View source for Physics equations/Equations

<div style="text-align: center;"></div>

<div style="text-align: center;">PLEASE DO NOT EDIT THIS WITHOUT FIRST POSTING ON THE TALK PAGE.  See '''[[Talk:Physics_equations/Equations|talk page]]''' for why this request in necessary. If you want to use these equations, see'''[[:Category:Physics equations/Sheet]]''' for versions that are suitable for printing as pdf files.</div>
==Physics equations==
====00-Mathematics for this course====
<section begin=00-Mathematics_for_this_course/>
[[File:Theta in radians minimalistic.svg|right|130px]][[File:Vector components with unit vectors.svg|right|180px]]
&nbsp;&nbsp;&nbsp;Measured in radians, <math>\theta = s/r</math> defines angle (in radians), where ''s'' is arclength and ''r'' is radius. The circumference of a circle is <math>C_\odot = \,2\pi r</math> and the circle's area is <math>A_\odot = \,\pi r^2</math> is its area.  The surface area of a sphere is <math>A_\bigcirc = 4\pi r^2</math> and sphere's volume is <math>V_\bigcirc = \frac{4}{3}\pi r^3</math> 

{{spaces|4}}A [[w:vector|vector]] can be expressed as, <math>\vec A = A_x \, \hat i+ A_y \, \hat j</math>, where   <math>A_x=A\cos\theta</math>, and  <math>A_y=A\sin\theta</math> are the x and y [[w:Tangential and normal components|components]].  Alternative notation for the [[w:unit vectors|unit vectors]] <math>(\hat i, \hat j)</math> include <math>(\hat x, \hat y)</math> and  <math>(\widehat {e_1}, \widehat {e_2})</math>.  An important vector is the [[w:Position (vector)|displacement]] from the origin, with components are typically written without subscripts: <math>\vec r = x\hat x + y\hat y</math>. The [[w:magnitude|magnitude]] (or ''absolute value'' or ''norm'') of a vector is is <math>A \equiv |\vec A|= \sqrt{A_x^2 + A_y^2} \quad</math>, where the [[w:angle|angle]] (or [[w:phase (waves)|phase]]), <math>\theta</math>, obeys <math>\tan\theta=y/x</math>, or (almost) equivalently, <math>\theta = \arctan{(y/x)} \quad</math>.  As with any function/[[w:inverse function|inverse function]] pair, the [[w:Trigonometric_functions#Sine.2C_cosine_and_tangent|tangent]] and [[w:Inverse trigonometric functions|arctangent]] are related by <math>\tan(\tan^{-1}\mathcal X) = \mathcal X</math> where <math>\mathcal X = y/x</math>.  The arctangent is not a true function because it is [[w:Multivalued function|multivalued]], with <math>\tan^{-1}(\tan \theta) = \theta\; or\; \theta+\pi</math>.
[[File:Vectors A+B=C.png|right|150px]]
{{spaces|4}}The geometric interpretations of <math>\vec A +\vec B =\vec C</math> and <math>\vec B = \vec C -\vec A</math> are shown in the figure.  Vector addition and subtraction can also be defined through the components: <math>\vec A +\vec B =\vec C \Leftrightarrow</math> <math>A_x + B_x =C_x</math> AND <math>A_y + B_y =C_y</math>
<section end=00-Mathematics_for_this_course/>

===01-Introduction===
<section begin=01-Introduction/>
{| class="infobox" style="padding: 0; text-align: center; width: 0"
|
{| style="border: 1px #aaa solid"
!style="background:#edf; text-align: center"|Text
!style="background:#edf; text-align: center"|Symbol
!style="background:#edf; text-align: center"|Factor
!style="background:#edf; text-align: center"|<small>Exponent</small>
|-
|giga
|align="center"|G
|align="right"|<small>{{gaps|1|000|000|000}}</small>
|align="right"|<small>E</small>9
|-
|mega
|align="center"|M
|align="right"|<small>{{gaps|1|000|000}}</small>
|align="right"|<small>E</small>6
|-
|kilo
|align="center"|k
|align="right"|<small>{{gaps|1|000}}</small>
|align="right"|<small>E</small>3
|-
|(none)
|align="center"|(none)
|1
|align="right"|<small>E</small>0
|-
|centi
|align="center"|c
|align="left"|0.01
|align="right"|<small>E</small>−2
|-
|milli
|align="center"|m
|align="left"|0.001
|align="right"|<small>E</small>−3
|-
|micro
|align="center"|μ
|align="left"|<small>{{gaps|0.000|001}}</small>
|align="right"|<small>E</small>−6
|-
|nano
|align="center"|n
|align="left"|<small>{{gaps|0.000|000|001}}</small>
|align="right"|<small>E</small>−9
|-
|pico
|align="center"|p
|align="left"|<small>{{gaps|0.000|000|000|001}}</small>
|align="right"|<small>E</small>−12
|}
|}
* 1 kilometer = .621 miles and  1 MPH = 1 mi/hr ≈ .447 m/s
* Typically  air density is 1.2kg/m<sup>3</sup>, with pressure 10<sup>5</sup>Pa. The density of water is 1000kg/m<sup>3</sup>. 
*Earth's mean radius ≈ 6371km, mass ≈ {{val|6|e=24|u=kg}}, and gravitational acceleration = {{nowrap begin}}g ≈ 9.8m/s<sup>2</sup>{{nowrap end}}
*Universal gravitational constant = G ≈ {{val|6.67|e=-11|u=m<sup>3</sup>·kg<sup>−1</sup>·s<sup>−2</sup>}}
*Speed of sound ≈ 340m/s and {{nowrap begin}}the speed of light = c ≈ 3×10<sup>8</sup>m/s{{nowrap end}}
*One light-year ≈ 9.5×10<sup>15</sup>m ≈ 63240AU (Astronomical unit) 
*The electron has charge, e ≈ 1.6 × 10<sup>−19</sup>C and mass ≈ 9.11 × 10<sup>-31</sup>kg. 1eV = {{nowrap begin}}1.602 × 10<sup>-19</sup>J{{nowrap end}} is a unit of energy, defined as the work associated with moving one electron through a potential difference of one volt.
* 1 amu =  1 u ≈ 1.66 × 10<sup>-27</sup> kg is the approximate mass of a proton or neutron.
*'''Boltzmann's constant''' = k<sub>B</sub>≈ {{nowrap|1.38 × 10<sup>-23</sup> [[w:Joule|J]] [[w:Kelvin|K]]<sup>−1</sup> }}, and the '''gas constant''' is R&nbsp;= {{nowrap|N<sub>A</sub>k<sub>B</sub>}}≈{{nowrap|8.314 [[w:Joule|J]]   [[w:Kelvin|K]]<sup>−1</sup>  [[w:Mole (unit)|mol]]<sup>−1</sup>}}, where N<sub>A</sub>≈ {{nowrap|6.02 × 10<sup>23</sup>}} is the Avogadro number.
*<math>k_{\mathrm{e}} = \frac{1}{4\pi\varepsilon_0} \,</math>≈ 8.987× 10<sup>9</sup> N·m²·C<sup>−2</sup> is a fundamental '''constant of electricity'''; also <math> \varepsilon_0 = \frac{1}{  4\pi k_{\mathrm{e}}  }  \, </math> ≈ 8.854 × 10<sup>−12</sup> F·m<sup>−1</sup> is the [[w:vacuum permittivity|vacuum permittivity]] or the [[electric constant]].
*<math> \mu_0 \,</math> = 4π × 10<sup>−7</sup> NA ≈ 1.257 × 10<sup>−6</sup> N A (magnetic permeability) is the fundamental constant of {{nowrap begin}}magnetism: <math>\sqrt{\varepsilon_0\mu_0}=1/c</math>.{{nowrap end}}
*<math>\hbar </math> = h/(2π) ≈ 1.054×10<sup>−34</sup> J·s the reduced Planck constant, and <math>a_0 = \frac{\hbar^2}{k_e m_e e^2} </math> {{nowrap begin}}≈ .526 × 10<sup>−10</sup> m is the Bohr radius.{{nowrap end}}
<section end=01-Introduction/>

===02-One dimensional kinematics===
<section begin=02-One_dimensional_kinematics/>
[[w:difference operator|Difference]] is denoted by <math>d\mathcal X</math>, <math>\delta\mathcal X</math>, or the [[w:Delta (letter)|''Delta'']]. <math>\Delta\mathcal X = \mathcal X_f - \mathcal X_i</math> or <math>\mathcal X - \mathcal X_0</math>.  Average, or [[w:mean|mean]], is denoted by <math>\bar{\mathcal X}=\langle \mathcal X\rangle = \mathcal X_\text{ave}=\Sigma \mathcal X_i\mathcal/N \ \text{or} \ \Sigma \mathcal P_i\mathcal X_i</math>, where <math>\mathcal N</math> is number and <math>\mathcal P_i</math> are probabilities. The average [[w:velocity|velocity]] is <math>\bar v = \Delta x/\Delta t</math>, and the average [[w:acceleration|acceleration]] is <math>\bar a = \Delta v/\Delta t</math>, where <math>x</math> denotes position. <small>In CALCULUS, </small>[[wikt:instantaneous values|<small>instantaneous values</small>]]<small> are denoted by {{nowrap begin}}''v(t)=dx/dt''{{nowrap end}} and {{nowrap begin}}''a=dv/dt=d<sup>2</sup>x/dt<sup>2</sup>''.{{nowrap end}}</small> 

The equations of motion for [[w:Acceleration#Uniform_acceleration|uniform acceleration]] are: <math>x= x_0 + v_0t+\tfrac 1 2 at^2</math>, and, <math>v = v_0+at</math>. Also, <math>v^2  = v_0^2 + 2a\left( x - x_0 \right)</math>, and, <math>x - x_0   = \tfrac 1 2 (v_0+v)= \bar vt</math>.  Note that <math>\bar v = \tfrac 1 2 (v_0+v)</math> only if the acceleration is uniform. 
<section end=02-One_dimensional_kinematics/>

===03-Two-Dimensional Kinematics===
<section begin=03-Two-Dimensional_Kinematics/>
[[File:Free fall velocity vectors.gif|right]]
{|
|-
| <math>x=x_0+v_{0x}\Delta t+\frac{1}{2}a_x\Delta t^2</math>
||{{spaces|5}}<math>v_x=v_{0x}+a_x \Delta t</math>
||{{spaces|5}}<math>v_x^2=v_{x0}^2+ 2a_x\Delta x </math>
|-
| <math>y=y_0+v_{0y}\Delta t+\frac{1}{2}a_y\Delta t^2</math>
||{{spaces|5}}<math>v_y=v_{0y}+a_y \Delta t</math>
||{{spaces|5}}<math>v_y^2=v_{x0}^2+ 2a_y\Delta y </math>
|}
<math>v^2=v_0^2+ 2a_x\Delta x + 2a_y\Delta y</math> &nbsp; ...<small>in advanced notation this becomes</small>  <math>\Delta (v^2) = 2\vec a\cdot\Delta\vec\ell</math>.

In [[w:free fall|free fall]] we often set, ''a<sub>x</sub>=0''  and  ''a<sub>y</sub>= -g''.  If angle is measured with respect to the x axis:

<math> v_x =  v\cos\theta</math> {{spaces|6}}<math> v_y =  v\sin\theta</math> {{spaces|6}}<math> v_{x0} =  v_0\cos\theta_0</math> {{spaces|6}}<math> v_{y0} =  v_0\sin\theta_0</math>
[[File:Relative motion man on train.gif|360px|right]]
The figure shows a Man moving relative to Train with velocity, <math>\vec v_{M|T}</math>, where the velocity of the train relative to Earth is, <math>\vec v_{T|E}</math> is the velocity of the Train relative to Earth.  The velocity of the Man relative to Earth is,

{{spaces|5}}<math>\underbrace{\vec v_{M|E}}_{50\;km/hr}=\underbrace{\vec v_{M|T}}_{10\;km/hr}+\underbrace{\vec v_{T|E}}_{40\;km/hr}\,</math> If the speeds are relativistic, define u=v/c where c is the speed of light, and this formula must be modified to: <math>u_{A|O}=  \frac{u_{A|O'}+ u_{O'|O}}{1+ (u_{A|O\,'})(u_{O\,'|O})}</math>
<section end=03-Two-Dimensional_Kinematics/>

===04-Dynamics: Force and Newton's Laws===
<section begin=04-Dynamics:_Force_and_Newton's_Laws/>
[[File:Free body1.3.svg|200px|right]][[File:Threetensions.jpg|250px|right]]
[[w:Newton's laws of motion|Newton's laws of motion]],  can be expressed with two equations, <math> m\vec a = \sum\vec F_j  \,</math> and <math>\vec F_{ij}=-\vec F_{ji}</math>.  The second represents the fact that the force that the ''{{nowrap|i-th}}'' object exerts one object exerts on the ''{{nowrap|j-th}}'' object is equal and opposite the force that the ''{{nowrap|j-th}}'' exerts on the ''{{nowrap|i-th}}'' object. Three [[w:Force#Non-fundamental_forces|non-fundamental]] forces are:  
#The [[w:normal force|normal force]], <math>N</math>, is a [[w:contact force|contact forces]]  that is perpendicular to the surface,
#The force of [[w:friction|friction]], <math>f</math>, is a [[w:contact force|contact force]]  that is parallel to the surface.  
#[[w:Tension (physics)|Tension]], <math>T</math>, is often associated with ropes and strings.  If the rope has sufficiently low [[w:weight|weight]] and of all external forces act at the two ends, then this [[w:tension (physics)|tension]] is distributed uniformly along the rope.  
#The fourth force is [[w:Force#Fundamental_forces|fundamental]]: [[w:Weight|Weight]] equals <math>mg</math>, and is the force of gravity acting on an object of mass, <math>m</math>.  At Earth's surface, <math>g\approx9.8m/s^2</math>.

{{spaces|5}}The x and y components of the three forces of tension on the small grey circle where the three "massless" ropes meet are:
:<math>T_{1x}=-T_1\cos\theta_1 </math> ,&nbsp;  &nbsp;  &nbsp;  &nbsp; <math> T_{1y}=T_1\sin\theta_1</math>
:<math>T_{2x}=0 </math> ,&nbsp; &nbsp;  &nbsp; &nbsp;  &nbsp; &nbsp;  &nbsp; &nbsp;  &nbsp; &nbsp; &nbsp;  &nbsp; &nbsp;  &nbsp;  &nbsp;<math>T_{2y}=-mg</math>
:<math>T_{3x}=T_3\cos\theta_3 </math> ,&nbsp;  &nbsp;  &nbsp;  &nbsp;  &nbsp; <math>T_{3y}=T_3\sin\theta_3</math>
<section end=04-Dynamics:_Force_and_Newton's_Laws/>

===05-Friction, Drag, and Elasticity===
<section begin=05-Friction,_Drag,_and_Elasticity/>
*<math>f_k=\mu_k N</math> is an approximation for the force friction when an object is sliding on a surface, where μ<sub>k</sub> ("mew-sub-k") is  the kinetic coefficient of friction, and ''N'' is the normal force.
*<math>f_s\le \mu_sN</math> approximates the maximum possible friction (called static friction) that can occur before the object begins to slide. Usually ''μ<sub>s</sub>''&nbsp;>&nbsp;''μ<sub>k</sub>''.  <small>Also, </small>[[w: Drag (physics)|<small>air drag</small>]]<small> often depends on speed, an effect this model fails to capture.</small>
<section end=05-Friction,_Drag,_and_Elasticity/>

===06-Uniform Circular Motion and Gravitation===
<section begin=06-Uniform_Circular_Motion_and_Gravitation/>
[[File:UniCircMot gv52.png|thumb|200px|uniform circular motion (here the Latin ''d'' was used instead of the Greek ''Δ'']]
* <math>2\pi\;rad=360\;deg=1\;rev</math> relates the radian, degree, and revolution.

*<math>f = \frac{\#\,\text{revs}}{\#\,\text{secs}}</math> is the number of revolutions per second, called '''[[w:frequency|frequency]]'''.

*<math>T = \frac{\#\,\text{secs}}{\#\,\text{revs}}</math> is the number of seconds per revolution, called '''[[w:period|period]]'''.  Obviously <math>fT = 1</math>.

*<math>\omega = \frac{\Delta\theta}{\Delta t}</math> is called '''[[w:angular frequency|angular frequency]]''' (ω is called omega, and &theta; is measured in radians).  Obviously <math>\omega T = 2\pi</math>

* <math>a=\frac{v^2}{r}=\omega v =\omega^2 r</math> is the acceleration of '''[[w:Circular_motion#Uniform|uniform circular motion]]''', where v is speed, and r is radius.
* <math>v=\omega r =2\pi r/T</math>, where T is period.

* <math>F = G \frac{mM}{r^2}= mg^*</math> is the force of gravity between two objects, where the universal constant of gravity is {{nowrap begin}}G ≈ 6.674 × 10<sup>-11</sup> m<sup>3</sup>·kg<sup>−1</sup>·s<sup>−2</sup>{{nowrap end}}.  

<section end=06-Uniform_Circular_Motion_and_Gravitation/>

===07-Work and Energy===
<section begin=07-Work_and_Energy/>
*<math>KE = \frac 1 2 m v^2\,</math> is kinetic energy, where ''m'' is mass and ''v'' is speed..
*<math>U_g = mgy</math> is gravitational potential energy,where y is height, and <math>g = 9.80\;\frac{m}{s^2}</math> is the gravitational acceleration at Earth's surface.
*<math>U_s = \frac 1 2 k_s x^2</math> is the potential energy stored in a spring with [[w:Hooke's_law|spring constant]] <math>k_s</math>.
*<math>\sum KE_f + \sum PE_f = \sum KE_i + \sum PE_i -Q</math> relates the final energy to the initial energy.  If energy is lost to heat or other nonconservative force, then Q>0.
*<math>W = F \ell\cos\theta=\vec F\cdot\vec\ell</math> (measured in Joules) is the work done by a force <math>F</math> as it moves an object a distance <math>\ell</math>.  The angle between the force and the displacement is &theta;.
*<math>\sum \vec F\cdot\Delta\ell</math> describes the work if the force is not uniform.  The steps, <math>\Delta\vec\ell</math>, taken by the particle are assumed small enough that the force is approximately uniform over the small step.  If force and displacement are parallel, then the work becomes the area under a curve of F(x) versus x.
*<math>P=\frac{\vec F \cdot \vec\Delta \ell}{\Delta t}=\vec F\cdot\vec v</math> is the power (measured in Watts) is the rate at which work is done.  ('''''v''''' is velocity.)
<section end=07-Work_and_Energy/>

===08-Linear Momentum and Collisions===
<section begin=08-Linear_Momentum_and_Collisions/>
*<math> \vec p = m\vec v</math> is momentum, where m is mass and <math> \vec v</math> is velocity.  The net momemtum is conserved if all forces between a system of particles are internal (i.e., come equal and opposite pairs):
*<math>\sum \vec p_f = \sum \vec p_i</math>.  
*<math>\bar F\Delta t = \Delta p</math> is the [[w:impulse (physics)|impulse]], or change in momentum associated with a brief force acting over a time interval <math>\Delta t</math>.  (Strictly speaking, <math>\bar F</math> is a time-averaged force defined by integrating over the time interval.)
<section end=08-Linear_Momentum_and_Collisions/>

===09-Statics and Torque===
<section begin=09-Statics_and_Torque/>
[[File:Torque formulas with lever arm OR perp- F and r.jpg|right|400px]]
* <math>\tau = rF\sin\theta,\!</math> is the [[w:torque|torque]] caused by a force, F, exerted at a distance ,r, from the axis.  The angle between '''''r''''' and '''''F''''' is &theta;. 

The {{lw|SI units|SI unit}} for torque is the {{lw|newton metre|}} (N·m). It would be inadvisable to call this a Joule, even though a Joule is also a (N·m). The symbol for torque is typically ''τ'', the [[Greek alphabet|Greek letter]] ''[[w:tau|tau]]''. When it is called moment, it is commonly denoted ''M''.<ref>https://en.wikipedia.org/w/index.php?title=Torque&oldid=582917749</ref>    The lever arm is defined as either ''r'', or ''r''<sub>⊥</sub>.  Labeling ''r'' as the ''[[wikt:lever arm|lever arm]]'' allows ''[[wikt:moment arm|moment arm]]'' to be reserved for ''r''<sub>⊥</sub>. 
<section end=09-Statics_and_Torque/>

===10-Rotational Motion and Angular Momentum===
<section begin=10-Rotational_Motion_and_Angular Momentum/>
{{#lst:Physics equations/Equations/Rotational and linear motion}}
{{#lst:Physics equations/Equations/Rotational and linear motion analogy}}
{{#lst:Physics equations/Equations/Moments of inertia (small table)}}
<section end=10-Rotational_Motion_and_Angular Momentum/>

===11-Fluid statics===
<section begin=11-Fluid_statics/>
[[File:Archimedes gv52.png|thumb|120px|Pressure is the weight per unit area of the fluid above a point.]]
'''Pressure versus Depth''':  A fluid's '''[[w:pressure|pressure]]''' is ''F/A'' where ''F'' is force and A is a (flat) area.  The pressure at depth, <math>h</math> below the surface is the weight (per area) of the fluid above that point.  As shown in the figure, this implies:

:<math> P = P_0 + \rho g h  </math>  

where <math>P_0</math> is the pressure at the top surface, <math>h</math> is the depth, and <math>\rho</math>  is the mass density of the fluid.  In many cases, only the difference between two pressures appears in the final answer to a question, and in such cases it is permissible to set the pressure at the top surface of the fluid equal to zero.  In many applications, it is possible to artificially set <math>P_0</math> equal to zero, for example at atmospheric pressure.  The resulting pressure is called the gauge pressure, for <math> P_{gauge} = \rho gh  </math> below the surface of a body of water.   

'''Buoyancy and Archimedes' principle''' Pascal's principle does not hold if two fluids are separated by a seal that prohibits fluid flow (as in the case of the piston of an internal combustion engine).  Suppose the upper and lower fluids shown in the figure are not sealed, so that a fluid of mass density <math>\rho_{flu}</math> comes to equilibrium above and below an object.  Let the object have a mass density of  <math>\rho_{obj}</math> and a volume of <math>A\Delta h</math>, as shown in the figure.  The net (bottom minus top) force on the object due to the fluid is called the buoyant force:

:<math>\rm{buoyant}\;\rm{force}=(A\Delta h )(\rho_{flu})g\,</math>, 

and is directed upward.  The volume in this formula, AΔh, is called the volume of the displaced fluid, since placing the volume into a fluid at that location requires the removal of that amount of fluid.  Archimedes principle states:

:'''''A body wholly or partially submerged in a fluid is buoyed up by a force equal to the weight of the displaced fluid.'''''

Note that if <math>\rho_{obj}=\rho_{flu}</math>, the buoyant force exactly cancels the force of gravity.  A fluid element within a stationary fluid will remain stationary.  But if the two densities are not equal, a third force (in addition to weight and the buoyant force) is required to hold the object at that depth.  If an object is floating or partially submerged, the volume of the displaced fluid equals the volume of that portion of the object which is below the waterline.
<section end=11-Fluid_statics/>

===12-Fluid dynamics===
<section begin=12-Fluid_dynamics/>
[[File:FlowRate gv52.png|thumb|180px|A fluid element speeds up if the area is constricted.]]
*<math>\frac{\Delta V}{ \Delta t } =\dot V=Av = Q</math> the volume flow for incompressible fluid flow if viscosity and turbulence are both neglected.  The average velocity is <math>v</math> and <math>A</math> is the cross sectional area of the pipe.  As shown in the figure, <math>v_1A_1=v_2A_2</math> because <math>Av</math> is constant along the developed flow.  To see this, note that the volume of pipe is <math>\Delta V=A\Delta x</math> along a distance <math>\Delta x</math>. And, <math>v = \Delta x / \Delta t</math> is the volume of fluid that passes a given point in the pipe during a time <math>\Delta t</math>.
*<math>P_1 + \rho g y_1 + \frac{1}{2}\rho v_1^2 = P_2 + \rho g y_2 + \frac{1}{2}\rho v_2^2</math> is [[w:Bernoulli's principle|Bernoulli's equation]], where <math>P</math> is pressure, <math>\rho</math> is density, and <math>y</math> is height.  This holds for [[w:inviscid flow|inviscid flow]].
<section end=12-Fluid_dynamics/>

===13-Temperature, Kinetic Theory, and Gas Laws===
<section begin=13-Temperature,_Kinetic Theory,_and_Gas_Laws/>
* <math>T_C=T_K-273.15</math> converts from Celsius to Kelvins, and <math>T_F=\frac 9 5 T_C+32</math> converts from Celsius to Fahrenheit.
* <math>PV=nRT=Nk_BT</math> is the '''[[w:ideal gas law|ideal gas law]]''', where P is pressure, V is volume, n is the number of moles and N is the number of atoms or molecules.  Temperature must be measured on an absolute scale (e.g. Kelvins).  
*N<sub>A</sub>k<sub>B</sub>=R where N<sub>A</sub>= {{nowrap|6.02 × 10<sup>23</sup>}} is the Avogadro number. Boltzmann's constant can also be written in eV and Kelvins: k<sub>B</sub> ≈{{nowrap|8.6 × 10<sup>-5</sup> eV/deg}}.
* <math>\frac 3 2 k_B T =\frac 1 2 mv_{rms}^2</math> is the average translational kinetic energy per "atom" of a 3-dimensional ideal gas. 
*<math>v_{rms}=\sqrt{\frac{3k_BT}{m}}=\sqrt{\overline{v^2}}</math> is the root-mean-square speed of atoms in an ideal gas.
* <math>E = \frac{\varpi}{2}Nk_BT</math> is the total energy of an ideal gas, where <math>\varpi=3\;</math> 
<section end=13-Temperature,_Kinetic Theory,_and_Gas_Laws/>

===14-Heat and Heat Transfer===
<section begin=14-Heat_and_Heat_Transfer/>
Here it is convenient to define '''[[w:heat|heat]]''' as energy that passes between two objects of different temperature <math>Q</math> The SI unit is the Joule.  The rate of heat trasfer, <math>\Delta Q/\Delta t</math> or  <math>\dot Q</math> is "power": {{nowrap begin}}1 Watt = 1 W = 1J/s{{nowrap end}} 
* <math>Q = mc_S\Delta T</math> is the heat required to change the temperature of a substance of mass, m.  The change in temperature is &Delta;T.  The '''[[w:Heat capacity|specific heat]]''', c<sub>S</sub>, depends on the substance (and to some extent, its temperature and other factors such as pressure).  Heat is the transfer of energy, usually from a hotter object to a colder one.  The units of specfic heat are ''energy/mass/degree'', or {{nowrap begin}}J/(kg-degree){{nowrap end}}.
* <math>Q = mL</math> is the heat required to change the phase of a a mass, m, of a substance (with no change in temperature).  The '''[[w:latent heat|latent heat]]''', L, depends not only on the substance, but on the nature of the phase change for any given substance.  L<sub>F</sub> is called the latent heat of fusion, and refers to the melting or freezing of the substance.  L<sub>V</sub> is called the latent heat of vaporization, and refers to evaporation or condensation of a substance.
*<math>\dot Q= \frac{k_cA}{d}\Delta T</math> is rate of heat transfer for a material of area, A. The difference in temperature between two sides separated by a distance, d, is <math>\Delta T</math>.  The '''[[w:thermal conductivity|thermal conductivity]]''', k<sub>c</sub>, is a property of the substance used to insulate, or subdue, the flow of heat.
* <math>\dot Q= \sigma A\epsilon T^{4}</math> is the power radiated by a surface of area, A, at a temperature, T, measured on an absolute scale such as Kelvins.  The emissivity, <math>0\le\epsilon\le 1</math>, is 1 for a black body, and 0 for a perfectly reflecting surface.  The Stefan-Boltzmann constant is <math>\sigma\approx 5.67 \times 10^{-8}\, \mathrm{J\, s^{-1}m^{-2}K^{-4}}</math>.
<section end=14-Heat_and_Heat_Transfer/>

===15-Thermodynamics===
<section begin=15-Thermodynamics/>
[[File:Piston gv52.png|thumb]]
* Pressure (P), Energy (E), Volume (V), and Temperature (T) are state variables ([[w:State_function|state functions]]called state functions).  The number of particles (N) can also be viewed as a state variable.
*  Work (W), Heat (Q) '''are not state variables.'''
* <math> S (V,T)= \frac{3 Nk_B}{2}\ln T+ Nk_B\ln V + constant</math>, is the [[w:entropy|entropy]] of an [[w:Ideal_gas|ideal ]], [[w:monatomic_gas|monatomic]] gas. The constant is arbitrary only in [[w:classical_physics|classical]] (non-quantum) thermodynamics. Since it is a function of state variables, entropy is also a [[w:state function|state function]]. 
[[File:Simple heat engine.jpg|220px|right]]
[[File:PdV work cycle.gif|right|120px]]
A point on a [[w:Pressure volume diagram|PV diagram]] define's the system's pressure (P) and volume (V). Energy (E) and pressure (P) can be deduced from [[w:Category:Equations_of_state|equations of state]]: {{nowrap begin}}E=E(V,P) and T=T(V,P){{nowrap end}}. If the piston moves, or if heat is added or taken from the substance, energy (in the form of work and/or heat) is added or subtracted.  If the path returns to its original point on the PV-diagram (e.g., 12341 along the rectantular path shown), and if the process is [[w:Quasistatic_process|quasistatic]], all state variables {{nowrap|(P, V, E, T)}} return to their original values, and the final system is indistinguishable from its original state.   

The net work done per cycle is area enclosed by the loop. This work equals the net heat flow into the system, <math>Q_{in} - Q_{out} </math> (valid only for closed loops).

'''Remember:''' Area ''"under"'' is the work associated with a path; Area ''"inside"'' is the total work per cycle.

* <math>\Delta W=-F\Delta x= ( -P\cdot\mathrm{Area} ) \left( \frac{\Delta V}{\mathrm{Area}} \right) = -P\Delta V</math> is the work done on a system of pressure P by a piston of voulume V.  If ΔV>0 the substance is expanding as it exerts an outward force, so that ΔW<0 and the substance is doing work on the universe; ΔW>0 whenever the universe is doing work on the system. 
* <math>\Delta Q</math> is the amount of heat (energy) that flows into a system. It is positive if the system is placed in a ''heat bath'' of higher temperature.  If this process is reversible, then the heat bath is at an infinitesimally higher temperature and a finite &Delta;Q takes an infinite amount of time.
* <math>\Delta E=\Delta Q-P\Delta V</math> is the change in energy ([[w:First Law of Thermodynamics|First Law of Thermodynamics]]).

<small>'''CALCULUS:'''</small> <math> \oint P \ dV = Q_{in} - Q_{out}</math> .

:In an isothermal expansion (contraction), temperature, T, is constant.  Hence P=nRT/V and substitution yields,
::<math>\int_{Vi}^{Vf}PdV=\int_{Vi}^{Vf}nRT\frac{dV}{V}=nRT\int_{Vi}^{Vf}\frac{dV}{V}=nRT\ln\frac{V_f}{V_i}</math>

<section end=15-Thermodynamics/>

===16-Oscillatory Motion and Waves===
<section begin=16-Oscillatory_Motion_and_Waves/>
*<math> x = x_0\cos{   \frac{2\pi t}{T_0}   }</math> describes oscillatory motion with period <math>T_0</math> (here we use the zero-subscript to denote constants that do not vary with time).
* <math> x(t) = x_0\cos\left(\omega_0 t - \varphi\right)</math>.  For example, <math>\cos\left(\omega_0 t - \varphi\right)= \sin\omega_0 t</math>.
*<math>\omega_0=\sqrt{\frac{k_s}{m}}=\frac{2\pi}{T}</math> for a mass-spring system with mass, m, and spring constant, k<sub>s</sub>.
*<math>\omega_0=\sqrt{\frac{g}{L}}=\frac{2\pi}{T}</math> for a low amplitude pendulum of length, L, in a gravitational field, g.
*<math>PE=\frac 1 2 k_s x^2</math> is the potential energy of a mass spring system.  

'''Let''' <math> x(t) = x_0\cos\left(\omega_0 t - \varphi\right) =</math> '''describe position:'''
*  <math>v(t) = dx/dt = -\omega_0 x_0\sin\left(\omega_0 t - \varphi\right)=v_0\cos\left(\omega_0 t + ...\right)</math>, where <math>v_0=\omega_0 x_0</math> '''is maximum velocity'''.
*<math> a(t) = dv/dt =-\omega_0 v_0\cos\left(\omega_0 t - \varphi\right)
=a_0\cos\left(\omega_0 t + ...\right)</math>, where <math>a_0=\omega_0 v_0=\omega_0^2 x_0</math>, '''is maximum acceleration'''.
* <math>F_0=m a_0</math>, '''relates maximum force to maximum acceleration.'''
*<math>E=\frac 1 2 mv_0^2=\frac 1 2 k_s x_0^2</math> '''is the total energy'''.
* <small>'''CALCULUS:'''</small> x(t) obeys the [[w:Linear_homogeneous_differential_equation|linear homogeneous differential equation]] (ODE), <math>\frac{d^2x}{dt^2} = -\omega_0^2 x(t)</math>
[[File:StandingWaves.png|right|180px]]
*<math>f\lambda =  v_p</math> relates the frequency, f, wavelength, &lambda;,and the the phase speed, v<sub>p</sub> of the wave (also written as v<sub>w</sub>)   This phase speed is the speed of individual crests, which for sound and light waves also equals the speed at which a [[w:wave packet|wave packet]] travels. 
*<math>L = \frac{n\lambda_n}{2}</math> describes the n-th normal mode vibrating wave on a string that is fixed at both ends (i.e. has a node at both ends).  The mode number, n = 1, 2, 3,..., as shown in the figure.
*Beat frequency: The frequency of beats heard if two closely space frequencies, <math>f_1</math> and <math>f_2</math>, are played is <math>\Delta f = |f_2-f_1|</math>.
*Musical acoustics: Frequency ratios of 2/1, 3/2, 4/3, 5/3, 5/4, 6/5, 8/5 are called the ([[w:Just intonation|just]]) "[[w:Octave|octave]]", "[[w:Perfect fifth|fifth]]",     "[[w:Perfect fourth|fourth]]", "[[w:Major sixth|major-sixth]]", "[[w:Major third|major-third]]", "[[w:Minor third|minor-third]]", and "[[w:Minor sixth|minor-sixth]]",  respectively.
<section end=16-Oscillatory_Motion_and_Waves/>

===17-Physics of Hearing===
<section begin=17-Physics_of_Hearing/>
*<math>v_s =\sqrt{   \frac{T}{273}    }\cdot 331 \text{m/s}</math> is the the approximate speed near Earth's surface, where the temperature, T, is measured in Kelvins.  A theoretical calculation is <math>v_s=\sqrt{\frac{\gamma k_B T}{m}}</math> where <math>\gamma =\frac{\varpi+2}{\varpi}</math> for a semi-classical gas with <math>\varpi</math> [[w:degrees of freedom (physics and chemistry)|degrees of freedom]]. For a diatomic gas such as Nitrogen, {{nowrap begin}}&gamma; = 1.4.{{nowrap end}}
*<math>v_s =\sqrt{   \frac{F}{\mu}    }</math> is the speed of a wave in a stretched string if <math>F</math> is the tension and <math>\mu</math> is the linear mass density (kilograms per meter).
<section end=17-Physics_of_Hearing/>

===18-Electric charge and field===
<section begin=18-Electric_charge_and_field/> 
*<math>F=k_{\mathrm{e}} \frac{qQ}{r^2} = \frac{1}{  4\pi \epsilon_0  } \frac{qQ}{r^2}\,</math> is Coulomb's law for the force between two charged particles separated by a distance r: k<sub>e</sub>≈8.987×10<sup>9</sup>N·m²·C<sup>−2</sup>, and ε<sub>0</sub>≈8.854×10<sup>−12</sup> F·m<sup>−1</sup>. 
*<math>\vec F=q\vec E</math> is the electric force on a "test charge", q, where <math>E =k_{\mathrm{e}} \frac{Q}{r^2} \,</math> is the ''magnitude'' of the electric field situated a distance ''r'' from a charge, Q.

Consider a collection of <math>N</math> particles of charge <math>Q_i</math>, located at points <math>\vec r_i</math> (called ''source points''), the electric field at <math>\vec r </math> (called the ''field point'') is:
*<math>  \vec{E}(\vec r)
=\frac{1}{4\pi \varepsilon_0}\sum_{i=1}^N \frac{\widehat\mathcal{R}_i Q_i}{|\mathcal{\vec R}_i |^2} = \frac{1}{4\pi \varepsilon_0}\sum_{i=1}^N \frac{\vec\mathcal{R}_i Q_i}{|\mathcal{\vec R}_i |^3} 
</math> is the electric field at the field point, <math>\vec r </math>, due to point charges at the source points,<math>\vec r_i</math> , and <math> \vec\mathcal R_i =  \vec r - \vec r_i ,</math> points from source points  to the field point. 

'''<small>CALCULUS</small>''' supplement:[[File:3D_Spherical.svg|right|240px]]

<math>  \vec{E}(\vec r)= k_e\int\frac{\hat\mathcal R dQ}{\mathcal R^2}</math> is the electric field due to distributed charge, where <math> dQ\rightarrow\lambda d\ell\rightarrow \sigma dA\rightarrow\rho dV\;</math>, and  <math>(\lambda, \sigma, \rho)</math> denote ''[[w:linear density|linear]]'', ''[[w:surface density|surface]]'', and ''[[w:volume density|volume]]'' density (or ''[[w:charge density|charge density]]''), respectively.  

{{spaces|5}}''Cartesian coordinates (x, y, z).'' Volume element: <math>dV=dxdydz</math>. Line element:<math>d\vec\ell =\hat x dx+\hat y dy+\hat z dz</math>. Three basic area elements: <math>\hat ndA = </math><math>\hat z dxdy</math>, or,<math> \hat x dydz </math>, or,<math>\hat y dzdx</math>.

{{spaces|5}}''Cylindrical coordinates (&rho;, &phi;, z):''  Volume element: <math>dV =\rho\,dr\,d\varphi \,dz</math> . Line element:<math>d\vec\ell =\hat\varphi rd\varphi+\hat r dr+\hat zdz</math>. Basic area elements: <math>\hat ndA = \rho\,d\varphi\,dz\,\hat \rho</math> (side), and, <math>\rho\,d\rho\,d\varphi\,\hat z</math> (top end).

{{space|5}}''Spherical coordinates (r, &theta;, &phi;):'' Volume element: <math>dV = r^2\,dr\,\sin\theta\,d\theta\,d\varphi \rightarrow 4\pi r^2dr</math> (if symmetry holds).  Line element:<math> d\vec\ell= \hat r \,dr+  \hat\theta\, r\, d\theta +   \hat \varphi\, r\, \sin{\theta} \, d\varphi</math>. Basic area element of a sphere: <math>\hat r dA = \hat r\,r^2 d\Omega</math>, where ''d&Omega;'' is a [[w:solid angle|solid angle]].  
<section end=18-Electric_charge_and_field/>

===19-Electric Potential and Electric Field===
<section begin=19-Electric_Potential_and_Electric_Field/>
*<math>U=qV</math> is the potential energy of a particle of charge, q, in the presence of an electric potential V.
*<math>\Delta V = -E \ell\cos\theta=\vec E\cdot\vec\ell</math> (measured in Volts) is the variation in electric potential as one moves through an electric field <math>E</math>.  The angle between the field and the displacement is &theta;.  The electric potential, V, decreases as one moves parallel to the electric field.
*<math>\Delta V = -\sum \vec E\cdot\Delta\ell</math> describes the electric potential if the field is not uniform.  
*<math>V(\vec r) = k\sum\frac{Q_j}{\mathcal R_j}</math> due to a set of charges <math>Q_j</math> at <math>\vec r_j</math> where <math>\vec\mathcal R_j = \vec r - \vec r_j</math>.

* <math>Q=CV</math> is the (equal and opposite) charge on the two terminals of a capacitor of capicitance, C, that has a voltage drop, V, across the two terminals.
*<math>C=\varepsilon A/d</math> is the capacitance of a parallel plate capacitor with surface area, A, and plate separation, d.  This formula is valid only in the limit that {{nowrap begin}}d<sup>2</sup>/A{{nowrap end}} vanishes.  If a dielectric is between the plates, then {{nowrap begin}}ε>ε<sub>0</sub>≈ 8.85 × 10<sup>−12</sup>{{nowrap end}} due to shielding of the applied electric field by [[w:dielectric|dielectric polarization]] effects.
*<math>U=\frac 1 2 QV = \frac 1 2 CV^2 = \frac{Q^2}{2C} </math> is the energy stored in a capacitor.
*<math>u=\frac{\varepsilon}{2}E^2</math> is the ''energy density'' (energy per unit volume, or Joules per cubic meter) of an electric field.

'''CALCULUS supplement'''

[[File:SurfacesWithAndWithoutBoundary.svg|thumb|right|200px|closed surfaces <nowiki>|</nowiki>
<span style="color:#0000ff"> '''&Omega;'''</span> & 
<span style="color:#ff0000"> '''&part;&Omega;'''</span><br>
To the left are closed surfaces.  To the right are open surfaces, <span style="color:#0000ff">'''&Omega;'''</span>, that possess closed boundaries, <span style="color:#ff0000"> '''&part;&Omega;'''.</span>]]
*<math> \int_\vec p^\vec q  \vec\nabla f\cdot d\vec\ell =  f (\vec{q}\,)- f(\vec{p}\,)   </math> is the [[w:Gradient Theorem|gradient theorem]].

*<math> \int_{\Sigma} \vec\nabla \times \vec F \cdot \vec{dA} =  \oint_{\partial\Sigma}\vec F\cdot d\hat\ell</math> is [[w:Stokes theorem|Stokes' theorem]]

*<math>\int_{\Omega} \vec\nabla \cdot \vec F  \;dV=
\oint_{\partial \Omega} \vec F \cdot d\vec A </math> is the [[w:divergence theorem|divergence theorem]] 

Here, &Omega; is a (3-dimensional) volume and &part;&Omega; is the boundary of the volume, which is a (two-dimensional) surface.  Also a surface is &Sigma;, which, if open, has the boundary &part;&Sigma;, which is a (one-dimensional) curve.  

*<math> V(\vec b) -V(\vec a)=-\int_{\vec a}^{\vec b} \vec E\cdot d\ell</math> in the limit that the Riemann sum becomes an integral. 
*<math>\vec E = -\vec\nabla V</math> where <math>\vec\nabla </math> <math>= \hat x\partial/\partial x +\hat y\partial/\partial y +\hat z\partial/\partial z </math> is the [[w:del operator|del operator]]. 

*<math> \varepsilon_0\int \vec E \cdot \vec{dA} =  Q_{encl}</math> is [[w:Gauss's_law|Gauss's law]] for the surface integral of the electric field over ''any'' closed surface, and <math>Q_{encl}</math> is the total charge inside that surface. 
*<math> \int \vec D \cdot \vec{dA} =  Q_{free}</math> is a useful variant if the medium is [[w:dielectric|dielectric]].  '''''D'''''=ε'''''E''''' is the [[w:electric displacement field|electric displacement field]].  The '''[[w:permittivity|permittivity]]''', {{nowrap begin}}ε = (1+χ)ε<sub>0</sub>,{{nowrap end}} where  {{nowrap begin}}ε<sub>0</sub>≈ 8.85 × 10<sup>−12</sup>{{nowrap end}}, and the electric susceptibility, χ, represents the degree to which the medium can be polarized by an electric field.  The free charge, Q<sub>free</sub>, represents all charges except those represented by the susceptibility, χ.

<section end=19-Electric_Potential_and_Electric_Field/>

===20-Electric Current, Resistance, and Ohm's Law===
<section begin=20-Electric_Current,_Resistance,_and_Ohm's_Law/>
*<math>I = \frac{\mathrm{d}Q}{\mathrm{d}t} \ </math> defines the electric current as the rate at which charge flows past a given point on a wire.  The direction of the current matches the flow of positive charge (which  is opposite the flow of electrons if electrons are the carriers.)
*<math>V = I R </math> is [[w:Ohm's Law|Ohm's Law]] relating [[w:current|current]], I,  and [[w:resistance|resistance]], R, to the difference in voltage, V, between the terminals.  The resistance, R, is positive in virtually all cases, and if R&nbsp;>&nbsp;0, the current flows from larger to smaller voltage.  Any device or substance that obeys this linear relation between I and V is called [[wikt:ohmic|ohmic]].
*<math>I=nqAv_{drift}</math> relates the density (n), the charge(q), and the average drift velocity (v<sub>drift</sub>) of the carriers.  The area (A) is measured by imagining a cut across the wire oriented such that the drift velocity is perpendicular to the surface of the (imaginary) cut.  
*<math>R=\frac{\rho L}{A} \ </math> expresses the resistance of a sample of ohmic material with a length (L) and area (A).  The 'resistivity', &rho; ("row"), is an [[w:Intensive_and_extensive_properties|intensive property]] of matter.
* Power is energy/time, measured in joules/second or J/s.  Often called P (never p).  It is measured in watts (W)
* Current is charge/time, measured in coulombs/second or C/s.  Often called I or i.  It is measured in amps or ampheres (A)
* Electric potential (or voltage) is energy/charge, measured in  joules/coulomb or J/C. Often called V (sometimes E, ''emf'', <math>\mathcal E</math>).  It is measured in volts (V)
* Resistance is voltage/current , measured in volts/amp or V/A.  Often called R (sometimes r, Z) It is measured in Ohms (Ω).
*<math>P = IV = I^2R=\frac{V^2}{R}</math> is the power dissipated as current flows through a resistor
<section end=20-Electric_Current,_Resistance,_and_Ohm's_Law/>

===21-Circuits, Bioelectricity, and DC Instruments===
<section begin=21-Circuits,_Bioelectricity,_and_DC_Instruments/>
{|
|-
  |rowspan="2"|[[File:KCL - Kirchhoff's circuit laws.svg|thumb|160px|<big>The current entering any junction is equal to the current leaving that junction. ''i''<sub>2</sub> + ''i''<sub>3</sub> = ''i''<sub>1</sub> + ''i''<sub>4</sub></big>]]
  |rowspan="2"|[[File:Kirchhoff voltage law.svg|thumb|215px|<big>The sum of all the voltages around the loop is equal to zero. v<sub>1</sub> + v<sub>2</sub> + v<sub>3</sub> - v<sub>4</sub> = 0</big>]]
 |[[File:Resistors in parallel.svg|thumb|145px|<big>Resistors in parallel</big>]]
   |-
 |[[File:Resistors in series.svg|thumb|180px|<big>Resistors in series</big>]] 
|}
[[File:Voltage divider.svg|thumb|120px|voltage divider]]
*<math>\sum_{k=1}^n {I}_k = 0</math> and <math>\sum_{k=1}^n V_k = 0</math> are Kirchoff's Laws<ref>https //en.wikipedia.org/w/index.php?title=Kirchhoff%27s_circuit_laws&oldid=579357795</ref>
*<math>V_{\rm out}=\frac{R_2}{R_1+R_2}V_{\rm in}</math> for the voltage divider shown.
[[Image:Capacitor_resistor_series.svg|thumb|left|180px|In this example, we assume that the rectangular element is a resistor, R, and that the internal resistance of the voltage source (not shown) is also R. The ammeter and voltmeter shown are ideal.]]
[[File:Capacitor Square wave charge-discharge.svg|thumb|Charging and discharging a capacitor with the alternating switch shown to the left (i.e. where the RC rise and discharge times are equal).|360px]]
* '''Simple RC circuit'''<ref>From https://en.wikipedia.org/w/index.php?title=RC_circuit&oldid=598786790</ref> The figure to the right depicts a capacitor being charged by an ideal voltage source.  If, at t=0, the switch is thrown to the other side, the capacitor will discharge, with the  voltage, ''V'' , undergoing [[w:exponential decay|exponential decay]]: 
:<math>
V(t)=V_0 e^{-\frac{t}{RC}} \ ,
</math>

where ''V<sub>0</sub>'' is the capacitor voltage at time ''t = 0'' (when the switch was closed).  The time required for the voltage to fall to <math>\frac{V_0}{e}\approx .37V_0 </math> is called the [[w:RC time constant|RC time constant]] and is given by

:<math> \tau = RC \ . </math>
<section end=21-Circuits,_Bioelectricity,_and_DC_Instruments/>

===22-Magnetism===
<section begin=22-Magnetism/>{{Physeq2|transcludesection=VectorMagneticForce}}
*Defining magnetic force and field without calculus:[[File:Manoderecha.svg|140px|right]]
# &nbsp; <math>B=\frac{\mu_0 I}{2\pi r}</math> is the magnetic field at a distance ''r'' from an infinitely long wire carrying a current, where {{nowrap begin}}μ<sub>0</sub> ={{nowrap end}} {{nowrap begin}}4π × 10<sup>−7</sup> N A{{nowrap end}}. This field points azimuthally around the wire in a direction defined by the right hand rule.  Application of the force law on a current element, we have
# &nbsp; <math>F=\frac{\mu_0 I_1 I_2\,\ell}{2\pi r}</math> is the force between two long wires of length <math>\ell</math> separated by a short distance <math>r<<\ell</math>.  The currents are I<sub>1</sub> and I<sub>2</sub>, with the force being attractive if the currents are flowing in the same direction. 

'''Cyclotron motion''': For a particle moving perpendicular to '''B''', we have cyclotron motion.  Recall that for uniform circular motion, the acceleration is ''a=v<sup>2</sup>/r'', where ''r'' is the radius.  Since sin&nbsp;&theta; =1, Newton's second law of motion (F=ma) yields,

:<math>ma=\frac{mv^2}{r}=qvB</math>

Since, sin&nbsp;&theta; =0, for motion parallel to a magnetic field, particles in a uniform magnetic field move in spirals at a radius which is determined by the perpendicular component of the velocity:

:<math>r =\frac{mv_\perp}{qB}</math>

'''Hall effect''': The Hall effect occurs when the magnetic field, velocity, and electric field are mutually perpendicular.  In this case, the electric and magnetic forces are aligned, and can cancel if ''qE=qvB'' (since sinθ = 1).  Since both terms are porportional to charge, q, the appropriate ratio of electric to magnetic field for null net force depends only on velocity:

:<math>E = vB = \frac{\mathrm{emf}}{\ell}</math>,

where we have used the fact that voltage (i.e. emf or potential) is related to the electric field and a displacement parallel to that field: ''ΔV&nbsp;=&nbsp;-E&nbsp;Δs&nbsp;cosθ''

'''CALCULUS supplement:'''
* <math>\vec B=\oint\frac{\mu_0}{4\pi}\frac{I d\vec{\ell} \times \hat r}{|r|^2}</math> and the volume integral <math>\int\frac{\mu_0}{4\pi}\frac{\vec J \times \hat r}{|r|^2} d\tau'</math>, where <math>\vec J</math> is [[w:current density|current density]].
* <math>\oint \vec B \cdot d\vec\ell = \mu_0 I_{encl}</math> is Ampere's law relating a closed integral involving magnetic field to the total current enclosed by that path.

<section end=22-Magnetism/>

===23-Electromagnetic Induction, AC Circuits, and Electrical Technologies===
<section begin=23-Electromagnetic_Induction,_AC_Circuits,_and_Electrical_Technologies/>
[[File:Moving_Conductor_In_Magnetic_Field.jpg|thumb|200px|In rod's frame the force on carriers is electric, not magnetic.  (See '''[[w:railgun|railgun]]''')]]
*<math>\vec E = \vec v \times \vec B</math>  is a consequence of the magnetic force law as seen in the reference frame of a moving charged object, where ''E'' is the electric field perceived by an observer moving at velocity ''v'' in the presence of a magnetic vield, ''B''.  Also written as, {{nowrap begin}}''E = vBsin&theta;''{{nowrap end}}, this can be used to derive [[w:Faraday's law of induction|Faraday's law of induction]]. (Here, &theta; is the angle between the velocity and the magnetic field.)
*<math>\Phi = \vec A\cdot\vec B = AB\cos\theta</math> is the magnetic flux, where &theta; is the angle between the magnetic field and the '''[[w:Normal (geometry)|normal]]''' to a surface of area, '''A'''.
*<math>emf=-N\frac{\Delta \Phi}{\Delta t}</math> is '''[[w:Faraday's law of induction|Faraday's law]]''' where ''t'' is time and ''N'' is the number of turns.  The minus sign reminds us that the emf, or '''''[[w:electromotive force|electromotive force]]''''', acts as a "voltage" that opposes the change in the magnetic field or flux.
<section end=23-Electromagnetic_Induction,_AC_Circuits,_and_Electrical_Technologies/>

===24-Electromagnetic Waves===
<section begin=24-Electromagnetic_Waves/>
[[File:Displacement_current_in_capacitor.svg|right|240px]]
* <math>\epsilon_0 \partial\vec E/\partial t</math> is called the [[w:displacement current|displacement current]] because it replaces the [[w:current density|current density]] when using [[w:Ampère's circuital law|Ampère's circuital law]] to calculate the [[w:line integral|line integral]] of the magnetic field around a closed loop.
Maxwell's equations hold for all volumes and closed surfaces. In vacuum, electromagnetic waves travel at the speed, <math>c = \frac{1}{\sqrt{\epsilon_0\mu_0}}</math>.
{|class="wikitable" 
|-align="center"
| <math>\oint_S  \mathbf{E} \cdot \mathrm{d}\mathbf{A} = \frac{1}{\epsilon_0} \int_V \rho\, \mathrm{d}V</math>
|| <math>\oint_C \mathbf{E} \cdot \mathrm{d}\mathbf{l} = -\int_S \frac{\partial\mathbf{B}}{\partial t} \cdot \mathrm{d} \mathbf{A}</math>
|-align="center"
| <math>\oint_S \mathbf{B} \cdot \mathrm{d}\mathbf{A} = 0</math>
|| <math>\oint_C \mathbf{B} \cdot \mathrm{d}\mathbf{l} = \mu_0 \int_S \mathbf{J} \cdot \mathrm{d} \mathbf{A} +
           \epsilon_0\mu_0 \int_S \frac{\partial\mathbf{E}}{\partial t} \cdot \mathrm{d} \mathbf{A}</math>
|}
<section end=24-Electromagnetic_Waves/>

===25-Geometric Optics===
<section begin=25-Geometric_Optics/>
[[File:lens3.svg|500px|right]]<math>\frac{1}{S_1} + \frac{1}{S_2} = \frac{1}{f}  </math> relates the focal length ''f'' of the lens, the image distance ''S<sub>1</sub>'', and the object distance ''S<sub>2</sub>''.  The figure depicts the situation for which {{nowrap|(S<sub>1</sub>, S<sub>2</sub>, f)}} are all positive: (1)The lens is converging ([[w:convex|convex]]); (2) The [[w:real image|real image]] is to the right of the lens; and (3) the object is to the left of the lens.  If the lens is diverging ([[w:concave|concave]]), then f&nbsp;<&nbsp;0.  If the image is to the left of the lens ([[w:virtual image|virtual image]]), then ''S<sub>2</sub>&nbsp;<&nbsp;0'' .
<section end=25-Geometric_Optics/>

<!-- HIDING UNUSED CHAPTER 26 BEGIN-->
{{hidden begin}} <nowiki>===26-Vision and Optical Instruments===<section begin=26-Vision_and_Optical_Instruments/>*foo<section end=26-Vision_and_Optical_Instruments/></nowiki>{{hidden end}}<!-- HIDING UNUSED CHAPTERS BEGIN-->

===27-Wave Optics===
<section begin=27-Wave_Optics/>
[[File:TwoSlitYoung's.png|120px|right]]
* <math>S\sin\theta = n\lambda</math>&nbsp;where <math>n = 1, 2, 3, 4...</math> describes the constructive interference associated with two slits in the [[w:Fraunhoffer|Fraunhoffer]] (far field) approximation.
[[File:Beats52.png|right|240px]]
*<math>\cos\left(\omega_1t\right)+\cos\left(\omega_2t\right)=A(t)\cos\left(\bar\omega t\right)</math> where <math>\bar\omega</math> is the high frequency carrier and <math>A(t)=2\cos\left(\frac{\Delta\omega}{2}t\right)</math> is the slowly varying envelope.  Here,
:<math>\bar\omega=\frac{\omega_1+\omega_2}{2}</math> and <math>\Delta\omega=\omega_2-\omega_1</math>.  Consequently, the beat frequency heard when two tones of frequency <math>f_1</math> and <math>f_2</math> is <math>\Delta f = f_2-f_1</math>.
*<math> 
cos  \left(  k\ell_1   -  \omega t  \right)+    cos  \left(   k\ell_2   -   \omega t   \right)
=2cos\left(k\Delta\ell\right)\cos(\omega t -\phi)
</math> models the addition of two waves of equal amplitude but different path length, <math>\Delta\ell=\ell_2-\ell_1</math>.
<section end=27-Wave_Optics/>
==Astronomy==
====MassRadiusDistanceEarthSunMoon====
<section begin=MassRadiusDistanceEarthSunMoon/>
*Earth's Radius:  R<sub>⊕</sub> ≈ 6.37x10<sup>6</sup>m<br> 
*Earth's Mass: M<sub>⊕</sub> ≈ 5.97×10<sup>24</sup>kg<br> 
*Solar and Lunar radius and mass: 
*Solar radius and mass: R<sub>☉</sub>≈110R<sub>⊕</sub> and  M<sub>☉</sub> ≈ 330,000M<sub>⊕</sub>. 
*Lunar radius and mass: R<sub>L</sub> ≈ 0.273R<sub>⊕</sub> and  M<sub>L</sub> ≈ 0.0123M<sub>⊕</sub>
*Earth-moon distance ≈ 60R<sub>⊕</sub><br> 
*Earth-Sun distance = 1AU ≈ 1.496x10<sup>11</sup>m ≈ 23481R<sub>⊕</sub>
*One light-year&nbsp;≈ 9.5×10<sup>15</sup>&nbsp;m =&nbsp;63240&nbsp;AU
*One parsec ≈ 3.26 light-years
<section end=MassRadiusDistanceEarthSunMoon/>
<ref>Taken from [https://www.google.com/ Google], [https://en.wikipedia.org/w/index.php?title=Orbit_of_the_Moon&oldid=599621026 Orbit of the Moon].</ref>

====KeplerNewtonMassPeriodDistanceNorm====
<section begin=KeplerNewtonMassPeriodDistanceNorm/>
*<math>a_\mathrm{AU}^3=\tilde M_\mathrm{net}P_\mathrm{year}^2</math>, where ''P'' is the period of orbit in years, and ''a'' is the semi-major axis measured in [[w:astronomical unit|AU]].  The net mass, <math>\tilde M_\mathrm{net}</math>, is the sum of the mass of both bodies, and is normalized to the mass of the Sun.  For a planet of mass, <math>m</math>, orbiting a star of much larger mass, <math>M>>m</math>, the normalized net mass is <math>\tilde M_\mathrm{net}=\,</math><math>(M+m)/M_\odot\approx\,</math><math>M/M_\odot</math>.  The mass of the Sun, <math>M_\odot</math>, is  1.99×10<sup>30</sup> kilograms. If <math>M_\odot =2</math> for some object, then that object is twice as massive as the Sun. One year is 3.15×10<sup>7</sup> seconds.
<section end=KeplerNewtonMassPeriodDistanceNorm/>

====AstrParallax====
<section begin=AstrParallax/>
*<math>D_\mathrm{parsec}=\frac{b_\mathrm{AU}}{\theta_\mathrm{arcsec}}</math>, where D is the distance to the object in parsecs, θ is the parallax angle in [[w:Minute_of_arc|arcseconds]], and b is the baseline in [[w:AU|AU]]; b=1 for observations taken from Earth.  One degree is 60 arcminutes and one arcminute is 60 arseconds.  One AU&nbsp;≈ 1.5x10<sup>11</sup> meters, and one parsec&nbsp;≈ 3.26&nbsp;light-years, and one light-year&nbsp;≈ 9.5×10<sup>15</sup>&nbsp;meters.
<section end=AstrParallax/>

====AstrInverseSquare====
<section begin=AstrInverseSquare/>
[[Image:Inverse square law.svg|thumb|200px]] 
*<math>4\pi \tilde I=\frac{\tilde L}{D^2}</math> is a "normalized intensity", closely related to [[w:relative magnitude|relative magnitude]], that allows students to combine equations and solve problems without resorting to the logarithmic magnitude scale.  If the distance to the stellar object, D, is measured in parsecs, it is the power per square parsec that enters a telescope on Earth.  The luminosity, <math>\tilde L</math>, (in solar units) is a measure of the [[w:absolute magnitude|absolute magnitude]]. In general, <math>\mbox{Intensity} \ \propto \ \frac{1}{\mbox{distance}^2} \, </math> is the [[w:inverse-square law|inverse-square law]]. 
<section end=AstrInverseSquare/>

====PhotonsWavesParticles====
<section begin=PhotonsWavesParticles/>
[[file:TheElectromagneticSpectrum.jpg|thumb|850px|Here, energy is measured in electron volts: 1&nbsp;eV=1.6×10<sup>-19</sup>J.  Also, hc=1240nm-eV.]]

*<math>E=hf=hc/\lambda</math> is the energy of a photon, where ''f'' is frequency and h&nbsp;&nbsp;≈6.6×10<sup>-34</sup>m<sup>2</sup>kg/s is Plank's constant, and c&nbsp;≈3×10<sup>8</sup>m/s is the speed of light.  Also, E=<big>ℏ</big>&nbsp;&omega; where <big>ℏ</big>&nbsp;≈1.05×10m<sup>2</sup>kg/s and &omega;&nbsp;=2π&nbsp;f.  
*<math>f \lambda=c</math> relates frequency, wavelength, and the speed (or [[w:phase_velocity|phase velocity]]).  Using [[w:wavenumber|wavenumber]], k&nbsp;=2π/λ, this can also be represented as &omega;&nbsp;=ck. 
<section end=PhotonsWavesParticles/>

====AstrBlackbody====
<section begin=AstrBlackbody/>
[[File:H-r diagram-black white.png|thumb|left|350px]]
[[File:Black_body_realization.svg|thumb|250px]]
     
* <math>\lambda_\text{max} T_K = .003\text{nm} </math> is '''[[w:Wein's law|Wein's law]]''' that relates the peak emission wavelength, λ<sub>max</sub>,  of a [[w:black body|black body]] to temperature, T measured in Kelvins.  Peak wavelength, λ<sub>max</sub>, is measured in nanometers (1nm=10<sup>-9</sup>m).  If temperature is measured in units normalized to the Sun's temperature, <math>T_\odot=5778K</math>, then 
* <math>\lambda_\text{max} \tilde T = 502\text{nm} </math> where <math>\tilde T=T/T_\odot</math> is the temperature normalized to the Sun's temperature.

&nbsp; &nbsp; The '''[[w:Stefan–Boltzmann_law|Stefan-Boltzmann law]]''' is usually written as  ''P=σAT<sup>4</sup>'', where A is surface area, T is temperature (in Kelvins), and σ is the [[w:Stefan-Boltzmann_constant|Stefan-Boltzmann constant]].  The power, P, can be written as normalized luminosity, <math>\tilde L = P/L_\odot</math>, where ''L<sub>☉</sub>''&nbsp; ''=3.85×10<sup>26</sup>W'' is the power output (or ''luminosity'') of the Sun.  In these normalized units, the Stefan-Boltzmann law is: 
* <math>\tilde L=\tilde R^2\tilde T^4</math>, where <math>\tilde R=R/R_\odot</math> is the radius and temperature normalized to the Sun's radius and <math>\tilde T=T/T_\odot</math> is the temperature normalized to the Sun's temperature.
<div style="page-break-before:always"></div>
[[Image:Wiens law.svg|thumb|500px]] 
<section end=AstrBlackbody/>

==Hidden chapters from openstax Physics==
<!-- HIDING UNUSED CHAPTERS BEGIN-->
{{hidden begin|title=References and unfinished chapters are hidden}}
<nowiki>===28-Special Relativity===
<section begin=28-Special_Relativity/>
*foo
<section end=28-Special_Relativity/>
===29-Introduction to Quantum Physics===
<section begin=29-Introduction_to_Quantum_Physics/>
*foo
<section end=29-Introduction_to_Quantum_Physics/>
===30-Atomic Physics===
<section begin=30-Atomic_Physics/>
*foo
<section end=30-Atomic_Physics/>
===31-Radioactivity and Nuclear Physics===
<section begin=31-Radioactivity_and_Nuclear_Physics/>
*foo
<section end=31-Radioactivity_and_Nuclear_Physics/>
===32-Medical Applications of Nuclear Physics===
<section begin=32-Medical_Applications_of_Nuclear_Physics/>
*foo
<section end=32-Medical_Applications_of_Nuclear_Physics/>
===33-Particle Physics===
<section begin=33-Particle_Physics/>
*foo
<section end=33-Particle_Physics/>
===34-Frontiers of Physics===
<section begin=34-Frontiers_of_Physics/>
*foo
<section end=34-Frontiers_of_Physics/>
</nowiki>
<references/>
{{hidden end}}<!-- HIDING UNUSED CHAPTERS END-->

[[Category:Physics equations/Sheet]]

[[Category:Physics equations/Equations]]

[[Category:Physics equations/Equations/Transcluded]]
[[Category:Quizbank/Equations]]