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Physic C E&M (Electric (electron (Coulomb's Law (\[F_E=\frac{kq_1q…
Physic C E&M
Electric
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Electric Potential
\[V=\frac{U}{q}\]
Point Charge
\[\begin{aligned}
V&=\frac{1}{4\pi\epsilon_0} \int\frac{dq}{r}\\
&=\frac{1}{4\pi\epsilon_0}\frac{Q}{\sqrt{x^2+a^2}}
\end{aligned}\]
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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
\[\begin{aligned}
dV&=\vec{E}*d\vec{l}=E_xdx+E_ydy+E_zdz\\
E_x&=-\frac{\partial V}{\partial x}\\
E_y&=-\frac{\partial V}{\partial y}\\
E_z&=-\frac{\partial V}{\partial z}\\
\end{aligned}\]
electron
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like charge repel, unlike charge attract
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Magnesium
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Hall Effect
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\[\begin{aligned}
qE_z+qv_dB_y=0\ \&&\ J_x=nqv_d\\
\to E=&-v_dB_y\\
nq=\frac{-J_xB_y}{E_z}\
\end{aligned}\]
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Ampere's Law
The magnetic field in space around an electric current is proportional to the electric current which serves as its source
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Circuit
Capacitor and Capacity
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how much charge stored per unit volt
\[\begin{aligned}
C&=\frac{Q}{V_{ab}} \to definition\ of\ capacitance\\
&=\epsilon_0\frac{A}{d} (for\ parallel)\\
&=\epsilon_0\frac{4\pi r_Ar_B}{r_B-r_A} (for\ sphere)
\end{aligned}\]
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Kirchhoff's Rule
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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
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R-C Circuit
\[\begin{aligned}
\epsilon-v_{ab}-v_{bc}&=0\\
Q_f&=C\epsilon\\
\to Q_f\ does\ not&\ depend\ on\ R\\
i=\frac{dq}{dt}&=\frac{\epsilon}{R}-\frac{q}{RC}=-\frac{1}{RC}(q-C\epsilon)\\
\frac{dq}{q-C\epsilon}&=\frac{dt}{RC}\\\\
q&=Q_f(1-e^{-t/RC})
\end{aligned}\]
R-L Circuit
\[\begin{aligned}
\left(\frac{di}{dt}\right)_{initial}&=\frac{\epsilon}{L}-\frac{iR}{L}*0 =\frac{\epsilon}{L}
\end{aligned}\]
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\[\begin{aligned}
U&=\int I_0 Lidi=0.5LI^2\\
&=U_0e^{-2(R/L)t}\to increase\\
&\frac{1}{2}Li^2=U_0e^{-2Rt/L}\to decrease
\end{aligned}\]
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L-C Circuit
\[\begin{aligned}
q=Qcos(\omega t+\phi)\\
i=-\omega Qsin(\omega t+\phi)\\
\omega=\sqrt{\frac{1}{LC}}
\end{aligned}\]
