## E-Field (Continuous Charge Distribution in 3d)

` $$ \boldsymbol{E} (R) ~=~ \frac{1}{4\pi\,\varepsilon_0}\,\int_{V}\frac{\boldsymbol{R}-\boldsymbol{r}}{|\boldsymbol{R}-\boldsymbol{r}|^3}\,\rho(\boldsymbol{r})\,\text{d}v $$ `

Learn about the electromagnetic properties of our universe, for example about electric charges, currents, electromagnetic fields and so on.

` $$ \boldsymbol{E}(r_{\perp}) = \frac{\sigma\,R}{\varepsilon_{0}}\,\frac{1}{r_{\perp}} $$ `

` $$ E(r) ~=~ \frac{Q}{4\pi \varepsilon_0 \, R^3} \, r $$ `

` $$ E ~=~ \frac{\sigma}{2 \varepsilon_0} $$ `

` $$ E(r) ~=~ \frac{1}{4\pi \varepsilon_0}\frac{Q}{r^2} $$ `

` $$ F_{\text e} ~=~ \frac{1}{4\pi \varepsilon_0 \, \varepsilon_{\text r}} \, \frac{q_1 \, q_2}{r^2} $$ `

` $$ \boldsymbol{F} ~=~ q \, \left( \boldsymbol{E} ~+~ \boldsymbol{v} ~\times~ \class{violet}{\boldsymbol{B}} \right) $$ `

` $$ q ~=~ \frac{9\pi \, d}{U} \, \sqrt{ \frac{2 \, \eta^3 \, v_{\downarrow}^3}{g \left( \rho_{\text o} ~-~ \rho_{\text L} \right) } } $$ `

` $$ I(t) ~=~ \left( 1 - \mathrm{e}^{-\frac{R}{L}\,t} \right) \, \frac{U_0}{R} $$ `

Derivation of the displacement current as a correction to the fourth Maxwell equation. For this purpose we use a plate capacitor.

Derivation of the total inductance (equivalent inductance) of a parallel circuit and series circuit of two coils using the Faraday's law of induction.

Derivation of the Hall voltage (via Hall effect), which depends only on quantities that we can easily determine in an experiment.