OpenStax University Physics/V2

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

Temperature and Heat
$$T_C=\tfrac 5 9\left(T_F-32\right)$$ relates Celsius to Fahrenheit temperature scales. $$T_K=T_C+273.15$$ relates Kelvin to Celsius. &#9645; Linear thermal expansion: $$\Delta L=\alpha L\Delta T$$ relates a small change in length to the total length $$L$$, where $$\alpha$$ is the coefficient of linear expansion. &#9645; For expansion in two and three dimensions: $$\Delta A=2\alpha A\Delta T$$ and $$\Delta V=\beta V\Delta T$$, respectively. &#9645; Heat transfer is $$Q=mc\Delta T$$ where $$c$$ is the specific heat capacity. In a calorimeter, $$Q_{cold}+Q_{hot}=0$$ &#9645; Latent heat due to a phase change is $$Q=mL_f$$ for melting/freezing and $$Q=mL_v$$ for evaporation/condensation. &#9645; Heat conduction (power): $$P=\tfrac{kA(T_h-T_c)}{d}$$ where $$k$$ is heat conductivity and $$d$$ is thickness and $$A$$ is area. &#9645; $$P_{net}=\sigma eA\left(T_2^4-T_1^4\right)$$ is the radiative energy transfer rate where $$e$$ is emissivity and $$\sigma$$ is the Stefan–Boltzmann constant.

The Kinetic Theory of Gases
Ideal gas law: Pressure&times;Volume $$=pV=nRT=Nk_BT$$ where $$n$$ is the number of moles and $$T$$ is an absolute temperature. &#9645; $$N=nN_A$$ is the number of particles. Gas constant $$R$$ = 8.3 J K&minus;1/mol &#9645; Avegadro's number: $$N_A$$ = 6.02&times;1023. Boltzmann's constant: $$k_B$$ = 1.38&times;10&minus;23J/K. &#9645; Van der Waals equation $$\left[p+a(nV)^2\right](V-nb)=nRT$$ &#9645; RMS speed $$v_{rms} = \sqrt{\overline{v^2}} = \sqrt{\tfrac{3RT}{M}} = \sqrt{\tfrac{3k_BT}{m}}$$ where the overline denotes mean, $$m$$ is a particle's mass and $$M$$ is the molar mass. &#9645; Mean free path $$\lambda=\tfrac{V}{4\sqrt 2\pi r^2N}=\tfrac{k_BT}{4\sqrt 2\pi r^2p}=v_{rms}\tau$$ where $$\tau$$ is the mean-free-time &#9645; Internal energy of an ideal monatomic gas $$E_{int}=\tfrac 3 2 Nk_BT = N\overline K$$, where $$\overline K=$$ average kinetic energy of a particle. &#9645; $$Q=nC_V\Delta T$$ defines the molar heat capacity at constant volume. &#9645; $$C_V=\tfrac d 2 R$$ for ideal gas with $$d$$ degrees of freedom &#9645; Maxwell–Boltzmann speed distribution $$f(v)=\tfrac{4}{\sqrt\pi}\left(\tfrac{m} {2k_BT}\right)^{3/2} v^2e^{-mv^2/2k_BT}$$ &#9645; Average speed $$\bar v=\sqrt{\tfrac 8\pi\tfrac{RT}{M}}$$ &#9645; Peak velocity $$v_p= \sqrt{\tfrac{2RT}{M}}$$

The First Law of Thermodynamics
(Pressure, volume, temperature) remain constant in an (isobaric, isochoric, isothermal) process. Heat is not transferred in an adiabatic process. &#9645; Equation of state $$f(p,V,T)=0$$ &#9645; Work done by a system $$W=\int_{V_1}^{V_2}pdV$$ &#9645; Internal energy $$E_{int}=\sum_i\left(\overline K_i + \overline U_i\right)$$ is a sum over all particles of kinetic and potential energies &#9645; First law $$\Delta E_{int}=Q-W$$ (Q is heat going in and W is work done by as shown in the figure) &#9645; $$C_p=C_V+R$$ is the molar heat capacity at constant volume &#9645; $$pV^\gamma=\text{constant}$$ for an adiabatic process in an ideal gas, where the heat capacity ratio $$\gamma = C_p/C_V$$

The Second Law of Thermodynamics
$$W=Q_h-Q_c=$$ work done in a heat engine cycle. &#9645; Efficiency $$=e=\tfrac{W}{Q_h}= 1-\tfrac{Q_c}{Q_h}$$ &#9645; Coefficient of performance for a refrigerator $$K_R=\tfrac{Q_c}{W}=\tfrac{Q_c}{Q_h-Q_c}$$, and heat pump $$K_P=\tfrac{Q_h}{W}=\tfrac{Q_h}{Q_h-Q_c}$$ &#9645; Entropy change $$\Delta S =\tfrac Q T$$ (reversible process at constant temperature) $$\rightarrow \int_A^B\tfrac {dQ}{T}=S_B-S_A$$ &#9645; $$\oint \tfrac{dQ}{T}$$ for any cyclic process $$\rightarrow \int_A^B\tfrac {dQ}{T}=S_B-S_A$$ is path independent. &#9645; $$\Delta S \ge 0$$ for any closed system. $$\lim_{T\to 0}\Delta S = 0$$ for any isothermal process.

Electric Charges and Fields
Coulomb's Law $$\vec F=\tfrac{1}{4\pi\varepsilon_0}\tfrac{q_1q_2}{r_{12}^2}\hat r_{12}$$ where the vacuum permittivity $$\varepsilon_0=$$ 8.85×10−12 F/m. Elementary charge = e = 1.602×10−19C (electrons have charge q=&minus;e and protons have charge q=+e.) &#9645; By superposition, $$\vec F=\tfrac{1}{4\pi\varepsilon_0}Q\sum_{i=1}^N \tfrac{q_i}{r_{Qi}^2}\hat r_{Qi}$$ where $$\vec r_{Qi} = \vec r_Q- \vec r_i$$ &#9645; Electric field $$\vec F=Q\vec E$$ where $$\vec E(\vec r_P) = \tfrac{1}{4\pi\varepsilon_0}\sum_{i=1}^N \tfrac{q_i}{r_{Pi}^2}\hat r_{Pi}$$ is the field at $$\vec r_P$$ due to charges at $$\vec r_i$$ &#9645; The field above an infinite wire $$\vec E(z)= \tfrac{1}{4\pi\varepsilon_0}\tfrac{2\lambda}{z}\hat k$$ and above an infinite plane $$\vec E= \tfrac{\sigma}{2\varepsilon_0}\hat k$$ &#9645; An electric dipole $$\vec p = q\vec d$$ in a uniform electric field experiences the torque $$\tau=\vec p\times\vec E$$

Gauss's Law
Flux for a uniform electric field $$\Phi=\vec E\cdot\vec A$$ $$\to\Phi=\int\vec E\cdot d\vec{A}=\int\vec E\cdot \hat n\,dA$$ in general. &#9645; Closed surface integral $$\Phi=\oint\vec E\cdot d\vec{A}=\oint\vec E\cdot \hat n\,dA$$ &#9645; Gauss's Law $$ = q_{enc}=\varepsilon_0\oint \vec E\cdot d\vec A$$. In simple cases: $$E\int dA=EA^*=\tfrac{q_{enc}}{\varepsilon_0}$$ &#9645; Electric field just outside the surface of a conductor $$\vec E= \tfrac{\sigma}{\varepsilon_0}$$

Electric Potential
Electric potential $$\Delta V_{AB}=V_A-V_B=-\int_A^B\vec E\cdot d\vec\ell$$. Change in potential energy $$=q\Delta V=\Delta U$$ &#9645; Electron (proton) mass = 9.11×10−31kg (1.67× 10−27kg). Electron volt: 1 eV = 1.602×10−19J &#9645; Near an isolated point charge $$V(r)=k\tfrac q r$$ where $$k=\tfrac{1}{4\pi\varepsilon_0}$$ =8.99&times;109 N·m/C2 is the Coulomb constant. &#9645; Work done to assemble N particles $$W_{12...N}= \sum_{i=1}^{N}\sum_{j=1}^{i-1}\tfrac{q_iq_j}{r_{ij}}= \tfrac k 2 \sum_{i=1}^{N}\sum_{j=1}^{N}\tfrac{q_iq_j}{r_{ij}}\text{ for } i\ne j$$ &#9645; Electric potential due to N charges. $$V_P=k\sum_1^N \frac{q_i}{r_i}$$. For continuous charge $$V_P=k\int\frac{dq}{r}$$. For a dipole, $$V=k\tfrac{\vec p\cdot\vec\hat r}{r^2}$$. &#9645; Electric field as gradient of potential $$\vec E =-\tfrac{\partial V}{\partial x}\hat i -\tfrac{\partial V}{\partial y}\hat j -\tfrac{\partial V}{\partial z}\hat k = -\vec\nabla V $$ &#9645; Del operatornote: Cartesian $$\vec\nabla= \hat i\tfrac{\partial}{\partial x} +\hat j\tfrac{\partial}{\partial y} +\hat k\tfrac{\partial}{\partial z}\text{; } $$Cylindrical $$\vec\nabla= \hat r\tfrac{\partial }{\partial r} +\hat\phi\tfrac{\partial }{\partial\phi} +\hat z  \tfrac{\partial }{\partial z}\text{; } $$Spherical $$\vec\nabla= \hat r \tfrac{\partial }{\partial r} + \hat\theta\tfrac{\partial }{\partial\theta} + \hat\phi  \tfrac{\partial }{\partial\phi}\text{.}$$

Capacitance
$$Q=CV$$ defines capacitance. For a parallel plate capacitor, $$C=\varepsilon_0\tfrac A d$$ where A is area and d is gap length. &#9645; $$4\pi\varepsilon_0\tfrac{R_1R_2}{R_2-R_1}$$ and $$\tfrac{2\pi\varepsilon_0\ell}{\ln(R_2/R_1)}$$ for a spherical and cylindrical capacitor, respectively &#9645; For capacitors in series (parallel) $$\tfrac{1}{C_S} = \sum\tfrac{1}{C_i} \left(C_P=\sum C_i\right)$$ &#9645; $$u=\tfrac 1 2 QV = \tfrac 1 2 CV^2=\tfrac{1}{2C}Q^2$$ &#9645; Stored energy density $$u_E=\tfrac 1 2 \varepsilon_0E^2$$ &#9645; A dielectric with $$\kappa>1$$ will decrease the capacitor's electric field $$E=\tfrac{1}{\kappa}E_0$$ and stored energy $$U=\tfrac 1\kappa U_0$$, but increase the capacitance $$C=\kappa C_0$$ due to the induced electric field $$\vec E_i=\left(\tfrac 1\kappa -1\right)\vec E_0$$

Current and Resistance
Current (1A=1C/s) $$I=dQ/dt=nqv_dA$$ where $$(n,q,v_d)=$$ (density, charge, drift velocity) of the carriers. &#9645; $$I=JA\rightarrow\int\vec J\cdot d\vec A$$, $$A$$ is the perpendicular area, and $$J$$ is current density. $$\vec E= \rho\vec J$$ is electric field, where $$\rho$$ is resistivity. &#9645; Resistivity varies with temperature as $$\rho=\rho_0\left[1+\alpha (T-T_0)\right]$$. Similarily, $$R=R_0\left[1+\alpha \Delta T\right]$$ where $$R=\rho \tfrac L A$$ is resistance (&Omega;) &#9645; Ohm's law $$V=IR$$ &#9645; Power $$=P=IV=I^2R=V^2/R$$

Direct-Current Circuits
Terminal voltage $$V_{terminal}=\varepsilon -Ir_{eq}$$ where $$r_{eq}$$ is the internal resistance and $$\varepsilon$$ is the electromotive force. &#9645; Resistors in series and parallel: $$R_{series}=\sum_{i=1}^N R_i$$ &#9645; $$ R_{parallel}^{-1}=\sum_{i=1}^NR_{i}^{-1}$$ &#9645; Kirchoff's rules. Loop:$$\sum I_{in}=\sum I_{out}$$ Junction: $$\sum V = 0$$ &#9645; $$V_{terminal}^{series} = \sum_{i=1}^N\varepsilon_i-I\sum_{i=1}^Nr_i$$ &#9645; $$V_{terminal}^{parallel} = \varepsilon -I\sum_{i=1}^N\left(\frac{1}{r_i}\right)^{-1}$$ where $$r_i$$ is internal resistance of each voltage source. &#9645; Charging an RC (resistor-capacitor) circuit: $$ q(t)=Q\left(1-e^{-t/\tau}\right)$$ and $$ I=I_0e^{-t/\tau}$$ where $$\tau = RC$$ is RC time, $$ Q=\varepsilon C$$ and $$I_0=\varepsilon/R$$. &#9645; Discharging an RC circuit: $$q(t)=Qe^{-t/\tau}$$ and $$I(t)=-\tfrac{Q}{RC}e^{-t/\tau}$$

Magnetic Forces and Fields
&#9645; $$\vec F=q\vec v\times\vec B$$ is the force due to a magnetic field on a moving charge. &#9645; For a current element oriented along $$\overrightarrow{d\ell},\;d\vec F = I\overrightarrow{d\ell}\times\vec B$$. &#9645; The SI unit for magnetic field is the Tesla: 1T=104 Gauss. &#9645; Gyroradius $$r=\tfrac{mB}{qB}.\;$$ Period $$T=\tfrac{2\pi m}{qB}.\;$$ &#9645; Torque on current loop $$\vec\tau=\vec\mu\times\vec B$$ where $$\vec\mu=NIA\hat n$$ is the dipole moment. Stored energy $$U=\vec\mu\cdot\vec B.$$ &#9645; Drift velocity in crossed electric and magnetic fields $$v_d=\tfrac E B$$ &#9645; Hall voltage = $$V$$ where the electric field is $$E=V/\ell=Bv_d=\tfrac{IB}{neA}$$ &#9645; Charge-to-mass ratio $$q/m=\tfrac{E}{BB_0r}$$ where the $$E$$ and $$B$$ fields are crossed and $$E=0$$ when the magnetic field is $$B_0$$

Sources of Magnetic Fields
&#9645; Permeability of free space $$\mu_0=4\pi\times 10^{-7}$$ T·m/A &#9645; Force between parallel wires $$\tfrac F\ell=\tfrac{\mu_0I_1I_2}{2\pi r}$$ &#9645; Biot–Savart law $$\vec B=\tfrac{\mu_0}{4\pi}\int\limits_{wire}\frac{Id\vec\ell\times\hat r}{r^2}$$

&#9645; Ampère's Law:$$\oint\vec B\cdot d\vec\ell=\mu_0I_{enc}$$ &#9645; Magnetic field due to long straight wire $$B=\tfrac{\mu_0I}{2\pi R}$$ &#9645; At center of loop $$B=\tfrac{\mu_0I}{2R}$$ &#9645; Inside a long thin solenoid $$B=\mu_0nI$$ where $$n=N/\ell$$ is the ratio of the number of turns to the solenoid's length. &#9645; Inside a toroid $$B=\tfrac{\mu_0N}{2\pi r}$$ &#9645; The magnetic field inside a solenoid filled with paramagnetic material is $$B=\mu nI$$ where $$\mu=(1+\chi)\mu_0$$ is the permeability

Electromagnetic Induction
Magnetic flux $$ \Phi_m=\int_S\vec B\cdot\hat n dA$$ &#9645; Electromotive force $$\varepsilon=- N\tfrac{d\Phi_m}{dt,}$$ (Faraday's law) &#9645; Motional emf $$ \varepsilon=B\ell v,$$ &#9645; rotating coil $$ NBA\omega\sin\omega t$$ &#9645; Motional emf around circuit $$\varepsilon= \oint\vec E\cdot d\vec\ell=-\tfrac{d\Phi_m}{dt}$$

Inductance
The SI unit for inductance is the Henry: 1H=1V·s/A &#9645; Mutual inductance: $$ M\tfrac{dI_2}{dt}=N_1\tfrac{d\Phi_{12}}{dt}=-\varepsilon_1$$ where $$ \Phi_{12}$$ is the flux through 1 due to the current in 2 and $$ \varepsilon_1$$ is the emf in 1. Likewise, it can be shownSEE TALK that, $$M\tfrac{dI_1}{dt}=-\varepsilon_2$$. &#9645; Self-inductance $$N\Phi_m=LI\rightarrow\varepsilon=-L\tfrac{dI}{dt}$$ &#9645; $$L_\text{solenoid}\approx\mu_0N^2A\ell,\,$$$$L_\text{toroid}\approx \tfrac{\mu_0N^2h}{2\pi}\ln\tfrac{R_2}{R_1}.$$ Stored energy $$U=\tfrac 12 LI^2.$$ &#9645; $$I(t)=\tfrac\varepsilon R \left(1-e^{-t/\tau}\right)$$is the current in an LR circuit where $$\tau=L/R$$ is the LR decay time. &#9645; The capacitor's charge on an LC circuit $$q=q_0\cos(\omega t+\phi)$$ where $$\omega=\sqrt{\tfrac{1}{LC}}$$ is angular frequency &#9645; LRC circuit $$q(t)=q_0e^{-Rt/2L}\cos (\omega 't + \phi)$$ where $$ \omega ' = \sqrt{\tfrac 1 {LC} + \left(\tfrac R {2L}\right)^2}$$

Alternating-Current Circuits
AC voltage and current $$v=V_0\sin(\omega t-\phi)$$ if $$i=I_0\sin\omega t.$$ &#9645; RMS values $$I_{rms}=\tfrac{I_0}{\sqrt 2}$$ and $$V_{rms}=\tfrac{V_0}{\sqrt 2}$$ &#9645; Impedance $$V_0=I_0X$$ &#9645; Resistor $$V_0=I_0X_R,\;\phi=0,$$ where $$X_R=R$$ &#9645; Capacitor $$V_0=I_0X_C,\;\phi=-\tfrac\pi 2,$$ where $$X_C=\tfrac 1{\omega C}$$ &#9645; Inductor $$V_0=I_0X_L,\;\phi=+\tfrac\pi 2,$$ where $$X_L=\omega L$$ &#9645; RLC series circuit $$V_0=I_0 Z$$ where $$ Z=\sqrt{R^2 + \left(X_L-X_C\right)^2}$$ and $$\phi=\tan^{-1}\frac{X_L-X_C}{R}$$ &#9645; Resonant angular frequency $$\omega_0=\sqrt\tfrac{1}{LC}$$ &#9645; Quality factor $$Q=\tfrac{\omega_0}{\Delta\omega}=\tfrac{\omega_0L}{R}$$ &#9645; Average power $$P_{ave}=\frac 1 2 I_0V_0\cos\phi=I_{rms}V_{rms}\cos\phi$$, where $$\phi=0$$ for a resistor. &#9645; Transformer voltages and currents $$\tfrac{V_S}{V_P}=\tfrac{N_S}{N_P}=\tfrac{I_P}{I_S}$$

Electromagnetic Waves
Displacement current $$I_d=\varepsilon_0\tfrac{d\Phi_E}{dt}$$ where $$\Phi_E=\int\vec E\cdot d\vec A$$ is the electric flux. Maxwell's equations $$\begin{align} \oint_S \vec{E} \cdot \mathrm{d}\vec{A} &= \frac 1{\epsilon_0}Q_{in} \qquad & \oint_S \vec{B} \cdot \mathrm{d}\vec{A} &= 0\\ \oint_C \vec{E} \cdot \mathrm{d}\vec{\ell} &= -  \int_S \frac{\partial\vec{B}}{\partial t} \cdot \mathrm{d} \vec{A} \qquad & \oint_C \vec{B} \cdot \mathrm{d}\vec{\ell} &= \mu_0I + \epsilon_0\mu_0\frac{\mathrm{d}\Phi_E}{\mathrm{d}t} \end{align}$$ See also http://ethw.org/w/index.php?title=Maxwell%27s_Equations&oldid=157445 &#9645; Plane EM wave equation $$ \frac{\partial^2E_y}{\partial x^2}=\varepsilon_0\mu_0\frac{\partial^2E_y}{\partial t^2}$$ where $$c=\tfrac 1{\sqrt{\varepsilon_0\mu}}$$ is the speed of light &#9645; The ratio of peak electric to magnetic field is $$\tfrac{E_0}{B_0}=c$$ and the Poynting vector $$\vec S = \tfrac 1{\mu_0} \vec E\times\vec B$$ represents the energy flux &#9645; Average intensity $$I=S_{ave}=\tfrac{c\varepsilon_0}{2}E_0^2=\tfrac{c}{2\mu_0}B_0^2=\tfrac{1}{2\mu_0}E_0B_0$$ &#9645; Radiation pressure $$p=I/c$$ (perfect absorber) and $$p=2I/c$$ (perfect reflector).