General Physics II Review
Ch. 21 Coulomb's Law
Charge
- Plastic rod rubbed on fur: negatively charged
- Glass rod rubbed on silk: positively charged
The total (net) electric charge of an isolated system is conserved.
- Charging by induction (without losing its own charge)
\[ e=1.60\times10^{-19}\rm C \]
- Quantized Charge
- Millikin Oil-Drop Experiment
Coulomb's Law
\[ F=k\frac{|q_1q_2|}{r^2} \]
\[ k= \frac{1}{4\pi\epsilon_0} = 8.99\times10^9 \rm N\cdot m^2/C^2 \]
\[ \epsilon_0 = 8.85\times10^{-12}\rm C^2/N\cdot m^2 \]
- \(k\) is the electromagnetic constant
- \(\epsilon_0\) is the permittivity constant
- Principle of Superposition
Ch. 22 Electric Field
Definition
\[ \vec{F} = q\vec{E}. \]
\[ \vec{E} = \frac{\vec{F}}{q_0} = k\frac{|q|}{r^2} = \frac{1}{4\pi\epsilon_0}\frac{|q|}{r^2} \]
Cases
The key to integrate, is find the relation between \(\mathop{dq}\) and charge density (as well as length or area)
Uniformly Charged Line
\[ E_x = \frac{1}{4\pi\epsilon_0}\frac{Q}{2a} \int_{-a}^{+a} \frac{x\mathop{dy}}{(x^2+y^2)^{3/2}} = \frac{Q}{4\pi\epsilon_0} \frac{1}{x\sqrt{x^2+a^2}} \]
If the line is very long, \(a\gg x\)
\[ E = \frac{\lambda}{2\pi\epsilon_0 r} \]
Uniformly Charged Ring
\[ \begin{aligned} E & = \int\mathop{dE}\cos\theta \\ & = \frac{z}{\sqrt{z^2+R^2}}\cdot \frac{\lambda}{4\pi\epsilon_0 (z^2+R^2)} \int_0^{2\pi R}\mathop{ds} \\ & = \frac{qz}{4\pi\epsilon_0 (z^2+R^2)^{3/2}}. \end{aligned} \]
If the charged ring is at large distance, \(z\gg R\)
\[ E = \frac{1}{4\pi\epsilon_0}\frac{1}{z^2}. \]
Uniformly Charged Disk
\[ \mathop{dq} = \sigma\mathop{dA} = \sigma(2\pi r\mathop{dr}). \]
\[ \mathop{dE} = \frac{\sigma z}{4\epsilon_0} \frac{2r\mathop{dr}}{(z^2+r^2)^{3/2}}. \]
\[ E = \int\mathop{dE} = \frac{\sigma}{2\epsilon_0}\left( 1 - \frac{z}{\sqrt{z^2+R^2}} \right). \]
For \(R\rightarrow\infty\) (infinite sheet)
\[ E = \frac{\sigma}{2\epsilon_0}. \]
Electric Field Line
Electric Dipole *
Electric Field Due to Electric Dipole
\[ \begin{aligned} E & = E_{(+)} - E_{(-)} \\ & = \frac{1}{4\pi\epsilon_0} \frac{q}{r^2_{(+)}} - \frac{1}{4\pi\epsilon_0} \frac{q}{r^2_{(-)}} \\ & = \cdots \\ & = \frac{q}{2\pi\epsilon_0z^3} \frac{d}{\left( 1 - \left(\frac{d}{2z}\right)^2 \right)^2} \end{aligned}. \]
If \(z\gg d\)
\[ E = \frac{1}{2\pi\epsilon_0}\frac{qd}{z^3} \]
- \(qd\) is the electric dipole moment \(\vec{p}\)
On the axis of the dipole
\[ \vec{E}_\text{dipole} \approx \frac{1}{4\pi\epsilon_0}\frac{\vec{p}}{r^3} \]
On the perpendicular plane
\[ \vec{E}_\text{dipole} \approx -\frac{1}{4\pi\epsilon_0}\frac{\vec{p}}{r^3} \]
Dipole in Electric Field
\[ \vec{\tau} = Fd\sin\theta = \vec{p}\times\vec{E}. \]
\[ U = -W = -\int\tau\mathop{d\theta} = \int pE\sin\theta\mathop{d\theta} = -pE\cos\theta = -\vec{p}\cdot\vec{E} \]
Ch. 23 Gauss's Law
\[ \epsilon_0\Phi = \epsilon_0\oint \vec{E}\cdot\mathop{d\vec{A}} = q_{\text{enclosed}} \]
\[ \oint_A \vec{E}\cdot\mathop{d\vec{A}} = \int_V \nabla\cdot\vec{E}\mathop{dV} = \frac{q_{\text{enc}}}{\epsilon_0}. \]
- Gauss's Law is equivalent with Coulomb's Law.
- Planar Symmetry
- Cylindrical Symmetry
Charged Isolated Conductor
- There can be no excess charge at any point within a solid conductor
- Electric field inside a conductor needs to be zero
- Charge on the inner wall cannot produce and electric field in the shell to affect the charge on the outer wall
Electrostatic Shielding
Ch. 24 Electric Potential
\[ \Delta U = U_f - U_i = -W_{\text{by the field}} \]
\[ V = V_f - V_i = -\int_i^f \vec{E}\cdot\mathop{d\vec{s}}. \]
\(i\) is at \(\infty\), where \(U\) is \(0\), \(V_i\) is \(0\).
- Electric potential for a point charge
\[ V = \frac{1}{4\pi\epsilon_0}\frac{q}{r} \]
- Potential due to electric dipole
\[ V = V_{(+)} + V_{(-)} = \cdots = \frac{1}{4\pi\epsilon_0}\frac{p\cos\theta}{r^2} \]
- Field from Potential
\[ E_s = -\frac{\partial V}{\partial s}. \]
\[ \vec{E} = -\nabla V \]
Ch. 25 Capacitance
\[ q = CV \]
- Parallel capacitor
\[ C = \frac{q}{V} = \frac{\epsilon_0 EA}{Ed} = \frac{\epsilon_0 A}{d}. \]
- Cylindrical capacitor
\[ q = \epsilon_0 EA = \epsilon_0 E(2\pi rL) \]
\[ V = \int_-^+ E\mathop{ds} = -\frac{q}{2\pi\epsilon_0 L} \int_b^a \frac{\mathop{dr}}{r} = \frac{q}{2\pi\epsilon_0 L} \ln\left(\frac{b}{a}\right) \]
\[ C = \frac{q}{V} = 2\pi\epsilon_0 \frac{L}{\ln(b/a)} \]
- Spherical capacitor
\[ \cdots C = 4\pi\epsilon_0 \frac{ab}{b-a}. \]
Series & Parallel
Energy stored
\[ \mathop{dW} = v\mathop{dq} = \frac{q\mathop{dq}}{C} \]
\[ W = \int_0^W \mathop{dW} = \frac{1}{C}\int_0^Q q\mathop{dq} = \frac{Q^2}{2C} \Rightarrow U = \frac{1}{2}CV^2 = \frac{1}{2}QV \]
- Energy density
\[ u = \frac{\frac{1}{2}CV^2}{Ad} = \frac{1}{2}\epsilon_0E^2 \]
- Dielectric
\[ \epsilon = \kappa\epsilon_0 \]
\[ E = \frac{V}{d} = \frac{q}{Cd} = \frac{q}{d\epsilon_0A/d} = \frac{\sigma}{\epsilon_0} \]
\[ E = \frac{\sigma-\sigma_i}{\epsilon_0} = \frac{E_0}{K} \Rightarrow \sigma_i = \sigma\left(1-\frac{1}{K}\right) \quad\text{(induced surface charge density)} \]
\[ \epsilon_0 \oint\kappa\vec{E}\cdot\mathop{d\vec{A}} = q \]
Ch. 26 Current and Resistance
Current
\[ i = \frac{\mathop{dq}}{\mathop{dt}} \]
Current is NOT a vector
- Current density
\[ i = \frac{q}{t} = \frac{nALe}{L/v_d} = nAev_d. \]
\[ \vec{J} = (ne)\vec{v}_d \]
Resistance
- Resistivity
\[ \rho = \frac{E}{J} \]
- Resistance
\[ \rho = \frac{V/L}{I/A} \Rightarrow R = \frac{V}{I} = \frac{\rho L}{A}. \]
Ohm's Law
For Ohmic contact cases
\[ V = iR \]
Ch. 27 Circuits
- Kirchhoff's Rules
\[ \sum I = 0,\quad \sum V = 0 \]
- Mark for RC Circuit
Ch. 28 Magnetic Field
- On Moving Charge
\[ \vec{F} = q\vec{v}\times\vec{B} \]
- Hall Effect
\[ n = \frac{Bi}{Vle} \]
- On a Current Carrying Conductor
\[ \vec{F}_B = i\vec{l}\times\vec{B} \]
- Magnetic Dipoles
\[ \bm{\mu} = NiA \]
\[ \bm{\tau=\mu\times B} \]
Ch. 29 Magnetic Fields Due to Currents
- Biot-Savart Law
\[ \mathop{d\bm{B}} = \frac{\mu_0}{4\pi}\frac{i\mathop{d\bm{l}}\times\bm{e}_r}{r^2} \]
- For B-Field of a wire with steady current
\[ B = \frac{\mu_0 i}{2\pi R} \]
- B-Field at the center of a circular arc of wire
\[ B = \int\mathop{dB} = \int_0^\phi \frac{\mu_0}{4\pi}\frac{iR\mathop{d\phi}}{R^2} = \frac{\mu_0 i}{4\pi R} \int_0^\phi\mathop{d\phi} = \frac{\mu_0 i\phi}{4\pi R} \]
- Gauss' Law for magnetism
\[ \oint \vec{B}\cdot\mathop{d\vec{A}} = 0 \]
- Ampere's Law
\[ \oint\vec{B}\cdot\mathop{d\vec{l}} = \mu_0I_\text{encl} \]
\[ \oint_\mathbf{l} \mathbf{B}\cdot\mathop{d\mathbf{l}} = \mu_0 \int_\mathbf{A}\mathbf{J}\cdot\mathop{d\mathbf{A}} \]
- B-Field of solenoid
\[ BL = \mu_0 nLI \]
- B-Field of toroid
\[ B = \frac{\mu_0 NI}{2\pi r} \]
- Current sheet
\[ B = \frac{1}{2}\mu_0 J_s \]
Ch. 30 Induction and Inductance
Laws of Induction
- Lenz's Law
- Faraday's Law of Induction
\[ \mathscr{E} = -\frac{\mathop{d\Phi_B}}{\mathop{dt}} \]
\[ -\frac{d}{dt}\int\limits_A\vec{B}\cdot\mathop{d\vec{A}} = \oint\limits_l\vec{E}\cdot\mathop{d\vec{l}} \]
\[ \nabla\times\mathbf{E} = -\frac{d\mathbf{B}}{dt} \]
- Eddy Current
Inductors
- Self induction
\[ \mathscr{E}_L = -L\frac{di}{dt} \]
\[ L = \frac{N\Phi_B}{i} \]
- Inductance of an ideal solenoid
\[ L = \mu_0n^2lA \]
- Inductance of a toroid
\[ L = \frac{\mu_0N^2b}{2\pi}\ln\frac{r_2}{r_1} \]
RL Circuit
\[ \mathscr{E} -i(t)R - L\frac{di}{dt} = 0 \Rightarrow \frac{di}{dt} + \frac{iR}{L} = \frac{\mathscr{E}}{L} \]
\[ i(t) = \frac{\mathscr{E}}{R}(1 - e^{-t/\tau_L}), \quad \tau_L = L/R \]
- Energy stored in magnetic field
\[ i\mathscr{E} = Li\frac{di}{dt} + i^2R \Rightarrow U_B = \int_0^t Li\mathop{di} = \frac{1}{2}Li^2 \]
- Energy density of magnetic field
\[ u_B = \frac{Li^2}{2Al} = \frac{U_B}{Al} = \frac{\mu_0n^2Ai^2}{2A} = \frac{\mu_0n^2i^2}{2} = \frac{B^2}{2\mu_0} \]
Mutual Induction
\[ \mathscr{E}_2 = -M\frac{di_1}{dt},\quad \mathscr{E}_1 = -M\frac{di_2}{dt} \]
Ch. 31 EM Oscillation & AC Current *
\[ U_C = \frac{q^2}{2C}, \quad U_B = \frac{Li^2}{2}, \quad \]
\[ -L\frac{di}{dt} - \frac{q}{C} = 0 \Rightarrow \frac{d^2q}{dt^2} + \frac{q}{LC} = 0 \]
\[ \Rightarrow -A\omega^2\cos(\omega t + \phi) + \frac{A}{LC}\cos(\omega t + \phi) = 0, \quad \omega = \frac{1}{\sqrt{LC}} \]
Damped Oscillation in LRC Circuit
Forced Oscillations
- capacitive reactance
\[ X_C = \frac{1}{\omega_d C} \]
Voltage leads the current
- inductive reactance
\[ X_L = \omega_d L \]
Voltage lags the current
- Impedance
\[ Z = \sqrt{R^2 + (X_L - X_C)^2}, \quad I = \frac{\mathscr{E}}{Za} \]
Power in AC Circuits
\[ P_\text{avg} = \mathscr{E}_\text{rms}I_\text{rms}\cos\phi, \quad \cos\phi = \frac{V_R}{\mathscr{E}_m} = \frac{R}{Z} \]
Transformers
\[ V_1I_1 = V_2I_2, \quad \frac{V_2}{V_1} = \frac{N_2}{N_1} \]
Ch. 32 Maxwell’s Equations; Magnetism of Matter *
- Integral Form
\[ \left\{ \begin{aligned} & \oint_{\mathbf{A} = \partial V} \mathbf{E}\cdot\mathop{d\mathbf{A}} = \frac{q_\text{enc}}{\epsilon_0} & \quad\text{Gauss' Law} \\ & \oint_{\mathbf{A} = \partial V} \mathbf{B}\cdot\mathop{d\mathbf{A}} = 0 & \quad\text{Gauss' Law for B-Field} \\ & \oint_{\mathbf{l} = \partial \mathbf{A}} \mathbf{E}\cdot\mathop{d\mathbf{l}} = - \frac{d}{dt} \int_\mathbf{A} \mathbf{B}\cdot\mathop{d\mathbf{A}} & \quad\text{Faraday's Law} \\ & \oint_{\mathbf{l} = \partial\mathbf{A}} \mathbf{B}\cdot\mathop{d\mathbf{l}} = \mu_0\left( \int_\mathbf{A} \mathbf{J}\cdot\mathop{d\mathbf{A}} + \epsilon_0\frac{d}{dt} \int_\mathbf{A} \mathbf{E}\cdot\mathop{d\mathbf{A}} \right) & \quad\text{Ampere' Law} \end{aligned} \right. \]
- Maxwell's Law of Induction
\[ \oint_\mathbf{l} \mathbf{B}\cdot\mathop{d\mathbf{l}} = \mu_0\epsilon_0\frac{d\Phi_E}{dt} = \mu_0\epsilon_0\frac{d}{dt} \int_\mathbf{A} \mathbf{E}\cdot\mathop{d\mathbf{A}} \]
- displacement current
\[ i_d = \epsilon_0\frac{d\Phi_E}{dt} \]
- Differential Form
\[ \left\{ \begin{aligned} & \nabla\cdot\mathbf{E} = \frac{\rho}{\epsilon_0} \\ & \nabla\cdot\mathbf{B} = 0 \\ & \nabla\cdot\mathbf{E} = - \frac{\partial\mathbf{B}}{\partial t} \\ & \nabla\cdot\mathbf{B} = \mu_0\left( \mathbf{J} + \epsilon_0\frac{\partial\mathbf{E}}{\partial t} \right) \\ \end{aligned} \right. \]
Ch. 33 Electromagnetic Waves *
\[ \left\{ \begin{aligned} & \nabla^2\mathbf{B} - \mu_0\epsilon_0\frac{\partial^2\mathbf{B}}{\partial t^2} = 0 \\ & \nabla^2\mathbf{E} - \mu_0\epsilon_0\frac{\partial^2\mathbf{E}}{\partial t^2} = 0 \end{aligned} \right.\Rightarrow \left\{ \begin{aligned} & \mathbf{E}(\mathbf{r},t) = \mathbf{E}_0 \cos(\omega t - \mathbf{k}\cdot\mathbf{r} + \phi_0) \\ & \mathbf{B}(\mathbf{r},t) = \mathbf{B}_0 \cos(\omega t - \mathbf{k}\cdot\mathbf{r} + \phi_0) \end{aligned} \right. \]
\[ k = |\mathbf{k}| = \frac{\omega}{c} = \frac{2\pi}{\lambda} \]
\[ \left|\frac{\mathbf{E}_0}{\mathbf{B}_0}\right| = c = \frac{1}{\sqrt{\mu_0\epsilon_0}} \]
Energy Transfer in EM Waves
\[ u_\text{total} = \epsilon_0E^2 = \epsilon_0EBc = \frac{EB}{\mu_0} \]
- Poynting Vector
\[ \vec{S} = \frac{1}{\mu_0}\vec{E}\times\vec{B} \]
- Intensity
\[ I = S_\text{avg} = \frac{E_0B_0}{2\mu_0} = \frac{E_0^2}{2\mu_0c} = \frac{E^2_\text{rms}}{\mu_0c} \]
\[ \frac{\text{average power}}{4\pi r^2} \propto \frac{1}{r^2} \]
Polarization
When a unpolarized light passes through a polarizer, half of the intensity is loss
\[ I - \frac{1}{2}I_0 \]
\[ I_2 = I_1\cos^2\phi \]
Reflection & Refraction
- Snell's Law
\[ \frac{\sin\theta_1}{\sin\theta_2} = \frac{n_\text{water}}{n_\text{air}},\quad n = c/v = \frac{\lambda_\text{vacuum}}{\lambda_\text{medium}} \]
- Fermat's Principle in Optics
- Total Internal Reflection
\[ \sin\theta_\text{critical} = \frac{n_2}{n_1} \]
- Light in a Raindrop *
- Polarization bt reflection
\[ \theta_B = \tan^{-1}\frac{n_2}{n_1} \]
Ch. 35 Interference *
Young's Interference
\[ d\sin\theta = m\lambda\Rightarrow y_m = R\tan\theta_m\approx R\sin\theta_m = R\frac{m\lambda}{d} \]
- Intensity of interference pattern
\[ \phi = \frac{2\pi d}{\lambda}\sin\theta, \quad I = 4I_0\cos^2\frac{1}{2}\phi \]
Thin-Film Interference
- Half wavelength shift
Ch. 36 Diffraction
Single Slit Diffraction
(of dark fringes)
\[ \sin\theta = \frac{m\lambda}{a} \]
\[ I = I_0\left[\frac{\sin(\beta/2)}{\beta/2}\right]^2, \quad \beta = \frac{2\pi}{\lambda}a\sin\theta \]
Single-Slit diffraction envelope in double slit interference
Diffraction on Multiple Slits
- Principal Maxima
\[ \sin\theta_n = n\lambda/d \]
- Interference Minima
(\(Nd\) is the total width of N-slits)
\[ \sin\theta_s = s\lambda/(Nd) \]
- Subsidiary Maxima
- Single-Slit Diffraction Envelope
Diffraction Gratings
- Dispersion
The angular separation of two lines whose wavelengths differ by a certain amount
\[ D = \frac{\Delta\theta}{\Delta\lambda} = \frac{m}{d\cos\theta} \]
- Resolving Power
\[ \Delta\theta_\text{hw} = \frac{\lambda}{Nd\cos\theta} \Rightarrow R = \frac{\lambda}{\Delta\lambda} = Nm \]
- Rayleigh’s criterion of resolvability
Ch. 37 Special Relativity
- Lorentz's Factor
\[ \gamma = \frac{1}{\sqrt{1 - u^2/c^2}} \]
- Time dilation
\[ t = \frac{1}{\sqrt{1 - (v/c)^2}}t' = \gamma t' \]
- Length contraction
\[ L = L'\sqrt{1 - (v/c)^2} = L'/\gamma \]
- Lorentz Transformation
...
\[ u = \frac{u' + v}{1 + vu'/c^2} \]