User:Mathwizard1232/roughdraft/Physics

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

$$F = k_\mathrm{e} \frac{q_1q_2}{r^2}$$

$$\mathbf{E}= {1 \over 4\pi\varepsilon_0}{q \over r^2}\mathbf{\hat{r}} \ $$

$$ \varepsilon_0 =\frac {1}{\mu_0 c_0^2}$$ = 8.854 187  817...×10−12 A·s/(V·m) = 8.854  187  817...×10−12 F/m

$$ \begin{align} k_{\mathrm{e}} &= \frac{1}{4 \pi \varepsilon_0} = \frac{c^2 \ \mu_0}{4 \pi} = c^2 \cdot 10^{-7} \ \mathrm{H} \cdot \mathrm{m}^{-1}\\ &= 8.987\ 551\ 787\ 368\ 176\ 4 \times 10^9 \ \mathrm{N \cdot m^2 \cdot C^{-2}}. \\ \end{align} $$

Gravity
$$F = G \frac{m_1 m_2}{r^2}$$

G is approximately equal to $6.673 N m^{2} kg^{−2}$

$$ g_0 = G \, m_\mathrm{Earth} / r_\mathrm{Earth}^2 = 9.8331 \ m/s^2$$

Reactence
$$X_C = \frac {1} {\omega C} = \frac {1} {2\pi f C}\quad$$

$$X_L = \omega L = 2\pi f L\quad$$

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

Poynting Vector
$$\mathbf{S} = \frac{1}{\mu_0}\mathbf{E} \times \mathbf{B},$$

Solenoids
$$ B = \mu_0 \mu_r \frac{N I}{h} $$

Ampere's Law
$$\oint_C \mathbf{B} \cdot \mathrm{d}\boldsymbol{\ell} = \mu_0 I_{\mathrm{enc}} $$

µ0 = $4$ N·A&minus;2 ≈ $1.257 H*m-1$ or N·A&minus;2; in the SI system of units, or using tesla·meter per ampere (T·m/A).

Inductance
$$\displaystyle L= \frac{N\Phi}{i} = \frac{u_0 N^2 A}{l}$$

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

$$i(t) = I_P \sin(2 \pi f t)\,$$

$$v(t) = 2 \pi f L I_P \cos(2 \pi f t)\,$$

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

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
$$P_R = I^2 R = I V = \frac{V^2}{R}$$

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

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

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

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