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

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

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

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

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

Torque on Dipole

\[ \vec{\tau} = Fd\sin\theta = \vec{p}\times\vec{E}. \]

\[ U = -W = -\int\tau\mathop{d\theta} = \int pE\sin\theta\mathop{d\theta} = -pE\cos\theta = -\vec{p}\cdot\vec{E} \]


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
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
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
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
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
Magnetic Dipoles

\[ \bm{\mu} = NiA \]

\[ \bm{\tau=\mu\times B} \]


Ch. 29 Magnetic Fields Due to Currents

  • Biot-Savart Law

\[ \mathop{d\bm{B}} = \frac{\mu_0}{4\pi}\frac{i\mathop{d\bm{l}}\times\bm{e}_r}{r^2} \]

Biot-Savart Law
  • 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

\[ BL = \mu_0 nLI \]

Solenoid
  • B-Field of toroid
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 *

\[ \left\{ \begin{aligned} & \nabla^2\mathbf{B} - \mu_0\epsilon_0\frac{\partial^2\mathbf{B}}{\partial t^2} = 0 \\ & \nabla^2\mathbf{E} - \mu_0\epsilon_0\frac{\partial^2\mathbf{E}}{\partial t^2} = 0 \end{aligned} \right.\Rightarrow \left\{ \begin{aligned} & \mathbf{E}(\mathbf{r},t) = \mathbf{E}_0 \cos(\omega t - \mathbf{k}\cdot\mathbf{r} + \phi_0) \\ & \mathbf{B}(\mathbf{r},t) = \mathbf{B}_0 \cos(\omega t - \mathbf{k}\cdot\mathbf{r} + \phi_0) \end{aligned} \right. \]

\[ k = |\mathbf{k}| = \frac{\omega}{c} = \frac{2\pi}{\lambda} \]

\[ \left|\frac{\mathbf{E}_0}{\mathbf{B}_0}\right| = c = \frac{1}{\sqrt{\mu_0\epsilon_0}} \]

EM Waves

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

\[ I - \frac{1}{2}I_0 \]

Polarizer

\[ I_2 = I_1\cos^2\phi \]

Reflection & Refraction

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
Brewster Angle

\[ \theta_B = \tan^{-1}\frac{n_2}{n_1} \]


Ch. 35 Interference *

Young's Interference

image-20200820205311296
image-20200820205339402

\[ d\sin\theta = m\lambda\Rightarrow y_m = R\tan\theta_m\approx R\sin\theta_m = R\frac{m\lambda}{d} \]

  • 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

Fraunhofer Single Slit

(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

...

\[ u = \frac{u' + v}{1 + vu'/c^2} \]