V$
linear charge density	$\lambda =Q/L$
acceleration due to charge	$a=qE/m$
conducting surface	$E=\sigma /\epsilon _{0}$
electric field of a line of charge	$E=\lambda /\epsilon _{0}2\pi r$
non-conducting sheet of charge	$E=\sigma /\epsilon _{0}2$
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 potential energy	$U$
Work done by electric potential energy	$\Delta U=-W$
finite potential energy of the system	$U-W_{\infty }$
electric potential	$V=U/q$
electric potential difference	$\Delta V=\Delta U/q=-W/q$
potential defined	$V=-W_{\infty }/q$
potential from the electric field	$\Delta V=-\int _{i}^{f}\mathbf {E} \cdot d\mathbf {s}$
potential of a point charge	$V=q/\epsilon _{0}4\pi r$
potential of a point charge group	$V=\Sigma V_{i}=(1/\epsilon _{0}4\pi )\Sigma q_{i}/r_{i}$
potential of a dipole	$V=p\cos \theta /\epsilon _{0}4\pi r^{2}$
potential of continuous charge	$V=\int dV=(1/\epsilon _{0}4\pi )\int dq/r$
field from potential	$\mathbf {E} =\nabla V$
potential of a pair of point charges	$U=W=q_{2}V=q_{1}q_{2}/\epsilon _{0}4\pi r$
capacitance	$C=q/V$
parallel plate capacitor	$C=\epsilon _{0}A/d$
cylindrical capacitor	$C=\epsilon _{0}2\pi L/\ln(b/a)$
spherical capacitor	$C=\epsilon _{0}4\pi ba/(b-a)$
isolated spherical capacitor	$C=\epsilon _{0}4\pi R$
parallel capacitors	$C_{eq}=\Sigma C_{i}$
series capacitors	$1/C_{eq}=\Sigma 1/C_{i}$
potential energy stored in a capacitor	$U=q^{2}/2C=CV^{2}/2$
energy density	$u=\epsilon _{0}E^{2}/2$
dielectric constant	$\kappa \geq 1$
dimensions of capacitance	$C=\epsilon _{0}{\mathcal {L}}$
dielectric	$\epsilon _{0}\to \epsilon _{0}\kappa$
Gauss' law with dialectric	$q=\epsilon _{0}\oint \kappa \mathbf {E} \cdot d\mathbf {A}$
electric displacement	$\mathbf {D} =\epsilon _{0}\kappa \mathbf {E}$
current	$i=dq/dt$
charge density	$\mathbf {J} =i/A$
?	$i=\int JdA$
drift speed	$\mathbf {v} _{d}$
?	$\mathbf {J} =ne\mathbf {v} _{d}/m^{3}$
resistance	$R=V/i$
resistivity in ohm-meters	$\rho =\mathbf {E} /\mathbf {J}$
conductivity	$\sigma =\mathbf {J} /\mathbf {E} =1/\rho$
?	$R/\rho =L/A$
variation of resistivity with temperature	$\rho -\rho _{0}=\rho _{0}\alpha (T-T_{0})$
temperature coefficient of resistivity	$\alpha$
Ohm's law	$V=iR$
electrical power	$P=iV$
resistive dissipation	$P=i^{2}R=V^{2}/R$
emf	${\mathcal {E}}=dW/dq=iR$
rules for calculating emf	loop, resistance, emf
internal resistance	$i={\mathcal {E}}/(R+r)$
resistors in series	$R_{eq}=\Sigma R_{i}$
resistors in parallel	$1/R_{eq}=1/\Sigma R_{i}$
potential difference across a real battery	$p={\mathcal {E}}-iR$
power of an emf device	$P_{emf}=i{\mathcal {E}}$
Kirchoff's junction rule	$i_{in}=i_{out}$
RC charging a capacitor	$q=C{\mathcal {E}}(1-e^{-t/RC}$
RC charging a capacitor	$i=({\mathcal {E}}/R)e^{-t/RC}$
RC charging a capcitor	$V_{C}={\mathcal {E}}(1-e^{-t/RC}$
capacitive time constant	$\tau =RC$
magnetic field	$\mathbf {F} _{B}=q\mathbf {v} \times \mathbf {B}$
Hall effect	$n=Bi/Vle$
circulating charged particle	$|q|vB=mv^{2}/r$
cyclotron resonance condition	$f=f_{osc}$
force on a current	$\mathbf {F} _{B}=i\mathbf {L} \times \mathbf {B}$
magnetic moment	$\mu =NiA$
magnetic dipole torque	$\mathbf {\tau } =\mathbf {\mu } \times \mathbf {B}$
magnetic potential energy	$U(\theta )=-\mathbf {\mu } \cdot \mathbf {B}$
magnetism constant	$\mu _{0}$, in Tm/A
Biot-Savart law	$d\mathbf {B} =(\mu _{0}/4\pi )(id\mathbf {s} \times {\hat {r}}/r^{2})$
magnetic field due to a long straight wire	$B=\mu _{0}i/2\pi R$
magnetic field due to a semi-infinite straight wire	$B=\mu _{0}i/4\pi R$
magnetic field at the center of a circular arc	$B=\mu _{0}i\phi /4\pi R$
Ampere's law	$\oint \mathbf {B} \cdot d\mathbf {s} =\mu _{0}i_{enc}$
ideal solenoid	$B=\mu _{0}in$
toroid	$B=\mu _{0}iN/2\pi r$
current carrying coil	$\mathbf {B} =\mu _{0}\mathbf {\mu } /2\pi z^{3}$
magnetic flux through A	$\Phi _{B}=\int \mathbf {B} \cdot d\mathbf {A}$
?	$\Phi _{B}=BA$
Faraday's law	${\mathcal {E}}=d\Phi _{B}/dt$
Lenz's law	$?$
Faraday's law	$\oint \mathbf {E} \cdot d\mathbf {s} =-d\Phi _{B}/dt$
inductance	$L=N\Phi _{B}/i$
solenoid	$L/l=\mu _{0}n^{2}A$
self-induced emf	${\mathcal {E}}_{L}=-Ldi/dt$
RL circuit	$Ldi/dt+Ri={\mathcal {E}}$
RL circuit rise of current	$i={\mathcal {E}}/R(1-e^{-t/\tau _{L}})$
RL circuit time constant	$\tau _{L}=L/R$
RL circuit decay of current	$i={\mathcal {E}}e^{-t/\tau _{L}}/R=i_{0}e^{-t/\tau _{L}}$
magnetic energy	$U_{B}=Li^{2}/2$
magnetic energy density	$u_{B}=B^{2}/2\mu _{0}$
mutual induction	${\mathcal {E}}_{1}=-Mdi_{2}/dt,{\mathcal {E}}_{2}=-Mdi_{1}/dt$
LC circuit	$\omega =1/{\sqrt {LC}}$
LC oscillations	$Ld^{2}q/dt^{2}+q/C=0$
LC charge	$q=Qcos(\omega t+\phi )$
LC current	$i=-\omega Qsin(\omega t+\phi )$
LC electrical energy	$U_{E}=q^{2}/2C=Q^{2}cos^{2}(\omega t+\phi )/2C$
[[]]	$U_{B}=Q^{2}sin^{2}(\omega t+\phi )/2C$
RLC circuit ODE	$Ld^{2}/q/dt^{2}+Rdq/dt+q/C=0$
RLC circuit ODE solution	$q=QeT^{-Rt/2L}cos(\omega 't+\phi )$
resistive load	$V_{R}=I_{R}R$
capacitive reactance	$X_{C}=1/\omega _{d}C$
capacitive load	$V_{C}=I_{C}X_{C}$
inductive reactance	$X_{L}=\omega _{d}L$
inductive load	$V_{L}=I_{L}X_{L}$
phase constant	$tan\phi =X_{L}-X_{C}/R$
electromagnetic resonance	$\omega _{d}=\omega =1/{\sqrt {LC}}$
rms current	$I_{rms}=I/{\sqrt {2}}$
rms voltage	$V_{rms}=V/{\sqrt {2}}$
rms emf	${\mathcal {E}}_{rms}={\mathcal {E}}_{m}/{\sqrt {2}}$
average power	$P_{avg}={\mathcal {E}}I_{rms}cos\phi$
transformation of voltage	$V_{s}N_{p}=V_{p}N_{s}$
transformation of currents	$I_{s}N_{s}=I_{p}N_{p}$
transformer reistance	$R_{e}q=(Np/Ns)^{2}R$
Gauss' law for magnetic fields	$\Phi _{B}=\oint \mathbf {B} \cdot d\mathbf {A} =0$
Maxwell's law of induction	$\oint \mathbf {B} /cdotd\mathbf {s} =\mu _{0}\epsilon _{0}d\Phi _{E}/dt$
Ampere-Maxwell law	$\oint \mathbf {B} \cdot d\mathbf {s} =\mu _{0}\epsilon _{0}d\Phi _{E}/dt+\mu _{0}i_{enc}$
displacement current	$i_{d}=\epsilon _{0}d\Psi _{E}/dt$
Ampere-Maxwell law	$\oint \mathbf {B} \cdot d\mathbf {s} =\mu _{0}i_{d,enc}+\mu _{0}i_{enc}$
induced magnetic field inside a circular capacitor	$B=(\mu _{0}i_{d}/2\pi R^{2})r$
induced magnetic field outside a circular capacitor	$B=\mu _{0}i_{d}/2\pi rr$
spin magnetic dipole moment	$\mathbf {\mu _{s}} =-e\mathbf {S} /m$
Bohr magneton	$\mu _{B}=eh/4\pi m$
?	$U=-\mathbf {\mu } _{s}\cdot \mathbf {B} _{ext}=-\mu _{s,z}B_{ext}$
orbital magnetic dipole moment	$\mathbf {\mu } _{orb}=-e\mathbf {L} _{o}rb/2m$
?	$U=-\mathbf {\mu } _{orb}\cdot \mathbf {B} _{ext}=-\mu _{orb,z}B_{ext}$

Light

electric wave component	$E=E_{m}sin(kx-\omega t)$
magnetic wave component	$B=B_{m}sin(kx-\omega t)$
celerity	$c=1/{\sqrt {\mu _{0}\epsilon _{0}}}$
magnitude ratio	$E/B=c$
Poynting vector	$\mathbf {S} =\mu _{0}^{-1}\mathbf {E} \times \mathbf {B}$
instantaneous energy flow rate	$S=E^{2}/c\mu _{0}$
intensity	$I=E_{rms}^{2}/c\mu _{0}$
intensity at the sphere	$I=P_{s}/4\pi r^{2}$
total absorption	$\Delta p=\Delta U/c$
total reflection back along path	$\Delta p=2\Delta U/c$
total absorption	$p_{r}=Ic$
total reflection back along path	$p_{r}=2I/c$
one-half rule	$I=I_{0}/2$
cosine-squared rule	$I=I_{0}cos^{2}\theta$
law of reflection	$\theta _{1}'=\theta _{1}$
law of refraction (snell's)	$n_{1}sin\theta _{1}=n_{2}sin\theta _{2}$
critical angle	$\theta _{c}=sin^{-1}n_{2}/n_{1}$
Brewster angle	$\theta _{B}=tan^{-1}n_{2}/n_{1}$
plane mirror	$i=-p$
spherical mirror	$f=r/2$
spherical mirror	$1/p+1/i=1/f$
lateral magnification	$|m|=h'/h$
lateral magnification	$m=-i/p$
?	$n_{1}/p+n_{2}/i=(n_{2}-n_{1})/r$
thin lens	$1/f=1/p+1/i$
thin lens in air	$1/f=(n-1)(1/r_{1}-1/r_{2})$
index of refraction	$n=c/v$
path length difference	$\Delta L=dsin\theta$
maxima - bright fringes	$dsin\theta =m\lambda$ for m natural
minima - dark fringes	$dsin\theta =(m+1/2)\lambda$ for m natural
double-slit interference intensity	$I=4I_{0}cos^{2}(\theta /2)$
?	$\theta =2\pi dsin\theta /\lambda$
bright film in air maxima	$2L=(m+1/2)\lambda /n_{2}$
bright film in air minima	$(m)\lambda /n_{2}$
single-slit minima	$asin\theta =m\lambda$
single-slit intensity	$I(\theta )=I_{m}(sin\alpha /\alpha )^{2}$
?	$\alpha =\phi /2=\pi asin\theta /\lambda$
circular aperture first minimum	$sin\theta =1.22\lambda /d$
Rayleigh's criterion	$\theta _{R}=1.22\lambda /d$
double slit intensity	$I(\theta )=I_{m}(cos^{2}\mathrm {B} )(sin\alpha /\alpha )^{2}$
diffraction grating maxima lines	$dsin\theta =m\lambda$, for natural m
diffraction grating half-width	$\Delta \theta _{hw}=\lambda /Ndcos\theta$
diffraction grating dispersion	$D=m/dcos\theta$
diffraction grating resolving power	$R=Nm$
Bragg's law	$2dsin\theta =m\lambda$

Relativity

The Relativity Postulate	${\displaystyle }$
The Speed of Light Postulate	${\displaystyle }$
Lorentz factor	$\gamma =1/{\sqrt {1-(v/c)^{2}}}$
speed parameter	$\mathrm {B} =v/c$
time dilation	$\Delta t=\gamma \Delta t_{0}$
length contraction	$L=L_{0}/\gamma$
Lorentz transformation	$x'=\gamma (x-vt)$
[[]]	$y'=y$
[[]]	$z'=z$
[[]]	$t'=\gamma (t-xv/c^{2})$
Doppler effect for light	$f=f_{0}{\sqrt {1-\mathrm {B} /1+\mathrm {B} }}$
wavelength Doppler shift	$v=|\Delta \lambda |c/\lambda _{0}$
momentum	$\mathbf {p} =\gamma m\mathbf {v}$
rest energy	$E_{0}=mc^{2}$
total energy	$E=E_{0}+K=mc^{2}+K=\gamma mc^{2}$
[[]]	$Q=-\Delta mc^{2}$
kinetic energy	$K=E-mc^{2}=\gamma mc^{2}-mc^{2}=mc^{2}(\gamma -1)$
momentum and kinetic energy triangle	$E={\sqrt {(pc)^{2}+(mc^{2})^{2}}}$

Particles

quantum constant	$h$, in energy/frequency
photon energy	$E=hf$
photoelectric equation	$hf=K_{max}+\Phi$
photon momentum	$p=hf/c=h/\lambda$
de Broglie wavelength	$\lambda =h/p$
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 energy well	$E_{n}=h^{2}n^{2}/8mL^{2}$
energy changes	$\Delta E=hf$
wave function of a trapped electron	$\psi _{n}(x)=Asin(n\pi x/L)$, for positive int n
probability of detection	$p(x)=\psi _{n}^{2}(x)dx$
normalization	$\int \psi _{n}^{2}(x)dx=1$
hydrogen orbital energy	$E_{n}=13.61eV/n^{2}$, for positive int n
hydrogen energy changes	$1/\lambda =R(1/n_{low}^{2}-1/n_{high}^{2}$
hydrogen radial probability density	$P(r)=4r^{2}/a^{3}e^{2r/a}$
spin magnetic 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	$\mathbf {\mu } _{orb,z}=-m_{\mathcal {L}}\mu _{B}$
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 radii	$r=r_{0}A^{1/3}$
mass excess	$\Delta =M-A$
radioactive decay	$N=N_{0}e^{-\lambda t}$
conservation of lepton number	${\displaystyle }$
conservation of baryon number	${\displaystyle }$
conservation of strangeness	${\displaystyle }$
the eightfold way	${\displaystyle }$
weak force	${\displaystyle }$
strong force	${\displaystyle }$
Hubble constant	$H=71.0km/s$
Hubble's law	$v=Hr$