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Formulas

Would you like to solve a physical or mathematical problem, but you just do not get into the quantitative context? And that, even though you have dealt with the respective topic in the universal book of knowledge and have looked at a learning video?

That's why the formula wand of the universal thinker comes to your aid! Just as I disarm the opponent with "Expelliarmus", so do you apply an appropriate spell formula to your problem to defeat it.

Formula Cylindrical Capacitor

\[ C ~=~ 2\pi \, \varepsilon_0 \, \varepsilon_{\text r} \, l \, \frac{ 1 }{\ln{ \left( \frac{R_2}{R_1} \right) }} \]

Formula Energy band gap of germanium (temperature dependence)

\[ E_{\text g} ~=~ 0.742\,\text{eV} - 3.90 \cdot 10^{-4} \, \frac{ \text{eV} }{ \text{K} } \cdot T \]

Formula Spherical Capacitor

\[ C ~=~ \frac{4\pi \, \varepsilon_{\text r} \, \varepsilon_0}{\frac{1}{R_1} ~-~ \frac{1}{R_2}} \]

Formula Energy band gap of GaAs (temperature dependence)

\[ E_{\text g} ~=~ 1.522\,\text{eV} - \frac{ 5.8 \cdot 10^{-4} \, \frac{ \text{eV} }{ \text{K} } \, T^2 }{ 300 \,\text{K} + T } \]

Formula Coriolis force (+ direction)

\[ \boldsymbol{F}_{\text c} ~=~ 2m \, \left( \boldsymbol{v} ~\times~ \boldsymbol{\omega} \right) \]

Formula Plate Capacitor

\[ C ~=~ \varepsilon_0 \, \varepsilon_{\text r} \, \frac{A}{d} \]

Formula Free fall / Inclined plane

\[ v ~=~ \sqrt{2g \, (h_0 ~-~ h)} \]

Formula Boltzmann distribution function

\[ P(W) ~=~ \mathrm{e}^{ -\frac{ W - \mu }{ k_{\text B} \, T}} \]

Formula Infinite 1d square well

\[ \psi_n(x) ~=~ \begin{cases} \sqrt{\frac{2}{L}}\sin\left(\frac{n\pi}{L}\,x\right)~, & 0\leq x \leq L \\\\ 0~, & x\lt0,x\gt L \end{cases} \]

Formula Infinite square potential well (1d)

\[ W_{n} ~=~ \frac{h^2}{8m \, L^2} \, n^2 \]

Formula Coriolis force - magnitude (velocity and angular velocity orthogonal)

\[ F_{\text c} ~=~ 2m \, v \, \omega \]

Formula Fermi distribution function

\[ P(W) ~=~ \frac{1}{\mathrm{e}^{ \frac{ W - \mu }{ k_{\text B} \, T}} ~+~ 1} \]

Formula Bose distribution function

\[ P(W) ~=~ \frac{1}{\mathrm{e}^{ \frac{ W - \mu }{ k_{\text B} \, T}} ~-~ 1} \]

Formula Charging capacitor

\[ I(t) ~=~ I_0 \, \mathrm{e}^{-\frac{t}{R\,C}} \]

Formula Charging capacitor

\[ U(t) ~=~ U_0 \, \left(1 - \mathrm{e}^{-\frac{t}{R\,C}}\right) \]

Formula Discharging capacitor

\[ U(t) ~=~ U_0 \, \mathrm{e}^{-\frac{t}{R\,C} } \]

Formula Archimedes principle

\[ F_{\text A} ~=~ \rho \, V \, g \]

Formula Thin circular ring

\[ I ~=~ m \, r^2 \]

Formula Coriolis acceleration

\[ a_{\text c} ~=~ 2 \, v \, \omega \, \sin(\varphi) \]

Formula Energy band of silicon (temperature dependence)

\[ E_{\text g} ~=~ 1.165\,\text{eV} - 2.84 \cdot 10^{-4} \, \frac{ \text{eV} }{ \text{K} } \cdot T \]