Physics equations/Collection of equations

Organization: These equations are under reorganization:

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
$$\vec{A}\cdot \vec{B} = A_xB_x+A_yB_y+A_zB_z = AB\cos\theta$$

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

$$\hat \mathbf i\cdot \hat \mathbf i = \hat \mathbf j\cdot \hat \mathbf j =\hat \mathbf k\cdot \hat \mathbf k =1$$

$$\hat \mathbf j\cdot \hat \mathbf k = \hat \mathbf k\cdot \hat \mathbf i =\hat \mathbf j\cdot \hat \mathbf k =0$$

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
$$\vec{F}=q_e\left(\vec{E}+\vec{v}\times\vec{B}\right)$$

$$ \vec{B} = \frac{\mu_0}{4\pi}\int_C \frac{I d\vec{\ell} \times \hat r}{|r|^2}$$

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) $$b = h c k^{-1} / \,$$ 4.965 114 231... =2.897 7721(26) × 10−3 m·K