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)

$$e=1.60\times {10}^{-19}\mathrm{C}$$

• Quantized Charge
• Millikin Oil-Drop Experiment

Coulomb’s Law

$$F=k\frac{|q_1{q}_2|}{r^2}$$

$$k=\frac{1}{4\pi {ϵ}_0}=8.99\times {10}^9\mathrm{N}\cdot {\mathrm{m}}^2/{\mathrm{C}}^2$$

$${ϵ}_0=8.85\times {10}^{-12}{\mathrm{C}}^2/\mathrm{N}\cdot {\mathrm{m}}^2$$

• k is the electromagnetic constant
• ϵ₀ is the permittivity constant
• Principle of Superposition

Ch. 22 Electric Field

Definition

F⃗ = qE⃗.

$$\overrightarrow{E}=\frac{\overrightarrow{F}}{q_0}=k\frac{|q|}{r^2}=\frac{1}{4\pi {ϵ}_0}\frac{|q|}{r^2}$$

Cases

The key to integrate, is find the relation between dq  and charge density (as well as length or area)

Uniformly Charged Line

Uniformly Charged Line

$$E_x=\frac{1}{4\pi {ϵ}_0}\frac{Q}{2a}{\int }_{-a}^{+a}\frac{xdy}{(x^2+y^2{)}^{3/2}}=\frac{Q}{4\pi {ϵ}_0}\frac{1}{x\sqrt{x^2+a^2}}$$

If the line is very long, a ≫ x

$$E=\frac{\lambda }{2\pi {ϵ}_{0}r}$$

Uniformly Charged Ring

Uniformly Charged Ring

$$\begin{array}{rl}E& =\int dE\cos \theta \\ & =\frac{z}{\sqrt{z^2+R^2}}\cdot \frac{\lambda }{4\pi {ϵ}_0(z^2+R^2)}{\int }_0^{2\pi R}ds\\ & =\frac{qz}{4\pi {ϵ}_0(z^2+R^2{)}^{3/2}}.\end{array}$$

If the charged ring is at large distance, z ≫ R

$$E=\frac{1}{4\pi {ϵ}_0}\frac{1}{z^2}.$$

Uniformly Charged Disk

Uniformly Charged Disk

dq  = σdA  = σ(2πrdr).

$$dE=\frac{\sigma z}{4{ϵ}_0}\frac{2rdr}{(z^2+r^2{)}^{3/2}}.$$

$$E=\int dE=\frac{\sigma }{2{ϵ}_0}(1-\frac{z}{\sqrt{z^2+R^2}}).$$

For R → ∞ (infinite sheet)

$$E=\frac{\sigma }{2{ϵ}_0}.$$

Electric Field Line

Electric Dipole *

Electric Field Due to Electric Dipole

Electric Dipole

$$\begin{array}{rl}E& =E_{(+)}-E_{(-)}\\ & =\frac{1}{4\pi {ϵ}_0}\frac{q}{r_{(+)}^2}-\frac{1}{4\pi {ϵ}_0}\frac{q}{r_{(-)}^2}\\ & =\cdots \\ & =\frac{q}{2\pi {ϵ}_0{z}^3}\frac{d}{{(1-{\left(\frac{d}{2z}\right)}^2)}^2}\end{array}.$$

If z ≫ d

$$E=\frac{1}{2\pi {ϵ}_0}\frac{qd}{z^3}$$

• qd is the electric dipole moment p⃗

On the axis of the dipole

$${\overrightarrow{E}}_{\text{dipole}}\approx \frac{1}{4\pi {ϵ}_0}\frac{\overrightarrow{p}}{r^3}$$

On the perpendicular plane

$${\overrightarrow{E}}_{\text{dipole}}\approx -\frac{1}{4\pi {ϵ}_0}\frac{\overrightarrow{p}}{r^3}$$

Dipole in Electric Field

Torque on Dipole

τ⃗ = Fdsin θ = p⃗ × E⃗.

U =  − W =  − ∫τdθ  = ∫pEsin θdθ  =  − pEcos θ =  − p⃗ ⋅ E⃗

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Ch. 23 Gauss’s Law

ϵ₀Φ = ϵ₀∮E⃗ ⋅ d  = q_enclosed

$${\oint }_A\overrightarrow{E}\cdot d\overrightarrow{A}={\int }_V\mathrm{\nabla }\cdot \overrightarrow{E}dV=\frac{q_{\text{enc}}}{{ϵ}_0}.$$

• 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

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Ch. 24 Electric Potential

ΔU = U_f − U_i =  − W_{by the field}

V = V_f − V_i =  − ∫_i^fE⃗ ⋅ d .

i is at ∞, where U is 0, V_i is 0.

• Electric potential for a point charge

$$V=\frac{1}{4\pi {ϵ}_0}\frac{q}{r}$$

• Potential due to electric dipole

Potential Due to Electric Dipole

$$V=V_{(+)}+V_{(-)}=\cdots =\frac{1}{4\pi {ϵ}_0}\frac{p\cos \theta }{r^2}$$

• Field from Potential

$$E_s=-\frac{\partial V}{\partial s}.$$

E⃗ =  − ∇V

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Ch. 25 Capacitance

q = CV

• Parallel capacitor

$$C=\frac{q}{V}=\frac{{ϵ}_{0}EA}{Ed}=\frac{{ϵ}_{0}A}{d}.$$

• Cylindrical capacitor

Cylindrical Capacitor

q = ϵ₀EA = ϵ₀E(2πrL)

$$V={\int }_{-}^{+}Eds=-\frac{q}{2\pi {ϵ}_{0}L}{\int }_b^a\frac{dr}{r}=\frac{q}{2\pi {ϵ}_{0}L}\ln \left(\frac{b}{a}\right)$$

$$C=\frac{q}{V}=2\pi {ϵ}_0\frac{L}{\ln (b/a)}$$

• Spherical capacitor

$$\cdots C=4\pi {ϵ}_0\frac{ab}{b-a}.$$

• Series & Parallel

• Energy stored

$$dW=vdq=\frac{qdq}{C}$$

$$W={\int }_0^{W}dW=\frac{1}{C}{\int }_0^{Q}qdq=\frac{Q^2}{2C}\Rightarrow U=\frac{1}{2}C{V}^2=\frac{1}{2}QV$$

• Energy density

$$u=\frac{\frac{1}{2}C{V}^2}{Ad}=\frac{1}{2}{ϵ}_0{E}^2$$

• Dielectric

Dielectric

ϵ = κϵ₀

$$E=\frac{V}{d}=\frac{q}{Cd}=\frac{q}{d{ϵ}_{0}A/d}=\frac{\sigma }{{ϵ}_0}$$

$$E=\frac{\sigma -{\sigma }_i}{{ϵ}_0}=\frac{E_0}{K}\Rightarrow {\sigma }_i=\sigma (1-\frac{1}{K})\text{(induced surface charge density)}$$

ϵ₀∮κE⃗ ⋅ d  = q

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Ch. 26 Current and Resistance

Current

$$i=\frac{dq}{dt}$$

Current is NOT a vector

• Current density

Current Density

$$i=\frac{q}{t}=\frac{nALe}{L/v_d}=nAe{v}_d.$$

J⃗ = (ne)v⃗_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

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Ch. 27 Circuits

• Kirchhoff’s Rules

∑I = 0,  ∑V = 0

• Mark for RC Circuit

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Ch. 28 Magnetic Field

• On Moving Charge

F⃗ = qv⃗ × B⃗

• Hall Effect

$$n=\frac{Bi}{Vle}$$

• On a Current Carrying Conductor

F⃗_B = il⃗ × B⃗

• Magnetic Dipoles

Magnetic Dipoles

μ = NiA

τ = μ × B

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Ch. 29 Magnetic Fields Due to Currents

• Biot-Savart Law

$$d\text{bm}B=\frac{{\mu }_0}{4\pi }\frac{id\text{bm}l\times \text{bm}{e}_r}{r^2}$$

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

$$B=\int dB={\int }_0^{\varphi }\frac{{\mu }_0}{4\pi }\frac{iRd\varphi }{R^2}=\frac{{\mu }_{0}i}{4\pi R}{\int }_0^{\varphi }d\varphi =\frac{{\mu }_{0}i\varphi }{4\pi R}$$

• Gauss’ Law for magnetism

∮B⃗ ⋅ d  = 0

• Ampere’s Law

∮B⃗ ⋅ d  = μ₀I_encl

∮_lB ⋅ d𝐥  = μ₀∫_AJ ⋅ d𝐀 

• B-Field of solenoid

BL = μ₀nLI

Solenoid

• B-Field of toroid

Toroid

$$B=\frac{{\mu }_{0}NI}{2\pi r}$$

• Current sheet

$$B=\frac{1}{2}{\mu }_0{J}_s$$

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Ch. 30 Induction and Inductance

Laws of Induction

• Lenz’s Law
• Faraday’s Law of Induction

$$\mathcal{E}=-\frac{d{\mathrm{\Phi }}_B}{dt}$$

$$-\frac{d}{dt}\underset{A}{\int }\overrightarrow{B}\cdot d\overrightarrow{A}=\underset{l}{\oint }\overrightarrow{E}\cdot d\overrightarrow{l}$$

$$\mathrm{\nabla }\times \mathbf{E}=-\frac{d\mathbf{B}}{dt}$$

• Eddy Current

Inductors

• Self induction

$${\mathcal{E}}_L=-L\frac{di}{dt}$$

$$L=\frac{N{\mathrm{\Phi }}_B}{i}$$

• Inductance of an ideal solenoid

L = μ₀n²lA

• Inductance of a toroid

$$L=\frac{{\mu }_0{N}^{2}b}{2\pi }\ln \frac{r_2}{r_1}$$

RL Circuit

$$\mathcal{E}-i(t)R-L\frac{di}{dt}=0\Rightarrow \frac{di}{dt}+\frac{iR}{L}=\frac{\mathcal{E}}{L}$$

$$i(t)=\frac{\mathcal{E}}{R}(1-e^{-t/{\tau }_L}),{\tau }_L=L/R$$

• Energy stored in magnetic field

$$i\mathcal{E}=Li\frac{di}{dt}+i^{2}R\Rightarrow U_B={\int }_0^{t}Lidi=\frac{1}{2}L{i}^2$$

• Energy density of magnetic field

$$u_B=\frac{L{i}^2}{2Al}=\frac{U_B}{Al}=\frac{{\mu }_0{n}^{2}A{i}^2}{2A}=\frac{{\mu }_0{n}^2{i}^2}{2}=\frac{B^2}{2{\mu }_0}$$

Mutual Induction

$${\mathcal{E}}_2=-M\frac{d{i}_1}{dt},{\mathcal{E}}_1=-M\frac{d{i}_2}{dt}$$

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Ch. 31 EM Oscillation & AC Current *

$$U_C=\frac{q^2}{2C},U_B=\frac{L{i}^2}{2},$$

$$-L\frac{di}{dt}-\frac{q}{C}=0\Rightarrow \frac{d^{2}q}{d{t}^2}+\frac{q}{LC}=0$$

$$\Rightarrow -A{\omega }^2\cos (\omega t+\varphi )+\frac{A}{LC}\cos (\omega t+\varphi )=0,\omega =\frac{1}{\sqrt{LC}}$$

Damped Oscillation in LRC Circuit

Forced Oscillations

• capacitive reactance

$$X_C=\frac{1}{{\omega }_{d}C}$$

Voltage leads the current

• inductive reactance

X_L = ω_dL

Voltage lags the current

• Impedance

$$Z=\sqrt{R^2+(X_L-X_C{)}^2},I=\frac{\mathcal{E}}{Za}$$

Power in AC Circuits

$$P_{\text{avg}}={\mathcal{E}}_{\text{rms}}{I}_{\text{rms}}\cos \varphi ,\cos \varphi =\frac{V_R}{{\mathcal{E}}_m}=\frac{R}{Z}$$

Transformers

$$V_1{I}_1=V_2{I}_2,\frac{V_2}{V_1}=\frac{N_2}{N_1}$$

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Ch. 32 Maxwell’s Equations; Magnetism of Matter *

• Integral Form

$$\{\begin{array}{rlr}& {\oint }_{\mathbf{A}=\partial V}\mathbf{E}\cdot d\mathbf{A}=\frac{q_{\text{enc}}}{{ϵ}_0}& \text{Gauss' Law}\\ & {\oint }_{\mathbf{A}=\partial V}\mathbf{B}\cdot d\mathbf{A}=0& \text{Gauss' Law for B-Field}\\ & {\oint }_{\mathbf{l}=\partial \mathbf{A}}\mathbf{E}\cdot d\mathbf{l}=-\frac{d}{dt}{\int }_{\mathbf{A}}\mathbf{B}\cdot d\mathbf{A}& \text{Faraday's Law}\\ & {\oint }_{\mathbf{l}=\partial \mathbf{A}}\mathbf{B}\cdot d\mathbf{l}={\mu }_0({\int }_{\mathbf{A}}\mathbf{J}\cdot d\mathbf{A}+{ϵ}_0\frac{d}{dt}{\int }_{\mathbf{A}}\mathbf{E}\cdot d\mathbf{A})& \text{Ampere' Law}\end{array}$$

• Maxwell’s Law of Induction

$${\oint }_{\mathbf{l}}\mathbf{B}\cdot d\mathbf{l}={\mu }_0{ϵ}_0\frac{d{\mathrm{\Phi }}_E}{dt}={\mu }_0{ϵ}_0\frac{d}{dt}{\int }_{\mathbf{A}}\mathbf{E}\cdot d\mathbf{A}$$

• displacement current

$$i_d={ϵ}_0\frac{d{\mathrm{\Phi }}_E}{dt}$$

• Differential Form

$$\{\begin{array}{rl}& \mathrm{\nabla }\cdot \mathbf{E}=\frac{\rho }{{ϵ}_0}\\ & \mathrm{\nabla }\cdot \mathbf{B}=0\\ & \mathrm{\nabla }\cdot \mathbf{E}=-\frac{\partial \mathbf{B}}{\partial t}\\ & \mathrm{\nabla }\cdot \mathbf{B}={\mu }_0(\mathbf{J}+{ϵ}_0\frac{\partial \mathbf{E}}{\partial t})\end{array}$$

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Ch. 33 Electromagnetic Waves *

$$\{\begin{array}{rl}& {\mathrm{\nabla }}^2\mathbf{B}-{\mu }_0{ϵ}_0\frac{{\partial }^2\mathbf{B}}{\partial t^2}=0\\ & {\mathrm{\nabla }}^2\mathbf{E}-{\mu }_0{ϵ}_0\frac{{\partial }^2\mathbf{E}}{\partial t^2}=0\end{array}\Rightarrow \{\begin{array}{rl}& \mathbf{E}(\mathbf{r},t)={\mathbf{E}}_0\cos (\omega t-\mathbf{k}\cdot \mathbf{r}+{\varphi }_0)\\ & \mathbf{B}(\mathbf{r},t)={\mathbf{B}}_0\cos (\omega t-\mathbf{k}\cdot \mathbf{r}+{\varphi }_0)\end{array}$$

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

$$\left|\frac{{\mathbf{E}}_0}{{\mathbf{B}}_0}\right|=c=\frac{1}{\sqrt{{\mu }_0{ϵ}_0}}$$

EM Waves

Energy Transfer in EM Waves

$$u_{\text{total}}={ϵ}_0{E}^2={ϵ}_{0}EBc=\frac{EB}{{\mu }_0}$$

• Poynting Vector

$$\overrightarrow{S}=\frac{1}{{\mu }_0}\overrightarrow{E}\times \overrightarrow{B}$$

• Intensity

$$I=S_{\text{avg}}=\frac{E_0{B}_0}{2{\mu }_0}=\frac{E_0^2}{2{\mu }_{0}c}=\frac{E_{\text{rms}}^2}{{\mu }_{0}c}$$

$$\frac{\text{average power}}{4\pi r^2}\propto \frac{1}{r^2}$$

Polarization

When a unpolarized light passes through a polarizer, half of the intensity is loss

$$I-\frac{1}{2}{I}_0$$

Polarizer

I₂ = I₁cos²ϕ

Reflection & Refraction

Reflection & Refraction

• Snell’s Law

$$\frac{\sin {\theta }_1}{\sin {\theta }_2}=\frac{n_{\text{water}}}{n_{\text{air}}},n=c/v=\frac{{\lambda }_{\text{vacuum}}}{{\lambda }_{\text{medium}}}$$

• 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

$${\theta }_B={\tan }^{-1}\frac{n_2}{n_1}$$

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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

$$\varphi =\frac{2\pi d}{\lambda }\sin \theta ,I=4I_0{\cos }^2\frac{1}{2}\varphi$$

Thin-Film Interference

• Half wavelength shift

Ch. 36 Diffraction

Single Slit Diffraction

Fraunhofer Single Slit

(of dark fringes)

$$\sin \theta =\frac{m\lambda }{a}$$

$$I=I_0{\left[\frac{\sin (\beta /2)}{\beta /2}\right]}^2,\beta =\frac{2\pi }{\lambda }a\sin \theta$$

Single-Slit diffraction envelope in double slit interference

Diffraction on Multiple Slits

• Principal Maxima

sin θ_n = nλ/d

• Interference Minima

(Nd is the total width of N-slits)

sin θ_s = sλ/(Nd)

• Subsidiary Maxima
• Single-Slit Diffraction Envelope

Diffraction Gratings

• Dispersion

The angular separation of two lines whose wavelengths differ by a certain amount

$$D=\frac{\mathrm{\Delta }\theta }{\mathrm{\Delta }\lambda }=\frac{m}{d\cos \theta }$$

• Resolving Power

$$\mathrm{\Delta }{\theta }_{\text{hw}}=\frac{\lambda }{Nd\cos \theta }\Rightarrow R=\frac{\lambda }{\mathrm{\Delta }\lambda }=Nm$$

• Rayleigh’s criterion of resolvability

Ch. 37 Special Relativity

• Lorentz’s Factor

$$\gamma =\frac{1}{\sqrt{1-u^2/c^2}}$$

• Time dilation

$$t=\frac{1}{\sqrt{1-(v/c{)}^2}}{t}^{\prime }=\gamma t^{\prime }$$

• Length contraction

$$L=L^{\prime }\sqrt{1-(v/c{)}^2}=L^{\prime }/\gamma$$

• Lorentz Transformation

…

$$u=\frac{u^{\prime }+v}{1+v{u}^{\prime }/c^2}$$