Imaginary Sphere Surface around a Point Charge Derivation Level 4 (for physics pros)Coulomb's Law using 1st Maxwell Equation Here you will learn how to derive Coulomb's law for two point charges from divergence theorem and Maxwell's first equation.
4. Maxwell-Gleichung: Stom erzeugt Magnetfeld Formula4. Maxwell Equation of Electrostatics (integral form) $$ \oint_{S} \class{violet}{\boldsymbol{B}} ~\cdot~ \text{d}\boldsymbol{s} ~=~ \mu_0 \, \class{red}{I} $$
Opposite current carrying wires FormulaInductance of Two Current-Carrying Wires $$ L ~=~ \frac{\mu_0 \, l}{\pi} \, \left[ \frac{1}{2} + \ln\left(\frac{d-R}{R}\right) \right] $$
Voltage divider - Circuit FormulaVoltage Divider (Output and Input Voltage, Resistors) $$ U_{\text{out}} = \frac{R_2}{R_1 + R_2} \, U_{\text{in}} $$
Helmholtz Coil with Dimensions Derivation Level 4 (for physics pros)Magnetic Field of a Helmholtz Coil Derivation of the homogeneous magnetic field in the center (on the symmetry axis) of the Helmholtz coil with radius R and distance d.
Helmholtz Coil with Dimensions FormulaHelmholtz Coil (B-Field, Same Current Direction) $$ \class{violet}{B}(z) = \frac{\mu_0 \, I \, R^2 \, N}{2} \, \left[ \left( (z-d/2)^2 + R^2 \right)^{-3/2} + \left( (z+d/2)^2 + R^2 \right)^{-3/2} \right] $$
E-Field - Continuous Charge Distribution (Linear Charge Density) FormulaE-Field (Continuous Charge Distribution in 1d) $$ \boldsymbol{E} ~=~ \frac{1}{4\pi\,\varepsilon_0}\,\int_{L}\frac{\boldsymbol{R}-\boldsymbol{r}}{|\boldsymbol{R}-\boldsymbol{r}|^3}\,\lambda(\boldsymbol{r})\,\text{d}l $$
E-Field - Continuous Charge Distribution (Surface Charge Density) FormulaE-Field (Continuous Charge Distribution in 2d) $$ \boldsymbol{E} ~=~ \frac{1}{4\pi\,\varepsilon_0}\,\int_{A}\frac{\boldsymbol{R}-\boldsymbol{r}}{|\boldsymbol{R}-\boldsymbol{r}|^3}\,\sigma(\boldsymbol{r})\,\text{d}a $$