E-Field - Continuous Charge Distribution (Volume Charge Density) FormulaE-Field (Continuous Charge Distribution in 3d) $$ \boldsymbol{E} (R) ~=~ \frac{1}{4\pi\,\varepsilon_0}\,\int_{V}\frac{\boldsymbol{R}-\boldsymbol{r}}{|\boldsymbol{R}-\boldsymbol{r}|^3}\,\rho(\boldsymbol{r})\,\text{d}v $$
Zylinderkoordinaten FormulaHollow Cylinder (E-Field Outside) $$ \boldsymbol{E}(r_{\perp}) = \frac{\sigma\,R}{\varepsilon_{0}}\,\frac{1}{r_{\perp}} $$
Electric Field Graph (inside / outside) - Solid sphere FormulaSolid Sphere (E-field Inside) $$ E(r) ~=~ \frac{Q}{4\pi \varepsilon_0 \, R^3} \, r $$
Gaussian Box for a Charged Infinite Plane FormulaCharged Infinite Plane (E-Field) $$ E ~=~ \frac{\sigma}{2 \varepsilon_0} $$
Electric Field Graph (inside / outside) - Solid sphere FormulaSolid Sphere (E-Field Outside) $$ E(r) ~=~ \frac{1}{4\pi \varepsilon_0}\frac{Q}{r^2} $$
Coulomb's Electric Force as a Function of the Distance Between the Charges FormulaCoulomb's Law (Electric Force, Charge, Distance) $$ F_{\text e} ~=~ \frac{1}{4\pi \varepsilon_0 \, \varepsilon_{\text r}} \, \frac{q_1 \, q_2}{r^2} $$
Electron movement at an angle to the magnetic field FormulaLorentz Force (E-Field, B-Field) $$ \boldsymbol{F} ~=~ q \, \left( \boldsymbol{E} ~+~ \boldsymbol{v} ~\times~ \class{violet}{\boldsymbol{B}} \right) $$
Suspended State of an Oil Droplet in the Oil Drop Experiment FormulaOil Drop Experiment (Charge, Velocity) $$ q ~=~ \frac{9\pi \, d}{U} \, \sqrt{ \frac{2 \, \eta^3 \, v_{\downarrow}^3}{g \left( \rho_{\text o} ~-~ \rho_{\text L} \right) } } $$
RL circuit - Switch-on process FormulaRL Circuit - Switching on (Current) $$ I(t) ~=~ \left( 1 - \mathrm{e}^{-\frac{R}{L}\,t} \right) \, \frac{U_0}{R} $$
The choice of the enclosing loop for the Ampere law is not unique Derivation Level 4 (for physics pros)Displacement Current for the 4th Maxwell Equation Derivation of the displacement current as a correction to the fourth Maxwell equation. For this purpose we use a plate capacitor.
Two coils connected in parallel Derivation Level 3 (with higher mathematics)Total (Equivalent) Inductance of a Series and Parallel Circuit of Coils Derivation of the total inductance (equivalent inductance) of a parallel circuit and series circuit of two coils using the Faraday's law of induction.
Hall plate with holes Derivation Level 3 (with higher mathematics)Hall Voltage due to the Hall Effect Derivation of the Hall voltage (via Hall effect), which depends only on quantities that we can easily determine in an experiment.