Physics
Physics is the study of matter, energy, motion, and fundamental forces, aiming to explain the behavior and structure of the universe through observation and theory.
Subsections of Physics
Constants & Conversions
Subsections of Constants & Conversions
Constants
Universal / Physical Constants
| Symbol | Quantity | Value | Unit | Dimension |
|---|---|---|---|---|
| $c$ | Speed of light in vacuum | $3 \times 10^{8}$ $\text{(exact) } 299,792,458$ | $\mathrm{m \, s^{-1}}$ | $\mathrm{LT^{-1}}$ |
| $h$ | Planck constant | $6.626 \times 10^{-34}$ | $\mathrm{J \, s}$ | $\mathrm{ML^{2}T^{-1}}$ |
| $hc$ | Photon Energy constant | $1242$ | $\mathrm{eV \cdot nm}$ $\mathrm{MeV \cdot fm}$ | $\mathrm{ML^{2}T^{-1}}$ |
| $\hbar = \dfrac{h}{2\pi}$ | Reduced Planck constant | $1.055 \times 10^{-34}$ | $\mathrm{J \, s}$ | $\mathrm{ML^{2}T^{-1}}$ |
| $\mu_0$ | Vacuum Magnetic permeability | $4\pi \times 10^{-7}$ | $\mathrm{N \, A^{-2}}$ | $\mathrm{MLT^{-2}I^{-2}}$ |
| $\varepsilon_0 = \dfrac{1}{\mu_0\,c^2}$ | Vacuum Electric permittivity | $8.854 \times 10^{-12}$ | $\mathrm{F \, m^{-1}}$ | $\mathrm{M^{-1}L^{-3}T^{4}I^{2}}$ |
| $Z_0 = \dfrac{1}{\varepsilon_0}$ | Characteristic impedance of vacuum | $3.77 \times 10^{2}$ | $\mathrm{\Omega}$ | $\mathrm{ML^{2}T^{-3}I^{-2}}$ |
| $G$ | Newtonian constant of gravitation | $6.67 \times 10^{-11}$ | $\mathrm{m^{3} \, kg^{-1} \, s^{-2}}$ | $\mathrm{L^{3}M^{-1}T^{-2}}$ |
| $R = \dfrac{PV}{nT}$ | Molar / Universal Gas constant | $8.314 \approx 25/3$ $0.082 \approx 1/12$ $62.36$ | $\mathrm{J \, mol^{-1} \, K^{-1}}$ $\mathrm{L \,atm \, mol^{-1} \, K^{-1}}$ $\mathrm{L \,torr \, mol^{-1} \, K^{-1}}$ | $\mathrm{ML^{2}T^{-2}K^{-1}}$ |
| $N_A$ | Avagadro constant | $6.022 \times 10^{-23}$ | $\mathrm{mol^{-1}}$ | $\mathrm{ML^{2}T^{-2}K^{-1}}$ |
| $N_A\,h$ | Molar Planck constant | $4 \times 10^{-10}$ | $\mathrm{J \, s \, mol^{-1}}$ | $\mathrm{ML^{2}T^{-2}K^{-1}}$ |
| $k_B = \dfrac{R}{N_A}$ | Boltzmann constant | $1.38 \times 10^{-23}$ | $\mathrm{J \, K^{-1}}$ | $\mathrm{ML^{2}T^{-2}K^{-1}}$ |
| $\sigma$ | StefanāBoltzmann constant | $5.67 \times 10^{-8}$ | $\mathrm{W \, m^{-2} \, K^{-4}}$ | $\mathrm{MT^{-3}K^{-4}}$ |
| $\mathrm{F} = N_A\,e$ | Faraday constant | $9.65 \times 10^{4}$ | $\mathrm{C \, mol^{-1}}$ | $\mathrm{MT^{-3}K^{-4}}$ |
| $e^-$ | Elementary Charge | $1.602 \times 10^{-19}$ | $\mathrm{C}$ | $\mathrm{TI}$ |
| $m_e$ | Electron mass | $9.11 \times 10^{-31}$ | $\mathrm{kg}$ | $\mathrm{M}$ |
| $m_p$ | Proton mass | $1.6726 \times 10^{-27}$ | $\mathrm{kg}$ | $\mathrm{M}$ |
| $m_n$ | Neutron mass | $1.6749 \times 10^{-27}$ | $\mathrm{kg}$ | $\mathrm{M}$ |
| $m_p / m_e$ | Proton-to-electron mass ratio | $1.84 \times 10^{3}$ | $-$ | Dimensionless |
| $m_{\mu}$ | Muon mass | $1.88 \times 10^{-28}$ | $\mathrm{kg}$ | $\mathrm{M}$ |
| $m_{\tau}$ | Tau mass | $3.16 \times 10^{-27}$ | $\mathrm{kg}$ | $\mathrm{M}$ |
| $\alpha = \dfrac{e^2}{2\epsilon_0hc}$ | Fine Structure constant | $7.297 \times 10^{-3} \approx \dfrac{1}{137}$ | $-$ | Dimensionless |
| $\alpha^{-1}$ | Inverse fine structure constant | $137.036$ | $-$ | Dimensionless |
| $m_u = \dfrac{m({}^{12}\mathrm{C})}{N_A}$ | Atomic mass unit | $1.66 \times 10^{-27}$ | $\mathrm{kg}$ | $\mathrm{M}$ |
| $\mu_B = \dfrac{he}{4\pi m_e}$ | Bohr Magneton | $9.274 \times 10^{-24}$ | $\mathrm{J \, T^{-1}}$ | $\mathrm{L^{2}IT^{-2}}$ |
| $R_{\infty} = \dfrac{m_e e^4}{8\epsilon_0 h^3c}$ | Rydberg constant | $1.10 \times 10^{7}$ | $\mathrm{m^{-1}}$ | $\mathrm{L^{-1}}$ |
| $R_{\infty}\,hc$ | Rydberg Unit of Energy | $2.18 \times 10^{-18}$ $13.6$ | $\mathrm{J}$ $\mathrm{eV}$ | $\mathrm{L^{-1}}$ |
| $a_0 = \dfrac{h\epsilon_0}{\pi e^2 m_e}$ | Bohr radius | $5.29 \times 10^{-11}$ | $\mathrm{m}$ | $\mathrm{L}$ |
| $b$ | Wien wavelength displacement constant | $2.90 \times 10^{-3}$ | $\mathrm{m \, K}$ | $\mathrm{LK}$ |
| $b^\prime$ | Wien frequency displacement law constant | $5.88 \times 10^{10}$ | $\mathrm{Hz \, K^{-1}}$ | $\mathrm{T^{-1}K^{-1}}$ |
| $b_{\text{entropy}}$ | Wien entropy displacement law constant | $3.00 \times 10^{-3}$ | $\mathrm{m \, K}$ | $\mathrm{LK}$ |
| $r_e$ | Classical electron radius | $2.82 \times 10^{-15}$ | $\mathrm{m}$ | $\mathrm{L}$ |
| $E_{\text{ion}}$ | Ionization Energy of hydrogen | $2.18 \times 10^{-18}$ | $\mathrm{J}$ | $\mathrm{ML^{2}T^{-2}}$ |
Derived / Composite Constants
| Symbol | Quantity | Value | Unit | Dimension |
|---|---|---|---|---|
| $V_{\text{molar}} = \dfrac{RT}{P}$ | Molar Volume of Ideal Gas at: | $\text{See Below}$ | $-$ | $-$ |
| (normal) $V_{\text{STP}}$ | $T = 0^{\circ}\mathrm{\,C} = 273.15\mathrm{\,K}\:,\quad P = 101.325\mathrm{\,kPa} = 1\mathrm{\,atm}$ | $22.4$ | $\mathrm{L}$ | $L^{3}$ |
| (new) $V_{\text{STP}}$ | $T = 0^{\circ}\mathrm{\,C} = 273.15\mathrm{\,K}\:,\quad P = 100\mathrm{\,kPa} = 0.987\mathrm{\,atm}$ | $22.7$ | $\mathrm{L}$ | $L^{3}$ |
| $V_{\text{NTP}}$ | $T = 20^{\circ}\mathrm{\,C} = 293.15\mathrm{\,K}\:,\quad P = 101.325\mathrm{\,kPa} = 1\mathrm{\,atm}$ | $24.0$ | $\mathrm{L}$ | $L^{3}$ |
| $V_{\text{SATP}}$ | $T = 25^{\circ}\mathrm{\,C} = 298.15\mathrm{\,K}\:,\quad P = 101.325\mathrm{\,kPa} = 1\mathrm{\,atm}$ | $24.5$ | $\mathrm{L}$ | $L^{3}$ |
Empirical / Local Constants
| Symbol | Quantity | Value | Unit | Dimension |
|---|---|---|---|---|
| $g_{\text{earth}}$ | Earth’s acceleration due to gravity | $9.81$ | $\mathrm{m \, s^{-2}}$ | $\mathrm{L^{3}M^{-1}T^{-2}}$ |
References
Note: This page includes several unconventional approximations, often tailored for exams where calculators arenāt permitted.
Conversions
Metric Prefixes
| Prefix | Symbol | Meaning | Prefix | Symbol | Meaning |
|---|---|---|---|---|---|
| quetta | $Q$ | $10^{30}$ | quecto | $q$ | $10^{-30}$ |
| ronna | $R$ | $10^{27}$ | ronto | $r$ | $10^{-27}$ |
| yotta | $Y$ | $10^{24}$ | yocto | $y$ | $10^{-24}$ |
| zetta | $Z$ | $10^{21}$ | zepto | $z$ | $10^{-21}$ |
| exa | $E$ | $10^{18}$ | atto | $a$ | $10^{-18}$ |
| peta | $P$ | $10^{15}$ | femto | $f$ | $10^{-15}$ |
| tera | $T$ | $10^{12}$ | pico | $p$ | $10^{-12}$ |
| giga | $G$ | $10^{9}$ | nano | $n$ | $10^{-9}$ |
| mega | $M$ | $10^{6}$ | micro | $\mu$ | $10^{-6}$ |
| kilo | $k$ | $10^{3}$ | milli | $m$ | $10^{-3}$ |
| hecto | $h$ | $10^{2}$ | centi | $c$ | $10^{-2}$ |
| deka | $da$ | $10^{1}$ | deci | $d$ | $10^{-1}$ |
| (base unit) | - | $10^{0}$ | (base unit) | - | $10^{0}$ |
Conversions
Length
$\begin{aligned} 1 \:m &= 39.37( \approx 243/8) \:in &&= 3.28( \approx 105/32) \:ft &&= 1.094( \approx 11/10) \:yd \\ 1 \:in &= 2.54 \:cm &&= 1/12 \:ft \\ 1 \:ft &= 12 \:in &&= 0.3048 \:m \\ 1 \:km &= 0.6214 \:mi &&= 3281 \:ft \\ 1 \:mi &= 5280 \:ft &&= 1.609 \:km \\ 1 \:\text{light-year} &= 9.461 \times 10^{12} \:km \end{aligned}$
Temperature
$\begin{aligned} \text{Kelvin, } & K &&= {}^\circ C + 273.15 \\ \text{Celsius, } & {}^\circ C &&= K - 273.15 &&= \dfrac{5}{9}({}^\circ F - 32) \\ \text{Fahrenheit, } & {}^\circ F &&= \dfrac{9}{5}{}^\circ C + 32 \\ \text{Rankine, } & {}^\circ R &&= {}^\circ F + 459.67&&= \dfrac{5}{9}K \end{aligned}$
Speed
$\begin{aligned} km/h &= \dfrac{5}{18} \:m/s , & m/s &= \dfrac{18}{5} \:km/h \\ mi/h &= 0.447 \:m/s , &ft/s &= 0.305 \:m/s \\ \end{aligned}$
Mass
$\begin{aligned} 1 \:kg &= 2.204 \:lb &&= 35.274 \:oz \\ 1 \:lb &= 0.4536 \:kg &&= 16 \:oz \\ 1 \:oz &= 0.0283 \:kg \\ 1 \:amu &= 1.66 \times 10^{-27} \:kg \end{aligned}$
Force
$\begin{aligned} 1 \:N &= 10^5 \:dyn &&= 0.2248 \:lbf \\ 1 \:dyn &= 10^{-5} \:N \\ 1 \:lbf &= 4.448 \:N \end{aligned}$
Area
$\begin{aligned} 1 \:m^2 &= 10.764 \:ft^2 &&= 1550 \:in^2 \\ 1 \:in^2 &= 6.45 \:cm^2 \\ 1 \:acre &= 4047 \:m^2 &&= 43560 \:ft^2 \\ 1 \:hectare &= 10^4 \:m^2 \\ 1 \:mi^2 &= 2.59 \:km^2 &&= 640 \:acres \end{aligned}$
Volume
$\begin{aligned} 1 \:m^3 &= 10^3 \:L &&= 35.315 \:ft^3 &&= 264.2 \:gal \\ 1 \:cm^3 &= 1 \:mL &&= 0.061 \:in^3 \\ 1 \:L &= 10^3 \:cm^3 &&= 0.264 \:gal \\ 1 \:ft^3 &= 7.48 \:gal &&= 28.317 \:L \\ 1 \:gal &= 3.785 \:L &&= 231 \:in^3 \end{aligned}$
Pressure
$\begin{aligned} 1 \:kPa &= 10^3 \:N/m^2 &&= 10^{-2} \:bar &&= 9.87 \times 10^{-3} \:atm \\ 1 \:atm &= 101.325 \:kPa &&= 1.013 \:bar &&= 760 \:\text{mmHg (Torr)} \\ 1 \:bar &= 10^2 \:kPa &&= 14.5 \:psi \\ 1 \:psi &= 6.895 \:kPa \\ 1 \:\text{Torr} &= 0.133 \:Pa && (\vec{g} = 9.80665 \:m/s^2) \end{aligned}$
Work/Heat
$\begin{aligned} 1 \:J &= 624.15 \times 10^{10} \:MeV &&= 10^7 \:erg \\ 1 \:eV &= 1.602 \times 10^{-19} \:J \\ 1 \:cal &= 4.184 \:J \\ 1 \:Btu &= 1055 \:J \\ 1 \:\text{kWh} &= 3.6 \times 10^6 \:J &&= 3412 \:Btu \end{aligned}$
Power
$\begin{aligned} 1 \:W &= 1 \:J/s &&= 0.7376 \:ft \cdot lbf/s \\ 1 \:hp &= 745.7 \:W \end{aligned}$
Angle
$\begin{aligned} 1^\circ \text{ (degree)} &= \dfrac{\pi}{180} \:\text{rad} &&= 0.01745 \:\text{rad} \\ 1^\circ &= 60'\text{ (minutes)} \\ 1' &= 60''\text{ (seconds)} \\ 1 \:\text{rad} &= \dfrac{180^\circ}{\pi} \: &&= 57.30^\circ \\ 1 \:\text{revolution} &= 360 \:{}^\circ &&= 2 \pi \:\text{rad} \\ 1 \:\text{rev/min (rpm)} &= 0.1047\:\text{rad/s} \end{aligned}$
References
Units
Derived Units
| Quantity | Name | Symbol | Other Units | Base Units |
|---|---|---|---|---|
| Plane Angle | Radian | $\text{rad}$ | $\dfrac{\text{m}}{\text{m}}$ | |
| Solid Angle | Steradian | $\text{sr}$ | $\dfrac{\text{m}^2}{\text{m}^2}$ | |
| Frequency | Hertz | $\text{Hz}$ | $\dfrac{1}{\text{s}}$ | |
| Force | Newton Dyne | $\text{N}$ $\text{dyne}$ | $\dfrac{\text{kg} \cdot \text{m}}{\text{s}^2}$ $\dfrac{\text{g} \cdot \text{cm}}{\text{s}^2}$ | |
| Pressure, Stress | Pascal Barye | $\text{Pa}$ $\text{Ba}$ | $\dfrac{\text{N}}{\text{m}^2}$ $\dfrac{\text{dyne}}{\text{cm}^2}$ | $\dfrac{\text{kg}}{\text{m} \cdot \text{s}^2}$ |
| Energy, Work, Heat | Joule Erg | $\text{J}$ $\text{erg}$ | $\text{N} \cdot \text{m}$ $\text{dyne} \cdot \text{cm}$ | $\dfrac{\text{kg} \cdot \text{m}^2}{\text{s}^2}$ $\dfrac{\text{g} \cdot \text{cm}^2}{\text{s}^2}$ |
| Power, Heat Flow | Watt | $\text{W}$ | $\dfrac{\text{J}}{\text{s}}$ | $\dfrac{\text{kg} \cdot \text{m}^2}{\text{s}^3}$ |
| Electric Charge | Coulomb | $\text{C}$ | $\text{A} \cdot \text{s}$ | |
| Electric Potential | Volt | $\text{V}$ | $\dfrac{\text{W}}{\text{A}}$ | $\dfrac{\text{kg} \cdot \text{m}^2}{\text{A} \cdot \text{s}^3}$ |
| Capacitance | Farad | $\text{F}$ | $\dfrac{\text{C}}{\text{V}}$ | $\dfrac{\text{A}^2 \cdot \text{s}^4}{\text{kg} \cdot \text{m}^2}$ |
| Resistance | Ohm | $\Omega$ | $\dfrac{\text{V}}{\text{A}}$ | $\dfrac{\text{kg} \cdot \text{m}^2}{\text{A}^2 \cdot \text{s}^3}$ |
| Conductance | Siemens | $\text{S}$ | $\dfrac{\text{A}}{\text{V}}$ | $\dfrac{\text{A}^2 \cdot \text{s}^3}{\text{kg} \cdot \text{m}^2}$ |
| Magnetic Flux | Weber | $\text{Wb}$ | $\text{V} \cdot \text{s}$ | $\dfrac{\text{kg} \cdot \text{m}^2}{\text{A} \cdot \text{s}^2}$ |
| Magnetic Flux Density | Tesla | $\text{T}$ | $\dfrac{\text{Wb}}{\text{m}^2}$ | $\dfrac{\text{kg}}{\text{A} \cdot \text{s}^2}$ |
| Inductance | Henry | $\text{H}$ | $\dfrac{\text{Wb}}{\text{A}}$ | $\dfrac{\text{kg} \cdot \text{m}^2}{\text{A}^2 \cdot \text{s}^2}$ |
| Celsius Temperature | Degree Celsius | ${}^{\circ} C$ | $K$ | |
| Luminous Flux | Lumen | $\text{lm}$ | $\text{cd} \cdot \text{sr}$ | $\dfrac{\text{cd} \cdot \text{m}^2}{\text{m}^2}$ |
| Illuminance | Lux | $\text{lx}$ | $\dfrac{\text{lm}}{\text{m}^2}$ | $\dfrac{\text{cd}}{\text{m}^2}$ |
| Activity | Becquerel | $\text{Bq}$ | $\dfrac{1}{\text{s}}$ |
Units Named After People
| Unit | Symbol | Scientist | Quantity |
|---|---|---|---|
| Becquerel | $\text{Bq}$ | Henri Becquerel | Activity |
| Bel* | $\text{B}$ | Alexander Graham Bell | Level |
| Coulomb | $\text{C}$ | Charles-Augustin Coulomb | Electric Charge |
| Degree Celsius | ${}^{\circ} \text{C}$ | Anders Celsius | Celsius Temperature |
| Dalton* | $\text{Da}$ | John Dalton | Mass |
| Farad | $\text{F}$ | Michael Faraday | Capacitance |
| Gray | $\text{Gy}$ | Louis Gray | Absorbed Dose |
| Henry | $\text{H}$ | Joseph Henry | Inductance |
| Hertz | $\text{Hz}$ | Heinrich Hertz | Frequency |
| Joule | $\text{J}$ | James Joule | Energy, Work, Heat |
| Kelvin | $K$ | William Thomson, Lord Kelvin | Temperature |
| Newton | $\text{N}$ | Isaac Newton | Force |
| Ohm | $\Omega$ | Georg Ohm | Resistance |
| Pascal | $\text{Pa}$ | Blaise Pascal | Pressure, Stress |
| Poise | $\text{P}$ | Jean Poiseuille | Dynamic Viscosity |
| Siemens | $\text{S}$ | Werner von Siemens | Conductance |
| Stokes | $\text{St}$ | George Stokes | Kinematic Viscosity |
| Tesla | $\text{T}$ | Nikola Tesla | Magnetic Field |
| Volt | $\text{V}$ | Alessandro Volta | Electric Potential |
| Watt | $\text{W}$ | James Watt | Power, Heat Flow |
| Weber | $\text{Wb}$ | Wilhelm Weber | Magnetic Flux |
References
Electromagnetism
Subsections of Electromagnetism
Induction
Electromagnetic Induction
Faraday’s Law
Whenever the flux of magnetic field through the area bounded by a closed conducting loop changes, and emf in produced in the loop.
EMF induced $(\mathcal{E})$:
$\mathcal{E} = -\dfrac{d\Phi}{dt}$Flux ($\Phi$):
$\Phi = \int{\vec{B}\cdot \vec{dS}} = BA \cos \theta$
Lenz’s Law
- The direction of induced current is such theat it opposes the change that has induced it.
Motional EMF
- EMF in a conductor moving with velocity $v$ in magnetic field $B$:
$\mathcal{E} = vBl$
Induced Electric Field
- Induced electric field $E$ around a loop: $\oint E \, dl = -\dfrac{d\Phi}{dt}$
Eddy Current
- Electromagnetic damping.
- Circular currents induced in conductors due to changing magnetic flux.
- $i \propto \left|\dfrac{d\Phi}{dt}\right|$.
Self-Induction
$\Phi = Li$
Induced EMF $(\mathcal{E})$ in coil due to its own current $I$:
$\mathcal{E} = -L \dfrac{di}{dt}$
Inductors
Self-Inductance of a Long Solenoid
$L = \mu_0\:n^2Al$
$n$: Turns per unit length,
$A$: Cross-sectional area,
$l$: Length of solenoid.
Growth and Decay of Current in an LR Circuit
- Growth:
$i = i_0 \biggr(1 - e^{-t/\tau} \biggr)$ - Decay:
$i = i_0\: e^{-t/\tau}$ - Time constant ($\tau$):
$\tau = \dfrac{L}{R}$
At $t = \tau$,
Growth: $i = i_0 (1-\dfrac{1}{e}) = 0.63 i_0$
Decay: $i = i_0 \dfrac{1}{e} = 0.37 i_0$
Energy Stored in an Inductor
- Energy $(U)$: $U = \dfrac{1}{2} L i^2$
Energy Density in a Magnetic Field
$B = \mu_0\:ni$
$u = \dfrac{B^2}{2 \mu_0}$
Mutual Induction
Mutual Inductance $(M)$:
$M = \dfrac{\mu_0 N_1 N_2 A}{l}$Induced EMF $\mathcal{E}_2$ in $\text{coil}_2$ due to change in current $i_1$ in $\text{coil}_1$:
$\mathcal{E}_2 = -M \dfrac{di_1}{dt}$