OpenStax University Physics/V2

http://cnx.org/content/col12074/latest/

Temperature and Heat

${\displaystyle T_{C}={\tfrac {5}{9}}\left(T_{F}-32\right)}$ relates Celsius to Fahrenheit temperature scales. ${\displaystyle T_{K}=T_{C}+273.15}$ relates Kelvin to Celsius.

▭ Linear thermal expansion: ${\displaystyle \Delta L=\alpha L\Delta T}$ relates a small change in length to the total length ${\displaystyle L}$, where ${\displaystyle \alpha }$ is the coefficient of linear expansion.
▭ For expansion in two and three dimensions: ${\displaystyle \Delta A=2\alpha A\Delta T}$ and ${\displaystyle \Delta V=\beta V\Delta T}$, respectively.
▭ Heat transfer is ${\displaystyle Q=mc\Delta T}$ where ${\displaystyle c}$ is the specific heat capacity. In a calorimeter, ${\displaystyle Q_{cold}+Q_{hot}=0}$
▭ Latent heat due to a phase change is ${\displaystyle Q=mL_{f}}$ for melting/freezing and ${\displaystyle Q=mL_{v}}$ for evaporation/condensation.
▭ Heat conduction (power): ${\displaystyle P={\tfrac {kA(T_{h}-T_{c})}{d}}}$ where ${\displaystyle k}$ is heat conductivity and ${\displaystyle d}$ is thickness and ${\displaystyle A}$ is area.
▭ ${\displaystyle P_{net}=\sigma eA\left(T_{2}^{4}-T_{1}^{4}\right)}$ is the radiative energy transfer rate where ${\displaystyle e}$ is emissivity and ${\displaystyle \sigma }$ is the Stefan–Boltzmann constant.

▭ ${\displaystyle N=nN_{A}}$ is the number of particles. Gas constant ${\displaystyle R}$ = 8.3 J K⁻¹/mol
▭ Avegadro's number: ${\displaystyle N_{A}}$ = 6.02×10²³. Boltzmann's constant: ${\displaystyle k_{B}}$ = 1.38×10⁻²³J/K.
▭ Van der Waals equation ${\displaystyle \left[p+a(nV)^{2}\right](V-nb)=nRT}$
▭ RMS speed ${\displaystyle v_{rms}={\sqrt {\overline {v^{2}}}}={\sqrt {\tfrac {3RT}{M}}}={\sqrt {\tfrac {3k_{B}T}{m}}}}$ where the overline denotes mean, ${\displaystyle m}$ is a particle's mass and ${\displaystyle M}$ is the molar mass.
▭ Mean free path ${\displaystyle \lambda ={\tfrac {V}{4{\sqrt {2}}\pi r^{2}N}}={\tfrac {k_{B}T}{4{\sqrt {2}}\pi r^{2}p}}=v_{rms}\tau }$ where ${\displaystyle \tau }$ is the mean-free-time
▭ Internal energy of an ideal monatomic gas ${\displaystyle E_{int}={\tfrac {3}{2}}Nk_{B}T=N{\overline {K}}}$, where ${\displaystyle {\overline {K}}=}$ average kinetic energy of a particle.
▭ ${\displaystyle Q=nC_{V}\Delta T}$ defines the molar heat capacity at constant volume.
▭ ${\displaystyle C_{V}={\tfrac {d}{2}}R}$ for ideal gas with ${\displaystyle d}$ degrees of freedom
▭ Maxwell–Boltzmann speed distribution ${\displaystyle f(v)={\tfrac {4}{\sqrt {\pi }}}\left({\tfrac {m}{2k_{B}T}}\right)^{3/2}v^{2}e^{-mv^{2}/2k_{B}T}}$
▭ Average speed ${\displaystyle {\bar {v}}={\sqrt {{\tfrac {8}{\pi }}{\tfrac {RT}{M}}}}}$ ▭ Peak velocity ${\displaystyle v_{p}={\sqrt {\tfrac {2RT}{M}}}}$

▭ Equation of state ${\displaystyle f(p,V,T)=0}$ ▭ Work done by a system ${\displaystyle W=\int _{V_{1}}^{V_{2}}pdV}$
▭ Internal energy ${\displaystyle E_{int}=\sum _{i}\left({\overline {K}}_{i}+{\overline {U}}_{i}\right)}$ is a sum over all particles of kinetic and potential energies
▭ First law ${\displaystyle \Delta E_{int}=Q-W}$ (Q is heat going in and W is work done by as shown in the figure)
▭ ${\displaystyle C_{p}=C_{V}+R}$ is the molar heat capacity at constant volume
▭ ${\displaystyle pV^{\gamma }={\text{constant}}}$ for an adiabatic process in an ideal gas, where the heat capacity ratio ${\displaystyle \gamma =C_{p}/C_{V}}$

▭ Coefficient of performance for a refrigerator ${\displaystyle K_{R}={\tfrac {Q_{c}}{W}}={\tfrac {Q_{c}}{Q_{h}-Q_{c}}}}$, and heat pump ${\displaystyle K_{P}={\tfrac {Q_{h}}{W}}={\tfrac {Q_{h}}{Q_{h}-Q_{c}}}}$
▭ Entropy change ${\displaystyle \Delta S={\tfrac {Q}{T}}}$ (reversible process at constant temperature) ${\displaystyle \rightarrow \int _{A}^{B}{\tfrac {dQ}{T}}=S_{B}-S_{A}}$
▭ ${\displaystyle \oint {\tfrac {dQ}{T}}}$ for any cyclic process ${\displaystyle \rightarrow \int _{A}^{B}{\tfrac {dQ}{T}}=S_{B}-S_{A}}$ is path independent.
▭ ${\displaystyle \Delta S\geq 0}$ for any closed system. ${\displaystyle \lim _{T\to 0}\Delta S=0}$ for any isothermal process.

Elementary charge = e = 1.602×10⁻¹⁹C (electrons have charge q=−e and protons have charge q=+e.)

Dipole moment

▭ By superposition, ${\displaystyle {\vec {F}}={\tfrac {1}{4\pi \varepsilon _{0}}}Q\sum _{i=1}^{N}{\tfrac {q_{i}}{r_{Qi}^{2}}}{\hat {r}}_{Qi}}$ where ${\displaystyle {\vec {r}}_{Qi}={\vec {r}}_{Q}-{\vec {r}}_{i}}$
▭ Electric field ${\displaystyle {\vec {F}}=Q{\vec {E}}}$ where ${\displaystyle {\vec {E}}({\vec {r}}_{P})={\tfrac {1}{4\pi \varepsilon _{0}}}\sum _{i=1}^{N}{\tfrac {q_{i}}{r_{Pi}^{2}}}{\hat {r}}_{Pi}}$ is the field at ${\displaystyle {\vec {r}}_{P}}$ due to charges at ${\displaystyle {\vec {r}}_{i}}$
▭ The field above an infinite wire ${\displaystyle {\vec {E}}(z)={\tfrac {1}{4\pi \varepsilon _{0}}}{\tfrac {2\lambda }{z}}{\hat {k}}}$ and above an infinite plane ${\displaystyle {\vec {E}}={\tfrac {\sigma }{2\varepsilon _{0}}}{\hat {k}}}$
▭ An electric dipole ${\displaystyle {\vec {p}}=q{\vec {d}}}$ in a uniform electric field experiences the torque ${\displaystyle \tau ={\vec {p}}\times {\vec {E}}}$

Gauss's Law

 closed .. open

Flux for a uniform electric field ${\displaystyle \Phi ={\vec {E}}\cdot {\vec {A}}}$ ${\displaystyle \to \Phi =\int {\vec {E}}\cdot d{\vec {A}}=\int {\vec {E}}\cdot {\hat {n}}\,dA}$ in general.

▭ Closed surface integral ${\displaystyle \Phi =\oint {\vec {E}}\cdot d{\vec {A}}=\oint {\vec {E}}\cdot {\hat {n}}\,dA}$
▭ Gauss's Law ${\displaystyle =q_{enc}=\varepsilon _{0}\oint {\vec {E}}\cdot d{\vec {A}}}$. In simple cases: ${\displaystyle E\int dA=EA^{*}={\tfrac {q_{enc}}{\varepsilon _{0}}}}$
▭ Electric field just outside the surface of a conductor ${\displaystyle {\vec {E}}={\tfrac {\sigma }{\varepsilon _{0}}}}$

▭  Electron (proton) mass = 9.11×10⁻³¹kg (1.67× 10⁻²⁷kg). Electron volt: 1 eV = 1.602×10⁻¹⁹J
▭  Near an isolated point charge ${\displaystyle V(r)=k{\tfrac {q}{r}}}$ where ${\displaystyle k={\tfrac {1}{4\pi \varepsilon _{0}}}}$ =8.99×10⁹ N·m/C² is the Coulomb constant.
▭ Work done to assemble N particles ${\displaystyle W_{12...N}=\sum _{i=1}^{N}\sum _{j=1}^{i-1}{\tfrac {q_{i}q_{j}}{r_{ij}}}={\tfrac {k}{2}}\sum _{i=1}^{N}\sum _{j=1}^{N}{\tfrac {q_{i}q_{j}}{r_{ij}}}{\text{ for }}i\neq j}$
▭ Electric potential due to N charges. ${\displaystyle V_{P}=k\sum _{1}^{N}{\frac {q_{i}}{r_{i}}}}$. For continuous charge ${\displaystyle V_{P}=k\int {\frac {dq}{r}}}$. For a dipole, ${\displaystyle V=k{\tfrac {{\vec {p}}\cdot {\vec {\hat {r}}}}{r^{2}}}}$.
▭ Electric field as gradient of potential ${\displaystyle {\vec {E}}=-{\tfrac {\partial V}{\partial x}}{\hat {i}}-{\tfrac {\partial V}{\partial y}}{\hat {j}}-{\tfrac {\partial V}{\partial z}}{\hat {k}}=-{\vec {\nabla }}V}$ ▭ Del operator^note: Cartesian ${\displaystyle {\vec {\nabla }}={\hat {i}}{\tfrac {\partial }{\partial x}}+{\hat {j}}{\tfrac {\partial }{\partial y}}+{\hat {k}}{\tfrac {\partial }{\partial z}}{\text{; }}}$Cylindrical ${\displaystyle {\vec {\nabla }}={\hat {r}}{\tfrac {\partial }{\partial r}}+{\hat {\phi }}{\tfrac {\partial }{\partial \phi }}+{\hat {z}}{\tfrac {\partial }{\partial z}}{\text{; }}}$Spherical ${\displaystyle {\vec {\nabla }}={\hat {r}}{\tfrac {\partial }{\partial r}}+{\hat {\theta }}{\tfrac {\partial }{\partial \theta }}+{\hat {\phi }}{\tfrac {\partial }{\partial \phi }}{\text{.}}}$

▭ ${\displaystyle 4\pi \varepsilon _{0}{\tfrac {R_{1}R_{2}}{R_{2}-R_{1}}}}$ and ${\displaystyle {\tfrac {2\pi \varepsilon _{0}\ell }{\ln(R_{2}/R_{1})}}}$ for a spherical and cylindrical capacitor, respectively
▭ For capacitors in series (parallel) ${\displaystyle {\tfrac {1}{C_{S}}}=\sum {\tfrac {1}{C_{i}}}\left(C_{P}=\sum C_{i}\right)}$
▭  ${\displaystyle u={\tfrac {1}{2}}QV={\tfrac {1}{2}}CV^{2}={\tfrac {1}{2C}}Q^{2}}$ ▭ Stored energy density ${\displaystyle u_{E}={\tfrac {1}{2}}\varepsilon _{0}E^{2}}$
▭ A dielectric with ${\displaystyle \kappa >1}$ will decrease the capacitor's electric field ${\displaystyle E={\tfrac {1}{\kappa }}E_{0}}$ and stored energy ${\displaystyle U={\tfrac {1}{\kappa }}U_{0}}$, but increase the capacitance ${\displaystyle C=\kappa C_{0}}$ due to the induced electric field ${\displaystyle {\vec {E}}_{i}=\left({\tfrac {1}{\kappa }}-1\right){\vec {E}}_{0}}$

▭ ${\displaystyle I=JA\rightarrow \int {\vec {J}}\cdot d{\vec {A}}}$ , ${\displaystyle A}$ is the perpendicular area, and ${\displaystyle J}$ is current density. ${\displaystyle {\vec {E}}=\rho {\vec {J}}}$ is electric field, where ${\displaystyle \rho }$ is resistivity.
▭ Resistivity varies with temperature as ${\displaystyle \rho =\rho _{0}\left[1+\alpha (T-T_{0})\right]}$. Similarily, ${\displaystyle R=R_{0}\left[1+\alpha \Delta T\right]}$ where ${\displaystyle R=\rho {\tfrac {L}{A}}}$ is resistance (Ω)
▭ Ohm's law ${\displaystyle V=IR}$ ▭  Power ${\displaystyle =P=IV=I^{2}R=V^{2}/R}$

▭ ${\displaystyle V_{terminal}^{series}=\sum _{i=1}^{N}\varepsilon _{i}-I\sum _{i=1}^{N}r_{i}}$ ▭ ${\displaystyle V_{terminal}^{parallel}=\varepsilon -I\sum _{i=1}^{N}\left({\frac {1}{r_{i}}}\right)^{-1}}$ where ${\displaystyle r_{i}}$ is internal resistance of each voltage source.
▭ Charging an RC (resistor-capacitor) circuit: ${\displaystyle q(t)=Q\left(1-e^{-t/\tau }\right)}$ and ${\displaystyle I=I_{0}e^{-t/\tau }}$ where ${\displaystyle \tau =RC}$ is RC time, ${\displaystyle Q=\varepsilon C}$ and ${\displaystyle I_{0}=\varepsilon /R}$.
▭ Discharging an RC circuit: ${\displaystyle q(t)=Qe^{-t/\tau }}$ and ${\displaystyle I(t)=-{\tfrac {Q}{RC}}e^{-t/\tau }}$

▭ The SI unit for magnetic field is the Tesla: 1T=10⁴ Gauss.
▭ Gyroradius ${\displaystyle r={\tfrac {mB}{qB}}.\;}$ Period ${\displaystyle T={\tfrac {2\pi m}{qB}}.\;}$
▭ Torque on current loop ${\displaystyle {\vec {\tau }}={\vec {\mu }}\times {\vec {B}}}$ where ${\displaystyle {\vec {\mu }}=NIA{\hat {n}}}$ is the dipole moment. Stored energy ${\displaystyle U={\vec {\mu }}\cdot {\vec {B}}.}$
▭ Drift velocity in crossed electric and magnetic fields ${\displaystyle v_{d}={\tfrac {E}{B}}}$
▭ Hall voltage = ${\displaystyle V}$ where the electric field is ${\displaystyle E=V/\ell =Bv_{d}={\tfrac {IB}{neA}}}$
▭ Charge-to-mass ratio ${\displaystyle q/m={\tfrac {E}{BB_{0}r}}}$ where the ${\displaystyle E}$ and ${\displaystyle B}$ fields are crossed and ${\displaystyle E=0}$ when the magnetic field is ${\displaystyle B_{0}}$

Sources of Magnetic Fields

▭ Permeability of free space ${\displaystyle \mu _{0}=4\pi \times 10^{-7}}$ T·m/A
▭ Force between parallel wires ${\displaystyle {\tfrac {F}{\ell }}={\tfrac {\mu _{0}I_{1}I_{2}}{2\pi r}}}$

▭ Biot–Savart law ${\displaystyle {\vec {B}}={\tfrac {\mu _{0}}{4\pi }}\int \limits _{wire}{\frac {Id{\vec {\ell }}\times {\hat {r}}}{r^{2}}}}$

▭ Ampère's Law:${\displaystyle \oint {\vec {B}}\cdot d{\vec {\ell }}=\mu _{0}I_{enc}}$
▭ Magnetic field due to long straight wire ${\displaystyle B={\tfrac {\mu _{0}I}{2\pi R}}}$ ▭ At center of loop ${\displaystyle B={\tfrac {\mu _{0}I}{2R}}}$
▭ Inside a long thin solenoid ${\displaystyle B=\mu _{0}nI}$ where ${\displaystyle n=N/\ell }$ is the ratio of the number of turns to the solenoid's length.
▭ Inside a toroid ${\displaystyle B={\tfrac {\mu _{0}N}{2\pi r}}}$

▭ The magnetic field inside a solenoid filled with paramagnetic material is ${\displaystyle B=\mu nI}$ where ${\displaystyle \mu =(1+\chi )\mu _{0}}$ is the permeability

Magnetic flux ${\displaystyle \Phi _{m}=\int _{S}{\vec {B}}\cdot {\hat {n}}dA}$ ▭ Electromotive force ${\displaystyle \varepsilon =-N{\tfrac {d\Phi _{m}}{dt,}}}$ (Faraday's law)
▭ Motional emf ${\displaystyle \varepsilon =B\ell v,}$ ▭ rotating coil ${\displaystyle NBA\omega \sin \omega t}$
▭ Motional emf around circuit ${\displaystyle \varepsilon =\oint {\vec {E}}\cdot d{\vec {\ell }}=-{\tfrac {d\Phi _{m}}{dt}}}$

▭ Self-inductance ${\displaystyle N\Phi _{m}=LI\rightarrow \varepsilon =-L{\tfrac {dI}{dt}}}$ ▭  ${\displaystyle L_{\text{solenoid}}\approx \mu _{0}N^{2}A\ell ,\,}$${\displaystyle L_{\text{toroid}}\approx {\tfrac {\mu _{0}N^{2}h}{2\pi }}\ln {\tfrac {R_{2}}{R_{1}}}.}$ Stored energy ${\displaystyle U={\tfrac {1}{2}}LI^{2}.}$ ▭ ${\displaystyle I(t)={\tfrac {\varepsilon }{R}}\left(1-e^{-t/\tau }\right)}$is the current in an LR circuit where ${\displaystyle \tau =L/R}$ is the LR decay time.
▭ The capacitor's charge on an LC circuit ${\displaystyle q=q_{0}\cos(\omega t+\phi )}$ where ${\displaystyle \omega ={\sqrt {\tfrac {1}{LC}}}}$ is angular frequency
▭ LRC circuit ${\displaystyle q(t)=q_{0}e^{-Rt/2L}\cos(\omega 't+\phi )}$ where ${\displaystyle \omega '={\sqrt {{\tfrac {1}{LC}}+\left({\tfrac {R}{2L}}\right)^{2}}}}$

Alternating-Current Circuits

RLC circiut

AC voltage and current ${\displaystyle v=V_{0}\sin(\omega t-\phi )}$ if ${\displaystyle i=I_{0}\sin \omega t.}$
▭ RMS values ${\displaystyle I_{rms}={\tfrac {I_{0}}{\sqrt {2}}}}$ and ${\displaystyle V_{rms}={\tfrac {V_{0}}{\sqrt {2}}}}$ ▭ Impedance ${\displaystyle V_{0}=I_{0}X}$

▭ Resistor ${\displaystyle V_{0}=I_{0}X_{R},\;\phi =0,}$ where ${\displaystyle X_{R}=R}$
▭ Capacitor ${\displaystyle V_{0}=I_{0}X_{C},\;\phi =-{\tfrac {\pi }{2}},}$ where ${\displaystyle X_{C}={\tfrac {1}{\omega C}}}$ ▭ Inductor ${\displaystyle V_{0}=I_{0}X_{L},\;\phi =+{\tfrac {\pi }{2}},}$ where ${\displaystyle X_{L}=\omega L}$
▭ RLC series circuit ${\displaystyle V_{0}=I_{0}Z}$ where ${\displaystyle Z={\sqrt {R^{2}+\left(X_{L}-X_{C}\right)^{2}}}}$ and ${\displaystyle \phi =\tan ^{-1}{\frac {X_{L}-X_{C}}{R}}}$
▭ Resonant angular frequency ${\displaystyle \omega _{0}={\sqrt {\tfrac {1}{LC}}}}$ ▭ Quality factor ${\displaystyle Q={\tfrac {\omega _{0}}{\Delta \omega }}={\tfrac {\omega _{0}L}{R}}}$
▭ Average power ${\displaystyle P_{ave}={\frac {1}{2}}I_{0}V_{0}\cos \phi =I_{rms}V_{rms}\cos \phi }$, where ${\displaystyle \phi =0}$ for a resistor.
▭ Transformer voltages and currents ${\displaystyle {\tfrac {V_{S}}{V_{P}}}={\tfrac {N_{S}}{N_{P}}}={\tfrac {I_{P}}{I_{S}}}}$

Maxwell's equations ${\displaystyle {\begin{aligned}\oint _{S}{\vec {E}}\cdot \mathrm {d} {\vec {A}}&={\frac {1}{\epsilon _{0}}}Q_{in}\qquad &\oint _{S}{\vec {B}}\cdot \mathrm {d} {\vec {A}}&=0\\\oint _{C}{\vec {E}}\cdot \mathrm {d} {\vec {\ell }}&=-\int _{S}{\frac {\partial {\vec {B}}}{\partial t}}\cdot \mathrm {d} {\vec {A}}\qquad &\oint _{C}{\vec {B}}\cdot \mathrm {d} {\vec {\ell }}&=\mu _{0}I+\epsilon _{0}\mu _{0}{\frac {\mathrm {d} \Phi _{E}}{\mathrm {d} t}}\end{aligned}}}$
See also http://ethw.org/w/index.php?title=Maxwell%27s_Equations&oldid=157445
▭ Plane EM wave equation ${\displaystyle {\frac {\partial ^{2}E_{y}}{\partial x^{2}}}=\varepsilon _{0}\mu _{0}{\frac {\partial ^{2}E_{y}}{\partial t^{2}}}}$ where ${\displaystyle c={\tfrac {1}{\sqrt {\varepsilon _{0}\mu }}}}$ is the speed of light
▭ The ratio of peak electric to magnetic field is ${\displaystyle {\tfrac {E_{0}}{B_{0}}}=c}$ and the Poynting vector ${\displaystyle {\vec {S}}={\tfrac {1}{\mu _{0}}}{\vec {E}}\times {\vec {B}}}$ represents the energy flux
▭ Average intensity ${\displaystyle I=S_{ave}={\tfrac {c\varepsilon _{0}}{2}}E_{0}^{2}={\tfrac {c}{2\mu _{0}}}B_{0}^{2}={\tfrac {1}{2\mu _{0}}}E_{0}B_{0}}$
▭ Radiation pressure ${\displaystyle p=I/c}$ (perfect absorber) and ${\displaystyle p=2I/c}$ (perfect reflector).