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: $A=4\pi r^2$;   $V=\frac{4}{3}\pi r^3$

$dA=r\mathrm{d}r\mathrm{d}\theta$

$dA=r^2\sin \theta \mathrm{d}\mathrm{\theta }\mathrm{d}\mathrm{\varphi }$

$\mathrm{d}V=r^2\sin \theta \mathrm{d}r\mathrm{d}\theta \mathrm{d}\phi$

Trigonometry

$\sin A=\frac{\text{opposite}}{\text{hypotenuse}}=\frac{a}{c}.$

$\cos A=\frac{\text{adjacent}}{\text{hypotenuse}}=\frac{b}{c}.$

$\tan A=\frac{\text{opposite}}{\text{adjacent}}=\frac{a}{b}=\frac{\sin A}{\cos A}.$

Vector dot product

$\overrightarrow{A}\cdot \overrightarrow{B}=A_x{B}_x+A_y{B}_y+A_z{B}_z=AB\cos \theta$

$A=\sqrt{\overrightarrow{A}\cdot \overrightarrow{A}}=\sqrt{A_x^2+A_y^2+A_z^2}$

$\widehat{𝐢}\cdot \widehat{𝐢}=\widehat{𝐣}\cdot \widehat{𝐣}=\widehat{𝐤}\cdot \widehat{𝐤}=1$

$\widehat{𝐣}\cdot \widehat{𝐤}=\widehat{𝐤}\cdot \widehat{𝐢}=\widehat{𝐣}\cdot \widehat{𝐤}=0$

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):   $V(\overrightarrow{b})-V(\overrightarrow{a})=-\underset{\overrightarrow{a}}{\overset{\overrightarrow{b}}{\int }}\overrightarrow{E}\cdot \mathrm{d}\overrightarrow{l}=\varphi (\overrightarrow{b})-\varphi (\overrightarrow{a})$

Electric potential due to point charges: $\varphi (\overrightarrow{r})=\frac{1}{4\pi {ϵ}_0}\frac{Q}{r}\to \frac{1}{4\pi {ϵ}_0}\sum \frac{q_j}{r_j}\to \frac{1}{4\pi {ϵ}_0}\int \frac{\rho \mathrm{d}V}{r}\to \frac{1}{4\pi {ϵ}_0}\int \frac{\mathrm{d}\rho }{r}$

Magnetic field: $\overrightarrow{B}=\frac{{\mu }_0}{4\pi }\underset{C}{\int }\frac{I(\mathrm{d}\overrightarrow{\ell }\times \hat{r})}{{\left|\overrightarrow{r}\right|}^2}$

Maxwell's equations

Gauss' Law relating electric field to charge: ${\oint }_S\overrightarrow{E}\cdot \mathrm{d}\overrightarrow{A}=\frac{1}{{ϵ}_0}{Q}_{enc}$

$\mathrm{\nabla }\cdot \overrightarrow{B}=0$

$\mathrm{\nabla }\times \overrightarrow{E}=-\frac{\partial \overrightarrow{B}}{\partial t}$

$\mathrm{\nabla }\times \overrightarrow{B}={\mu }_0\overrightarrow{J}+\frac{1}{c^2}\frac{\partial \overrightarrow{E}}{\partial t}$

$\mathrm{\nabla }\times \overrightarrow{H}=\overrightarrow{J}+\frac{\partial \overrightarrow{D}}{\partial t}$

$\mathrm{\nabla }\cdot \overrightarrow{D}=\rho$

to discard

$\mathrm{\nabla }\cdot \overrightarrow{E}=\frac{\rho }{{ϵ}_0}$

Maxwell's equations integral form

${\oint }_S\overrightarrow{E}\cdot \mathrm{d}\overrightarrow{A}=\frac{1}{{ϵ}_0}{Q}_{enc}$

${\oint }_S\overrightarrow{B}\cdot \mathrm{d}\overrightarrow{A}=0$

${\oint }_C\overrightarrow{E}\cdot \mathrm{d}\overrightarrow{\ell }=-\underset{S}{\int }\frac{\partial \overrightarrow{B}}{\partial t}\cdot \mathrm{d}\overrightarrow{A}$

${\oint }_C\overrightarrow{B}\cdot \mathrm{d}\overrightarrow{\ell }=\mu \underset{S}{\int }\overrightarrow{J}\cdot \mathrm{d}\overrightarrow{A}+\frac{1}{c^2}\underset{S}{\int }\frac{\partial \overrightarrow{D}}{\partial t}\cdot \mathrm{d}\overrightarrow{A}$

${\oint }_S\overrightarrow{D}\cdot \mathrm{d}\overrightarrow{A}=\underset{V}{\int }\rho \mathrm{d}V$

${\oint }_C\overrightarrow{H}\cdot \mathrm{d}\overrightarrow{\ell }=\underset{S}{\int }\overrightarrow{J}\cdot \mathrm{d}\overrightarrow{A}+\underset{S}{\int }\frac{\partial \overrightarrow{D}}{\partial t}\cdot \mathrm{d}\overrightarrow{A}$

permittivity and magnetic permeability

$\overrightarrow{B}=\mu \overrightarrow{H}={\mu }_0(\overrightarrow{H}+\overrightarrow{M})$

$\overrightarrow{D}=ϵ\overrightarrow{E}={ϵ}_0\overrightarrow{E}+\overrightarrow{P}$

$\overrightarrow{D}=ϵ\overrightarrow{E}$

$\overrightarrow{H}=\overrightarrow{B}/\mu$

Force equations

$\overrightarrow{F}=q_e(\overrightarrow{E}+\overrightarrow{v}\times \overrightarrow{B})$

$\overrightarrow{B}=\frac{{\mu }_0}{4\pi }\underset{C}{\int }\frac{Id\overrightarrow{\ell }\times \hat{r}}{|r{|}^2}$

Current and circuits

https://en.wikibooks.org/w/index.php?title=Electronics/Formulas&oldid=2348180

$\overrightarrow{J}=nqv=n_{q}q{\overrightarrow{v}}_{\text{d}}$

$I=JA=\overrightarrow{J}\cdot \overrightarrow{A}$

$\mathrm{\Delta }U=qV$

$I=\frac{\mathrm{d}\mathrm{Q}}{\mathrm{d}\mathrm{t}}$

$P=\frac{\mathrm{d}\mathrm{U}}{\mathrm{d}\mathrm{t}}=IV=I^{2}R=\frac{V^2}{R}$

$V=IR$

$\overrightarrow{E}\cdot \mathrm{\Delta }\overrightarrow{\ell }=\mathrm{\Delta }V$

$Q=CV$

$U=\frac{1}{2}QV=\frac{1}{2}C{V}^2=\frac{Q^2}{2C}$

resistivity

$R=\rho \frac{L}{A}$

$\alpha =\frac{\rho -{\rho }_0}{{\rho }_0}\frac{1}{T-T0}$

$\frac{\mathrm{\Delta }\rho }{\rho }=\alpha \mathrm{\Delta }T$  where  $\mathrm{\Delta }\rho =\rho -{\rho }_0;\mathrm{\Delta }T=T-T_0$

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= $\hslash =h/(2\pi )$= 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 =$\sigma ={\pi }^2{k}^4/60{\hslash }^3{c}^2$ =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) $b=hc{k}^{-1}/$ 4.965 114 231... =2.897 7721(26) × 10⁻³ m·K