Physics I Essentials

Cheatsheet Content

1. Thermodynamics: Ideal Gas Laws & Processes 1.1. Ideal Gas Equation of State Relates pressure ($P$), volume ($V$), and temperature ($T$) for an ideal gas: $$PV = nRT = \frac{m}{M}RT = NkT$$ $P$: Pressure (Pa) $V$: Volume ($m^3$) $T$: Temperature (K) $n$: Number of moles ($n = m/M$) $N$: Number of molecules ($N = nN_A$) $R$: Ideal gas constant ($8.314 J/(mol \cdot K)$) $k$: Boltzmann constant ($1.38 \cdot 10^{-23} J/K$) $N_A$: Avogadro's number ($6.022 \cdot 10^{23} mol^{-1}$) For any ideal gas process: $\frac{P_1V_1}{T_1} = \frac{P_2V_2}{T_2}$ 1.2. Ideal Gas Processes Summary Process Condition Relationship Work ($A$) Heat ($Q$) $\Delta U$ Isochoric $V = \text{const}$ $\frac{P_1}{T_1} = \frac{P_2}{T_2}$ $0$ $nC_V\Delta T$ $nC_V\Delta T$ Isobaric $P = \text{const}$ $\frac{V_1}{T_1} = \frac{V_2}{T_2}$ $-P\Delta V = -nR\Delta T$ $nC_P\Delta T$ $nC_V\Delta T$ Isothermal $T = \text{const}$ $P_1V_1 = P_2V_2$ $-nRT\ln\left(\frac{V_2}{V_1}\right)$ $nRT\ln\left(\frac{V_2}{V_1}\right)$ $0$ Adiabatic $Q = 0$ $PV^\gamma = \text{const}$ $\frac{P_2V_2 - P_1V_1}{1-\gamma}$ $0$ $nC_V\Delta T$ $\gamma = C_P/C_V$ (adiabatic index) $C_V = \frac{i}{2}R$ (molar specific heat at constant volume) $C_P = C_V + R = (\frac{i}{2}+1)R$ (molar specific heat at constant pressure) $i$: Degrees of freedom (Monoatomic $i=3$, Diatomic $i=5$, Polyatomic $i=6$) 1.3. Kinetic Theory of Ideal Gas Average Translational Kinetic Energy: $\bar{W} = \frac{3}{2}kT$ Root Mean Square (RMS) Speed: $v_{rms} = \sqrt{\frac{3kT}{m_0}} = \sqrt{\frac{3RT}{M}}$ Basic Equation: $P = \frac{2}{3} n_0 \bar{W}$ ($n_0$ is number density) 2. Thermodynamics: First & Second Laws 2.1. First Law of Thermodynamics Conservation of Energy: $\Delta U = Q + A_{on~system} = Q - A_{by~system}$ $\Delta U$: Change in internal energy ($J$) $Q$: Heat (positive if absorbed, negative if released) ($J$) $A$: Work (positive if done *on* system, negative if done *by* system) ($J$) Work done by system: $A_{by~system} = \int P dV$ Specific Heat: $Q = mc\Delta T$ ($c$: specific heat, $J/(kg \cdot K)$) Phase Change Heat: $Q = \pm mL$ ($L$: latent heat, $J/kg$) 2.2. Second Law of Thermodynamics & Efficiency Heat Engine Efficiency ($\epsilon$): $$\epsilon = \frac{W_{net}}{|Q_h|} = 1 - \frac{|Q_c|}{|Q_h|}$$ Carnot Efficiency (Maximum): $$\epsilon_{Carnot} = 1 - \frac{T_c}{T_h}$$ Refrigerator/Heat Pump COP: Refrigerator: $COP_{ref} = \frac{|Q_c|}{W_{net}} = \frac{|Q_c|}{|Q_h| - |Q_c|}$ Heat Pump: $COP_{HP} = \frac{|Q_h|}{W_{net}} = \frac{|Q_h|}{|Q_h| - |Q_c|}$ 2.3. Entropy ($S$) Measure of disorder. For reversible process: $dS = \frac{\delta Q_{rev}}{T}$ Isolated System: $\Delta S \ge 0$ (equals 0 for reversible, greater than 0 for irreversible) Phase Change: $\Delta S = \frac{mL}{T_{phase}}$ Ideal Gas: $\Delta S = nC_V \ln\left(\frac{T_2}{T_1}\right) + nR \ln\left(\frac{V_2}{V_1}\right)$ 3. Electrostatics: Fields & Potentials 3.1. Coulomb's Law Force between two point charges $q_1, q_2$ separated by $r$: $$F = k_e \frac{|q_1q_2|}{r^2}$$ $k_e = 8.9876 \times 10^9 N \cdot m^2/C^2 = \frac{1}{4\pi\epsilon_0}$ $\epsilon_0 = 8.854 \times 10^{-12} C^2/(N \cdot m^2)$ (permittivity of free space) 3.2. Electric Field ($\vec{E}$) Force per unit charge: $\vec{E} = \vec{F}/q_0$ Point Charge $q$: $\vec{E} = k_e \frac{q}{r^2}\hat{r}$ Continuous Charge Distribution: $\vec{E} = k_e \int \frac{dq}{r^2}\hat{r}$ Infinite Plane (surface charge density $\sigma$): $E = \frac{\sigma}{2\epsilon_0}$ Infinite Line (linear charge density $\lambda$): $E = \frac{\lambda}{2\pi\epsilon_0 r}$ Conductor in Equilibrium: $E=0$ inside; $E=\sigma/\epsilon_0$ just outside, perpendicular to surface. 3.3. Electric Potential ($V$) Potential energy per unit charge: $V = U/q_0$ Potential Difference: $\Delta V = V_B - V_A = -\int_A^B \vec{E} \cdot d\vec{s}$ Point Charge $q$: $V = k_e \frac{q}{r}$ Continuous Charge Distribution: $V = k_e \int \frac{dq}{r}$ Relation to Field: $\vec{E} = -\nabla V$ Conductor in Equilibrium: $V = \text{const}$ throughout and on surface. 3.4. Gauss's Law Electric flux through a closed surface is proportional to enclosed charge: $$ \oint \vec{E} \cdot d\vec{A} = \frac{q_{in}}{\epsilon_0} $$ 3.5. Dielectrics & Polarization Material with dielectric constant $\kappa_e$ (or $\epsilon = \kappa_e \epsilon_0$). Field Reduction: $E = E_0 / \kappa_e$ Displacement Field: $\vec{D} = \epsilon_0 \vec{E} + \vec{P} = \epsilon_0 \kappa_e \vec{E}$ Boundary Conditions: $E_{t1} = E_{t2}$; $D_{n1} = D_{n2}$ 4. Capacitance & Energy Storage 4.1. Capacitance ($C$) Ability to store charge: $C = \frac{Q}{\Delta V}$ (Unit: Farad, F) Parallel-Plate: $C = \frac{\epsilon A}{d} = \frac{\kappa_e \epsilon_0 A}{d}$ Cylindrical: $C = \frac{2\pi\epsilon L}{\ln(R_2/R_1)}$ Spherical: $C = \frac{4\pi\epsilon R_1 R_2}{R_2-R_1}$ 4.2. Capacitor Combinations Series: $\frac{1}{C_{eq}} = \sum_i \frac{1}{C_i}$ Parallel: $C_{eq} = \sum_i C_i$ 4.3. Energy Stored in Capacitor $$U_E = \frac{1}{2}Q\Delta V = \frac{1}{2}C(\Delta V)^2 = \frac{Q^2}{2C}$$ Energy Density: $u_E = \frac{1}{2}\epsilon_0 E^2$ (in vacuum/air) 5. Magnetostatics: Fields & Forces 5.1. Magnetic Force on Charged Particle Lorentz Force: $\vec{F}_B = q(\vec{v} \times \vec{B})$ Magnitude: $F_B = |q|vB\sin\theta$ Direction: Right-hand rule Unit of $\vec{B}$: Tesla (T) Circular Motion in Uniform $\vec{B}$ (if $\vec{v} \perp \vec{B}$): Radius: $r = \frac{mv}{|q|B}$ Angular Speed: $\omega = \frac{|q|B}{m}$ 5.2. Magnetic Force on Current-Carrying Wire Straight wire of length $\vec{L}$ carrying current $I$: $$\vec{F}_B = I(\vec{L} \times \vec{B})$$ Magnitude: $F_B = ILB\sin\theta$ 5.3. Sources of Magnetic Field Biot-Savart Law: $d\vec{B} = \frac{\mu_0}{4\pi} \frac{I d\vec{s} \times \hat{r}}{r^2}$ $\mu_0 = 4\pi \times 10^{-7} T \cdot m/A$ (permeability of free space) Long Straight Wire: $B = \frac{\mu_0 I}{2\pi r}$ Center of Circular Loop (radius $a$, $N$ turns): $B = \frac{\mu_0 N I}{2a}$ Solenoid (n turns/unit length): $B = \mu_0 n I$ (inside) 5.4. Ampere's Law Line integral of $\vec{B}$ around a closed loop: $$ \oint \vec{B} \cdot d\vec{s} = \mu_0 I_{enc} $$ 5.5. Magnetic Flux ($\Phi_B$) Total magnetic field passing through a surface: $$ \Phi_B = \int \vec{B} \cdot d\vec{A} $$ Unit: Weber (Wb) Gauss's Law for Magnetism: $\oint \vec{B} \cdot d\vec{A} = 0$ (no magnetic monopoles) 6. Hall Effect Voltage developed across a current-carrying conductor in a magnetic field, due to charge carrier deflection. Hall Voltage ($U_H$): $$U_H = E_H d = \frac{R_H B I}{t}$$ Where $d$ is width, $t$ is thickness. Hall Coefficient ($R_H$): $R_H = \frac{1}{n|q|}$ (determines carrier type and density $n$) Hall Electric Field ($E_H$): $E_H = v_d B$ ($v_d$: drift velocity)