# Physics equations/Collection of equations

Organization: These equations are under reorganization:

## Part II equations waiting to be converted into templates
### Math
#### circumference, area, volume, elements

Circumference and area of circle: $C_{\odot }=\,2\pi r$; $A_{\odot }=\,\pi r^{2}$
Area and volume of sphere: $A=4\pi r^{2}$; $V={\frac {4}{3}}\pi r^{3}$
$dA=r\mathrm {d} r\,\mathrm {d} \theta$
$dA=r^{2}\sin \theta \,{\rm {d\theta \,{\rm {d\phi }}}}$
$\mathrm {d} V=r^{2}\sin \theta \,\mathrm {d} r\,\mathrm {d} \theta \,\mathrm {d} \varphi$

#### Trigonometry

$\sin A={\frac {\textrm {opposite}}{\textrm {hypotenuse}}}={\frac {a}{\,c\,}}\,.$
$\cos A={\frac {\textrm {adjacent}}{\textrm {hypotenuse}}}={\frac {b}{\,c\,}}\,.$
$\tan A={\frac {\textrm {opposite}}{\textrm {adjacent}}}={\frac {a}{\,b\,}}={\frac {\sin A}{\cos A}}\,.$

#### Vector dot product

${\vec {A}}\cdot {\vec {B}}=A_{x}B_{x}+A_{y}B_{y}+A_{z}B_{z}=AB\cos \theta$
$A={\sqrt {{\vec {A}}\cdot {\vec {A}}}}={\sqrt {A_{x}^{2}+A_{y}^{2}+A_{z}^{2}}}$
${\hat {\mathbf {i} }}\cdot {\hat {\mathbf {i} }}={\hat {\mathbf {j} }}\cdot {\hat {\mathbf {j} }}={\hat {\mathbf {k} }}\cdot {\hat {\mathbf {k} }}=1$
${\hat {\mathbf {j} }}\cdot {\hat {\mathbf {k} }}={\hat {\mathbf {k} }}\cdot {\hat {\mathbf {i} }}={\hat {\mathbf {j} }}\cdot {\hat {\mathbf {k} }}=0$

### Electromagnetism

$_{fromhttps://en.wikiversity.org/w/index.php?title=Electromagnetism_Formulae&oldid=1121007https://en.wikiversity.org/w/index.php?title=Maxwell%27s_Equations&oldid=790747}$

#### Before Maxwell's Equations

Electric field and voltage (or potential):
 $V({\vec {b}})-V({\vec {a}})=-\int _{\vec {a}}^{\vec {b}}{\vec {E}}\cdot \mathrm {d} {\vec {l}}=\phi ({\vec {b}})-\phi ({\vec {a}})$
Electric potential due to point charges: $\phi ({\vec {r}})={\frac {1}{4\pi \epsilon _{0}}}{\frac {Q}{r}}\rightarrow {\frac {1}{4\pi \epsilon _{0}}}\sum {\frac {q_{j}}{r_{j}}}\rightarrow {\frac {1}{4\pi \epsilon _{0}}}\int {\frac {\rho \mathrm {d} {V}}{r}}\rightarrow {\frac {1}{4\pi \epsilon _{0}}}\int {\frac {\mathrm {d} \rho }{r}}$
Magnetic field: ${\vec {B}}={\frac {\mu _{0}}{4\pi }}\int _{C}{\frac {I\left(\mathrm {d} {\vec {\ell }}\times {\hat {r}}\right)}{\left|{\vec {r}}\,\right|^{2}}}$

#### Maxwell's equations

Gauss' Law relating electric field to charge: $\oint _{S}{\vec {E}}\cdot \mathrm {d} {\vec {A}}={\frac {1}{\epsilon _{0}}}Q_{enc}$
$\nabla \cdot {\vec {B}}=0$
$\nabla \times {\vec {E}}=-{\frac {\partial {\vec {B}}}{\partial t}}$
$\nabla \times {\vec {B}}=\mu _{0}\ {\vec {J}}+{\frac {1}{c^{2}}}{\frac {\partial {\vec {E}}}{\partial t}}$
$\nabla \times {\vec {H}}={\vec {J}}+{\frac {\partial {\vec {D}}}{\partial t}}$
$\nabla \cdot {\vec {D}}=\rho$

#### to discard

$\nabla \cdot {\vec {E}}={\frac {\rho }{\epsilon _{0}}}$

#### Maxwell's equations integral form

$\oint _{S}{\vec {E}}\cdot \mathrm {d} {\vec {A}}={\frac {1}{\epsilon _{0}}}Q_{enc}$
$\oint _{S}{\vec {B}}\cdot \mathrm {d} {\vec {A}}=0$
$\oint _{C}{\vec {E}}\cdot \mathrm {d} {\vec {\ell }}=-\int _{S}{\frac {\partial {\vec {B}}}{\partial t}}\cdot \mathrm {d} {\vec {A}}$
$\oint _{C}{\vec {B}}\cdot \mathrm {d} {\vec {\ell }}=\mu \int _{S}{\vec {J}}\cdot \mathrm {d} {\vec {A}}+{\frac {1}{c^{2}}}\int _{S}{\frac {\partial {\vec {D}}}{\partial t}}\cdot \mathrm {d} {\vec {A}}$
$\oint _{S}{\vec {D}}\cdot \mathrm {d} {\vec {A}}=\int _{V}\rho \,\mathrm {d} V$
$\oint _{C}{\vec {H}}\cdot \mathrm {d} {\vec {\ell }}=\int _{S}{\vec {J}}\cdot \mathrm {d} {\vec {A}}+\int _{S}{\frac {\partial {\vec {D}}}{\partial t}}\cdot \mathrm {d} {\vec {A}}$

### permittivity and magnetic permeability

${\vec {B}}=\mu {\vec {H}}=\mu _{0}\left({\vec {H}}+{\vec {M}}\right)$
${\vec {D}}=\epsilon {\vec {E}}=\epsilon _{0}{\vec {E}}+{\vec {P}}$
${\vec {D}}=\epsilon {\vec {E}}$
${\vec {H}}={\vec {B}}/\mu$

### Force equations

${\vec {F}}=q_{e}\left({\vec {E}}+{\vec {v}}\times {\vec {B}}\right)$
${\vec {B}}={\frac {\mu _{0}}{4\pi }}\int _{C}{\frac {Id{\vec {\ell }}\times {\hat {r}}}{|r|^{2}}}$

## Current and circuits

https://en.wikibooks.org/w/index.php?title=Electronics/Formulas&oldid=2348180
${\vec {J}}=nqv=n_{q}\,q\,{\vec {v}}_{\text{d}}$
$I=JA={\vec {J}}\cdot {\vec {A}}$
$\Delta U=q\,V$
$I={\frac {\rm {\,dQ}}{\rm {dt}}}$
$P={\frac {\rm {dU}}{\rm {dt}}}=I\,V=I^{2}\,R={\frac {V^{2}}{R}}$
$V=I\,R$
${\vec {E}}\cdot \,\Delta {\vec {\ell }}=\Delta V$
$Q=CV$
$U={\frac {1}{2}}Q\,V={\frac {1}{2}}CV^{2}={\frac {Q^{2}}{2C}}$

#### resistivity

$R=\rho {\frac {L}{A}}$
$\alpha ={\frac {\rho -\rho _{0}}{\rho _{0}}}{\frac {1}{T-T0}}$
${\frac {\Delta \rho }{\rho }}=\alpha \Delta T$ where $\Delta \rho =\rho -\rho _{0};\quad \Delta T=T-T_{0}\quad$

#### SI units

See https://en.wikipedia.org/w/index.php?title=SI_base_unit&oldid=579303285

### SI units

7 Base units (https://en.wikipedia.org/w/index.php?title=International_System_of_Units&oldid=584747081)
* metre m length
* kilogram kg mass
* second s time
* ampere A electric current
* kelvin K thermodynamic temperature (0°C = 273.15 K)
* Mole molamount of substance (6.02214×10$^{23}$)
* candela cd luminous intensity (typically 18 mW)
Derived units
* radian [rad ] called: angle. units: 1 = m/m
* steradian [ sr ] called: solid angle. units: 1 = m$^2$/m$^2$
* hertz [ Hz ] called: frequency. units: s$^{−1}$
* newton [ N ] called: force, weight. units: kg⋅m⋅s$^{−2}$
* pascal [ Pa ] called: pressure, stress. units: N/m$^2$ = kg⋅m$^{−1}$⋅s$^{−2}$
* joule [ J ] called: energy, work, heat. units: N⋅m = kg⋅m$^2$⋅s$^{−2}$
* watt [ W ] called: power, radiant flux . units:J/s = kg⋅m$^2$⋅s$^{−3}$
* coulomb [ C ] called: electric charge or quantity of electricity. units: s⋅A
* volt [V ] called: voltage (electrical potential difference), electromotive force . units:W/A = kg⋅m$^2$⋅s$^{−3}$⋅A$^{−1}$
* farad [ F ] called: electric capacitance . units:C/V = kg$^{−1}$⋅m$^{−2}$⋅s$^4$⋅A$^2$
* ohm [ Ω ] called: electric resistance, impedance, reactance. units: V/A = kg⋅m$^2$⋅s$^{−3}$⋅A$^{−2}$
* siemens [ S ] called: electrical conductance. units: A/V = kg−1⋅m−2⋅s3⋅A2
* weber [ Wb ] called: magnetic flux. units: V⋅s = kg⋅m$^2$⋅s$^{−2}$⋅A$^{−1}$
* tesla [ T ] called: magnetic field strength. units: Wb/m2 = kg⋅s−2⋅A−1
* henry [ H ] called: inductance . units:Wb/A = kg⋅m$^2$⋅s$^{−2}$⋅A$^{−2}$
* degree [ Celsius °C] called: temperature relative to 273.15 K. units: K
* lumen [ lm ] called: luminous flux . units:cd⋅sr cd
* lux [ lx ] called: illuminance . units:lm/m$^2$ = m$^{−2}$⋅cd
* becquerel [ Bq] called: radioactivity (decays per unit time) . units: s$^{−1}$
* gray [ Gy ] called: absorbed dose (of ionizing radiation). units: J/kg = m$^2$⋅s$^{−2}$
* sievert [ Sv] called: equivalent dose (of ionizing radiation) . units:J/kg = m$^2$⋅s$^{−2}$
* katal [ kat] called: catalytic activity . units:s$^{−1}$⋅mol

### Advanced Physical Constants

speed of light $c\,$ =2998×10$^8$m·s$^{−1}$
Planck constant=$h\,$ =6.626 × 10$^{−34}$ J·s
reduced Planck constant= $\hbar =h/(2\pi )$= 1.0546 × 10$^{−34}$ J·s
more constants that need sorting:
Atomic mass constant =$m_{\mathrm {u} }=1\,\mathrm {u} \,$ =1.660 538 921(73) × 10$^{−27}$ kg
Avogadro's number =$N_{\mathrm {A} },L\,$ =6.022 141 29(27) × 10$^{23}$ mol$^{−1}$ (number of atoms in a mole)
Boltzmann constant =$k=k_{\mathrm {B} }=R/N_{\mathrm {A} }\,$ =1.381 × 10$^{−23}$ J·K$^{−1}$ (converts Kelvins to Joules)
gas constant =$R\,$ =8.31446 J·K$^{−1}$·mol$^{−1}$ (converts Kelvins to Joules per mole)
Stefan–Boltzmann constant =$\sigma =\pi ^{2}k^{4}/60\hbar ^{3}c^{2}$ =5.670 373(21) × 10$^{−8}$ W·m$^{−2}$·K$^{−4}$ (black body power per unit area is $\sigma T^{4}$
Wien displacement law constant (needs to be better defined in terms of maximum lambda and temperature) $b=hck^{-1}/\,$ 4.965 114 231... =2.897 7721(26) × 10$^{−3}$ m·K