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

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

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

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

Poynting Vector

$\mathbf {S} ={\frac {1}{\mu _{0}}}\mathbf {E} \times \mathbf {B} ,$

Solenoids

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

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

$V=iR\,$

Power

$P_{R}=I^{2}R=IV={\frac {V^{2}}{R}}$

Problem 3

Bohr Model

$\Delta {E}=E_{2}-E_{1}=h\nu \ ,$

h = 1.054571628*10^-34 J·s

$L=n{h \over 2\pi }=n\hbar$

$n\lambda =2\pi r\,$

${m_{e}v^{2} \over r}={Zk_{e}e^{2} \over r^{2}}$

$E={1 \over 2}m_{e}v^{2}-{k_{e}e^{2} \over r}=-{k_{e}e^{2} \over 2r}$

$r_{n}={n^{2}\hbar ^{2} \over Zk_{e}e^{2}m_{e}}$

$r_{1}={\hbar ^{2} \over (k_{e}e^{2})m_{e}}=0.529\times 10^{-10}\mathrm {m}$

$E=-{k_{e}e^{2} \over 2r_{n}}=-{(k_{e}e^{2})^{2}m_{e} \over 2\hbar ^{2}n^{2}}={-13.6\mathrm {eV} \over n^{2}}$

Electrons

9.10938215×10^−31 kg=0.510 998 910(13) MeV = 8.187 104 38× 10^−14 J = 5.4857990943(23)×10−4 u

−1.602176487(40)×10−19 C

Problem 4

Charged Particle in Magnetic Field

$\mathbf {F} =q(\mathbf {v} \times \mathbf {B} ),$

$\mathbf {F_{c}} =m\mathbf {v} ^{2}/r\,$

$\mathbf {R} ={\frac {m\mathbf {v} }{q\mathbf {B} }}$

Cyclotron

$v={\sqrt {\frac {2Vq}{m}}}$

${\frac {v}{r}}={\frac {Bq}{m}}$

$f={\frac {Bq}{2\pi m}}$