Electric
Fields
Field of charge line segement
Gauss Law
Φ=→E⋅→A=Qenclosedϵ0
Electric Potential Energy
\(U=-Fd=-Eqd\)
For two point charge
\[U=\frac{1}{4\pi\epsilon_0}\frac{q_{1}q_{2}}{r}\]
Electric Potential
\[V=\frac{U}{q}\]
Point Charge
\[\begin{aligned} V&=\frac{1}{4\pi\epsilon_0} \int\frac{dq}{r}\\ &=\frac{1}{4\pi\epsilon_0}\frac{Q}{\sqrt{x^2+a^2}} \end{aligned}\]
Electric Potential Surface
surface that has the same Electric Potential
Gradient
\[\begin{aligned} \vec{\nabla}f &=(\hat{i}\frac{\partial}{\partial x}+\hat{j}\frac{\partial}{\partial y}+\hat{k}\frac{\partial}{\partial z})\\ \vec{E} &=-\vec{\nabla}V\\ \to E_r &=-\frac{\partial V}{\partial r}(radial\ electric\ field) \end{aligned}\]
Potential Gradient
\[\begin{aligned} dV&=\vec{E}*d\vec{l}=E_xdx+E_ydy+E_zdz\\ E_x&=-\frac{\partial V}{\partial x}\\ E_y&=-\frac{\partial V}{\partial y}\\ E_z&=-\frac{\partial V}{\partial z}\\ \end{aligned}\]
Circuit
Capacitor and Capacity
two conductor separated by an insulator (vacuum) from a capacitor
how much charge stored per unit volt
\[\begin{aligned} C&=\frac{Q}{V_{ab}} \to definition\ of\ capacitance\\ &=\epsilon_0\frac{A}{d} (for\ parallel)\\ &=\epsilon_0\frac{4\pi r_Ar_B}{r_B-r_A} (for\ sphere) \end{aligned}\]
\[\begin{aligned} 1F=1farad=\frac{C}{V}=1 coulomb/volt \end{aligned}\]
In Series
\[\begin{aligned} \frac{1}{C_{all}}=\frac{1}{C_1}+\frac{1}{C_2}+\frac{1}{C_3}... \end{aligned}\]
In Parallel
\[\begin{aligned} C_{all}=C_1+C_2+C_3... \end{aligned}\]
Energy Store
\[\begin{aligned} U=\frac{Q^2}{2C}=0.5CV^2=0.5QV\\ \end{aligned}\]
Magnesium
Dielectric
\[\begin{aligned} \kappa=\frac{C}{C_0}=\frac{V_0}{V}=\frac{E_0}{E} \end{aligned}\]
Energy Density
\[\begin{aligned} u=\frac{0.5CV^2}{Ad}=\frac{1}{2}\epsilon_0E^2 \end{aligned}\]
Kirchhoff's Rule
Junction Rule
The algebraic sum of the currents into any junction is zero
\[\begin{aligned} \sum I=0 \end{aligned}\]
loop rule
the algebraic sum of the potential differences in any loop, including those associated with EMFs and those of resistive elements, must equal zero
\[\begin{aligned} \sum V=0 \end{aligned}\]
EMFs and Resistors is in opposite direction
R-C Circuit
\[\begin{aligned} \epsilon-v_{ab}-v_{bc}&=0\\ Q_f&=C\epsilon\\ \to Q_f\ does\ not&\ depend\ on\ R\\ i=\frac{dq}{dt}&=\frac{\epsilon}{R}-\frac{q}{RC}=-\frac{1}{RC}(q-C\epsilon)\\ \frac{dq}{q-C\epsilon}&=\frac{dt}{RC}\\\\ q&=Q_f(1-e^{-t/RC}) \end{aligned}\]
Magnetic Force
Moving Charge
\[F=qv\times B=|q|vBsin(\phi)\]
Direction
Left Hand Rule
always perpendicular to the Magnetic field and the direction of the charge
Unit
\[1N/C*(m/s)=1N/A*m=1tesla=1T=1*10^4 G(gauss)\]
Magnetic Flux
\[\begin{aligned} \int \vec{B}*d\vec{A}=0 \end{aligned}\]
Current-Carrying Conductor
\[\begin{aligned} \vec{F} &=q\vec{v}_d\times \vec{B}\\ &=(nAl)(qv_dB)=(nq v_dA)(lB)\\ \to&=IlB\ \ \ \vec{F}=I*\vec{l}\times\vec{B} \end{aligned}\]
Hall Effect
\[\begin{aligned} qE_z+qv_dB_y=0\ \&&\ J_x=nqv_d\\ \to E=&-v_dB_y\\ nq=\frac{-J_xB_y}{E_z}\ \end{aligned}\]
Between Parallel Conductors
同种电流相互吸引，异种电流相互排斥
Magnetic Field
Straight Current-Carrying Conductor
\[\begin{aligned} B=\frac{\mu_0I}{4\pi}\int^a_{-a}\frac{xdy}{(x^2+y^2)^{3/2}}= \frac{\mu_0I}{2\pi}\frac{a}{x \sqrt{a^2 + x^2}}\\ a==\infty\to B=\frac{\mu_0I}{2\pi x} \end{aligned}\]
Ampere's Law
The magnetic field in space around an electric current is proportional to the electric current which serves as its source
\[\oint Bdl=\mu_0I\]
Magnetic Field of a Circular Current Loop
\[\begin{aligned} B_x &=\frac{\mu_0 N I a^2}{2(x^2+a^2)^{3/2}}\\\\ B_x &=\frac{\mu_0 NI}{2a} (center) \end{aligned}\]
Electro-Magnetic Induction
Faraday's Law
\[\epsilon=-\frac{d\Phi_E}{dt}\]
Lenz's Law
The induced EMF resulting from a changing magnetic flux has a polarity that leads to an induced current whose direction is such that the induced magnetic field opposes he original flux change.
Motional Electromotive Force
Disk
\[ \epsilon=\int\limits^R_0 \omega Brdr=\frac{1}{2}\omega BR^2 \]
Induced Electric Field
\[\begin{aligned} \Phi_B&=BA=\mu_0 nIA\\ \epsilon&=-\frac{d\Phi_B}{dt}=-\mu_0 nA\frac{dI}{dt}\\ \oint{\vec{E}}*d\vec{l}&=\epsilon=-\frac{d\Phi_B}{dt}\\ \end{aligned}\]
Mutual Inductance
\[\begin{aligned} \because&\ \begin{vmatrix} \epsilon_2=-M\frac{di_1}{dt} \\ \epsilon_1=-M\frac{di_2}{dt} \end{vmatrix}\\ M &=\frac{N_2\Phi_{B2}}{i_1}=\frac{N_1\Phi_{B1}}{i_2} \end{aligned}\]
Self-Inductance
\[\begin{aligned} L &=\frac{N\Phi_B}{i}\\ N\frac{d\Phi_B}{dt}&=L\frac{di}{dt}\\ \epsilon&=-Nd\Phi_B/dt\\ &=-L\frac{di}{dt} \end{aligned}\]
R-L Circuit
\[\begin{aligned} \left(\frac{di}{dt}\right)_{initial}&=\frac{\epsilon}{L}-\frac{iR}{L}*0 =\frac{\epsilon}{L} \end{aligned}\]
\[P=\epsilon i=i^2R+Li\frac{di}{dt}\]
\[\begin{aligned} U&=\int I_0 Lidi=0.5LI^2\\ &=U_0e^{-2(R/L)t}\to increase\\ &\frac{1}{2}Li^2=U_0e^{-2Rt/L}\to decrease \end{aligned}\]
\[i=\frac{\epsilon}{R}(1-e^{-Rt/L})\to increase\\i=I_0e^{Rt/L}\to decrease\]
L-C Circuit
\[\begin{aligned} q=Qcos(\omega t+\phi)\\ i=-\omega Qsin(\omega t+\phi)\\ \omega=\sqrt{\frac{1}{LC}} \end{aligned}\]
Displacement Current
\[i_D=\frac{d\Phi_E}{dt}=i_c\]
\[j_D=\epsilon\frac{dE}{dt}\]
complete Ampere's Law
Eddy Current
Current created by the change of partial magnetic flux
Moving Charge
\[B=\frac{\mu_0}{4\pi}\frac{|q|vsin\phi}{r^2}\]
electron
elementary charge
like charge repel, unlike charge attract
Coulomb's Law
\[F_E=\frac{kq_1q_2}{r^2}\]
only negative charge move
\[\vec{E}=\frac{1}{4\pi \epsilon_0}\frac{Q}{x^2}\hat{i}\]
Field of Disk
\[R>> E_x=\frac{\sigma}{2\epsilon_0}\]
close surface's integrate