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

_{from https://en.wikiversity.org/w/index.php?title=Electromagnetism_Formulae&oldid=1121007 https://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)

1. metre m length
2. kilogram kg mass
3. second s time
4. ampere A electric current
5. kelvin K thermodynamic temperature (0°C = 273.15 K)
6. Mole molamount of substance (6.02214×10²³)
7. candela cd luminous intensity (typically 18 mW)

Derived units

• radian  [rad ] called: angle. units: 1 = m/m
• steradian [ sr ] called: solid angle. units: 1 = m²/m²
• hertz [ Hz ] called: frequency. units: s⁻¹
• newton [ N ] called: force, weight. units: kg⋅m⋅s⁻²
• pascal [ Pa ] called: pressure, stress. units: N/m² = kg⋅m⁻¹⋅s⁻²
• joule [ J ] called: energy, work, heat. units: N⋅m = kg⋅m²⋅s⁻²
• watt [ W ] called: power, radiant flux . units:J/s = kg⋅m²⋅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⋅m²⋅s⁻³⋅A⁻¹
• farad [ F ] called: electric capacitance . units:C/V = kg⁻¹⋅m⁻²⋅s⁴⋅A²
• ohm [ Ω ] called: electric resistance, impedance, reactance. units: V/A = kg⋅m²⋅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⋅m²⋅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⋅m²⋅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/m² = 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 = m²⋅s⁻²
• sievert [ Sv] called: equivalent dose (of ionizing radiation) . units:J/kg = m²⋅s⁻²
• katal [ kat] called: catalytic activity . units:s⁻¹⋅mol

Advanced Physical Constants

speed of light $c\,$ =2998×10⁸m·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) × 10²³ 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