भौतिकी के सूत्र

Foundational equations in translation and rotation.

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

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.

Equations in translation and rotation, assuming constant acceleration.

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

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	${\displaystyle T=2\pi r/v}$

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

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	${\displaystyle \Delta E=f_{k}d}$

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

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

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

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

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

torsion constant	${\displaystyle \kappa =-\tau /\theta }$
phasor	${\displaystyle }$
node	${\displaystyle }$
antinode	${\displaystyle }$
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	${\displaystyle E_{mec}(t)=ke^{-bt/m}x_{m}^{2}/2}$

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

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	${\displaystyle }$
Pascal's principle	${\displaystyle }$
Archimedes' Principle	${\displaystyle }$
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	${\displaystyle }$
equation of continuity	${\displaystyle R_{V}=Av=}$ constant
Bernoulli's equation	${\displaystyle p+\rho v^{2}/2+\rho gy=}$ constant

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<R	${\displaystyle E=0}$
electric field of uniform charge r<=R	${\displaystyle E=qr/\epsilon _{0}4\pi R^{3}}$
electric field energy density	${\displaystyle u=\epsilon _{0}E^{2}/2}$
electric potential versus electric potential energy	(energy)/(charge) versus (energy)
electric potential energy	${\displaystyle U=-W_{\infty }}$
electric potential	${\displaystyle V=-W_{\infty }/q=U/q}$
electric potential difference	${\displaystyle \Delta V=-W/q=\Delta U/q}$
electric potential from electric field	${\displaystyle \Delta V=-\int _{i}^{f}\mathbf {E} \cdot d\mathbf {s} }$
electric field from electric potential	${\displaystyle \nabla V=-\mathbf {E} }$
electric potential of a charged point	${\displaystyle V=q/\epsilon _{0}4\pi r}$
electric potential of a set of charged points	${\displaystyle V=\Sigma V_{i}=(1/\epsilon _{0}4\pi )\Sigma q_{i}/r_{i}}$
electric potential of a dipole	${\displaystyle V=p\cos \theta /\epsilon _{0}4\pi r^{2}}$
electric potential of continuous charge	${\displaystyle V=\int dV=(1/\epsilon _{0}4\pi )\int dq/r}$
electric potential energy of a pair of charged points	${\displaystyle Vq_{2}=U=W=q_{1}q_{2}/\epsilon _{0}4\pi r}$
capacitance	${\displaystyle C=q/V}$ (charge)^2/(energy)
capacitance of parallel plates	${\displaystyle C=\epsilon _{0}A/d}$
capacitance of a cylinder	${\displaystyle C=\epsilon _{0}2\pi L/\ln(b/a)}$
capacitance of a sphere	${\displaystyle C=\epsilon _{0}4\pi ba/(b-a)}$
capacitance of an isolated sphere	${\displaystyle C=\epsilon _{0}4\pi R}$
capacitors in parallel	${\displaystyle C_{eq}^{+1}=\Sigma C_{i}^{+1}}$
capacitors in series	${\displaystyle C_{eq}^{-1}=\Sigma C_{i}^{-1}}$
capacitor potential energy	${\displaystyle U=q^{2}/C2=CV^{2}/2}$
current	${\displaystyle i=dq/dt}$
drift speed	${\displaystyle \mathbf {v} _{d}}$
current density	${\displaystyle \mathbf {J} =ne\mathbf {v} _{d}/m^{3}}$
current density magnitude	${\displaystyle J=i/A}$
current density to get current	${\displaystyle i=\int JdA}$
resistance	${\displaystyle R=V/i}$
resistivity	${\displaystyle \rho =\mathbf {E} /\mathbf {J} }$
resistivity temperature coefficient	${\displaystyle \alpha }$
resistivity across temperature	${\displaystyle \rho -\rho _{0}=\rho _{0}\alpha (T-T_{0})}$
resistivity and resistance	${\displaystyle RA=\rho L}$
electrical conductivity	${\displaystyle \sigma =1/\rho =\mathbf {J} /\mathbf {E} }$
resistor power dissipation	${\displaystyle P=i^{2}R=V^{2}/R}$
internal resistance	${\displaystyle i={\mathcal {E}}/(R+r)}$
resistors in series	${\displaystyle R_{eq}^{+1}=\Sigma R_{i}^{+1}}$
resistors in parallel	${\displaystyle R_{eq}^{-1}=\Sigma R_{i}^{-1}}$
Kirchoff's current law	${\displaystyle i_{in}=i_{out}}$
Ohm's law	${\displaystyle V=iR}$
emf	${\displaystyle {\mathcal {E}}=dW/dq=iR}$
emf rules	loop, resistance, emf
electrical power	${\displaystyle P=iV}$
emf power	${\displaystyle P_{emf}=i{\mathcal {E}}}$
electric potential difference across a real battery	${\displaystyle p={\mathcal {E}}-iR}$
magnetic field force on a moving charge	${\displaystyle \mathbf {F} _{B}=q\mathbf {v} \times \mathbf {B} }$
magnetic field force on a current	${\displaystyle \mathbf {F} _{B}=i\mathbf {L} \times \mathbf {B} }$
Hall effect	${\displaystyle n=Bi/Vle}$
circulating charged particle	${\displaystyle |q|vB=mv^{2}/r}$
cyclotron resonance condition	${\displaystyle f=f_{osc}}$
magnetic field of a line	${\displaystyle B=\mu _{0}i/2\pi R}$
magnetic field of a ray	${\displaystyle B=\mu _{0}i/4\pi R}$
magnetic field at the center of a circular arc	${\displaystyle B=\mu _{0}i\phi /4\pi R}$
magnetic field of a solenoid	${\displaystyle B=\mu _{0}in}$
magnetic field of a toroid	${\displaystyle B=\mu _{0}iN/2\pi r}$
magnetic field of a current carrying coil	${\displaystyle \mathbf {B} =\mu _{0}\mathbf {\mu } /2\pi z^{3}}$
self induction of emf	${\displaystyle {\mathcal {E}}_{L}=-Ldi/dt}$
magnetic energy	${\displaystyle U_{B}=Li^{2}/2}$
magnetic energy density	${\displaystyle u_{B}=B^{2}/2\mu _{0}}$
mutual induction	${\displaystyle {\mathcal {E}}_{1}=-Mdi_{2}/dt,{\mathcal {E}}_{2}=-Mdi_{1}/dt}$
transformation of voltage	${\displaystyle V_{s}N_{p}=V_{p}N_{s}}$
transformation of current	${\displaystyle I_{s}N_{s}=I_{p}N_{p}}$
transformation of reistance	${\displaystyle R_{eq}=(Np/Ns)^{2}R}$
induced magnetic field inside a circular capacitor	${\displaystyle B=(\mu _{0}i_{d}/2\pi R^{2})r}$
induced magnetic field outside a circular capacitor	${\displaystyle B=\mu _{0}i_{d}/2\pi rr}$
RC circuit ODE with respect to time	${\displaystyle Rq'+C^{-1}q={\mathcal {E}}}$
RC circuit capacitive time constant	${\displaystyle \tau =RC}$
RC circuit charging a capacitor	${\displaystyle q=C{\mathcal {E}}(1-e^{-t/RC})}$
RL circuit ODE with respect to time	${\displaystyle Li''+Ri'={\mathcal {E}}}$
RL circuit time constant	${\displaystyle \tau _{L}=L/R}$
RL circuit rise of current	${\displaystyle i={\mathcal {E}}/R(1-e^{-t/\tau _{L}})}$
RL circuit decay of current	${\displaystyle i={\mathcal {E}}e^{-t/\tau _{L}}/R=i_{0}e^{-t/\tau _{L}}}$
LC circuit ODE with respect to time	${\displaystyle Lq''+C^{-1}q={\mathcal {E}}}$
LC circuit	${\displaystyle \omega =1/{\sqrt {LC}}}$
LC circuit charge	${\displaystyle q=Qcos(\omega t+\phi )}$
LC circuit current	${\displaystyle i=-\omega Qsin(\omega t+\phi )}$
LC circuit electrical potential energy	${\displaystyle U_{E}=q^{2}/2C=Q^{2}cos^{2}(\omega t+\phi )/2C}$
LC circuit magnetic potential energy	${\displaystyle U_{B}=Q^{2}sin^{2}(\omega t+\phi )/2C}$
RLC circuit ODE with respect to time	${\displaystyle Lq''+Rq'+C^{-1}q={\mathcal {E}}}$
RLC circuit charge	${\displaystyle q=QeT^{-Rt/2L}cos(\omega 't+\phi )}$
resistive load	${\displaystyle V_{R}=I_{R}R}$
capacitive load	${\displaystyle V_{C}=I_{C}X_{C}}$
inductive load	${\displaystyle V_{L}=I_{L}X_{L}}$
resistive reactance	${\displaystyle X_{R}=?}$
capacitive reactance	${\displaystyle X_{C}=1/\omega _{d}C}$
inductive reactance	${\displaystyle X_{L}=\omega _{d}L}$
phase constant	${\displaystyle tan\phi =X_{L}-X_{C}/R}$
electromagnetic resonance	${\displaystyle \omega _{d}=\omega =1/{\sqrt {LC}}}$
AC current	${\displaystyle I_{rms}=I/{\sqrt {2}}}$
AC voltage	${\displaystyle V_{rms}=V/{\sqrt {2}}}$
AC emf	${\displaystyle {\mathcal {E}}_{rms}={\mathcal {E}}_{m}/{\sqrt {2}}}$
AC power	${\displaystyle P_{avg}={\mathcal {E}}I_{rms}cos\phi }$