User:Mathwizard1232/roughdraft/Physics

MC

Electric Force

$\mathbf {E} ={\frac {\mathbf {F} }{q}}$

$F=k_{\mathrm {e} }{\frac {q_{1}q_{2}}{r^{2}}}$

$\mathbf {E} ={1 \over 4\pi \varepsilon _{0}}{q \over r^{2}}\mathbf {\hat {r}} \$

$\varepsilon _{0}={\frac {1}{\mu _{0}c_{0}^{2}}}$ = 8.854187817...×10⁻¹² A·s/(V·m) = 8.854187817...×10⁻¹² F/m

${\begin{aligned}k_{\mathrm {e} }&={\frac {1}{4\pi \varepsilon _{0}}}={\frac {c^{2}\ \mu _{0}}{4\pi }}=c^{2}\cdot 10^{-7}\ \mathrm {H} \cdot \mathrm {m} ^{-1}\\&=8.987\ 551\ 787\ 368\ 176\ 4\times 10^{9}\ \mathrm {N\cdot m^{2}\cdot C^{-2}} .\\\end{aligned}}$

Gravity

$F=G{\frac {m_{1}m_{2}}{r^{2}}}$

G is approximately equal to 6.673×10⁻¹¹ N m² kg⁻²

$g_{0}=G\,m_{\mathrm {Earth} }/r_{\mathrm {Earth} }^{2}=9.8331\ m/s^{2}$

Reactence

$X_{C}={\frac {1}{\omega C}}={\frac {1}{2\pi fC}}\quad$

$X_{L}=\omega L=2\pi fL\quad$

$Z={\sqrt {R^{2}+X^{2}}}$

Problem 1

Ampere's Law

$\oint _{C}\mathbf {B} \cdot \mathrm {d} {\boldsymbol {\ell }}=\mu _{0}I_{\mathrm {enc} }$

µ₀ = 4π×10⁻⁷ N·A⁻² ≈ 1.2566370614...×10⁻⁶ H*m-1 or N·A⁻²; in the SI system of units, or using tesla·meter per ampere (T·m/A).

Inductance

$\displaystyle L={\frac {N\Phi }{i}}={\frac {u_{0}N^{2}A}{l}}$

$v(t)=L{\frac {di(t)}{dt}}$

$i(t)=I_{P}\sin(2\pi ft)\,$

$v(t)=2\pi fLI_{P}\cos(2\pi ft)\,$

$E_{\mathrm {stored} }={1 \over 2}LI^{2}$

Problem 2 (I swear to god this feels like cheating but I'm going off the professor's slides, not some leaked exam.)

Faraday's Law

$\textstyle {\mathcal {E}}=-{\frac {d\Phi _{B}}{dt}}$

Current

$I={\frac {dQ}{dt}}\,.$

Capacitors

$v(t)={\frac {q(t)}{C}}={\frac {1}{C}}\int _{t_{0}}^{t}i(\tau )\mathrm {d} \tau +v(t_{0})$

$i(t)={\frac {\mathrm {d} q(t)}{\mathrm {d} t}}=C{\frac {\mathrm {d} v(t)}{\mathrm {d} t}}$

Resistors

$V=iR\,$