FormulaFree electron gas in 1d (density of states) $$ D(W) ~=~ \frac{L}{\pi} \, \left(\frac{2m}{\hbar^2}\right)^{1/2} \, \frac{1}{\sqrt{W}} $$
FormulaFree electron gas in 3d (density of states) $$ D(W) ~=~ \frac{V}{2\pi^2} \, \left(\frac{2m}{\hbar^2}\right)^{3/2} \, \sqrt{W} $$
FormulaPhase shift (RLC series circuit) $$ \tan(\varphi) ~=~ \frac{ \omega \, L ~-~ \frac{1}{\omega \, C} }{ R } $$
FormulaResonant RLC series circuit (maximum current, voltage, total impedance) $$ I_0 ~=~ \frac{ U_0 }{ \sqrt{ R^2 ~+~ \left( \omega \, L ~-~ \frac{1}{\omega \, C} \right)^2 } } $$
FormulaImpedance (RLC series circuit) $$ |Z| ~=~ \sqrt{ R^2 ~+~ \left( \omega \, L ~-~ \frac{1}{\omega \, C} \right)^2 } $$
FormulaHorizontal throw (throw distance, initial velocity, initial height) $$ w ~=~ v_0 \, \sqrt{ \frac{2\, y_0}{ g } } $$
FormulaHorizontal throw (throw duration, initial height) $$ t_{\text d} ~=~ \sqrt{ \frac{2\, y_0}{ g } } $$
FormulaLong coil (magnetic field, number of turns, current) $$ \class{violet}{B} ~=~ \mu_0 \, \mu_{\text r} \, \frac{ \class{red}{I} \, N }{ l } $$
FormulaBiot-Savart law for thin conductors (Magnetic field) $$ \boldsymbol{B}(\boldsymbol{r}) ~=~ \frac{\mu_0 \, I}{4\pi} \int_{S} \frac{\boldsymbol{r}-\boldsymbol{R}}{|\boldsymbol{r}-\boldsymbol{R}|^3} \times \text{d}\boldsymbol{s} $$
