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Electromagnetism formulas


Remember that: \( \vec{a} = \mathbf{a} \), \(\dot{a}= \frac{da}{dt}\) and \(\Delta a = a_{final} - a_{initial} = a_{f} - a_{i} = a_{2} - a_{1} \). The \(\propto\) symbol is read as "is proportional to". While \(\hat{a} = \frac{\vec{a}}{\Vert \vec{a} \Vert}\) is the unit vector.

Symbols

name symbol
electric field $$E$$
fundamental electric charge unit $$e = 1.602177 \times 10^{-19}\ \mathrm{C}$$
force $$F$$
Coulomb constant $$k = 8.99 \times 10^{9} \mathrm{N \cdot m^{2}/C^{2}}$$
length $$L$$
length $$l$$
electric dipole moment $$p$$
electric charge $$q$$
small point electric charge $$q_0$$
distance $$r$$
surface $$s$$
potential energy $$U$$
volume $$v$$
electric permitivity of free space $$\epsilon_{0} = 8.85 \times 10^{-12} \frac{C^{2}}{N \cdot m^{2}}$$
charge per unit length $$\lambda$$
ratio of a circle's circumference to its diameter $$\pi = 3.14159$$
charge per unit volume $$\rho$$
charge per unit area $$\sigma$$
torque $$\tau$$

Electrostatics

name equation
ley de Coulomb $$F = k \frac{q_{1}q_{2}}{r^{2}}$$ $$F = \frac{1}{4\pi \epsilon_{0}} \frac{q_{1}q_{2}}{r^{2}}$$ $$\vec{F} = \frac{1}{4\pi \epsilon_{0}} \frac{q_{1}q_{2}}{\left(\vec{r} - \vec{r}^{\prime}\right)^{2}} \hat{r}$$
electric field $$\vec{E} = \vec{F}/q_{0}$$ $$E = \frac{1}{4\pi \epsilon_{0}} \sum^{n}_{i=1} \frac{q_{i}}{r^{2}}$$ $$\vec{E} \left( \vec{r} \right) = \frac{1}{4\pi \epsilon_{0}} \sum^{n}_{i=1} \frac{q_{i}}{\left(\vec{r} - \vec{r}^{\prime}_{i}\right)^{2}} \hat{r}$$
electric field over a region $$\vec{E} \left( \vec{r} \right) = \frac{1}{4\pi \epsilon_{0}} \int \frac{dq}{\left(\vec{r} - \vec{r}^{\prime}_{i}\right)^{2}} \hat{r}$$
electric field of a line charge $$\vec{E} \left( \vec{r} \right) = \frac{1}{4\pi \epsilon_{0}} \int_{\mathcal{L}} \frac{\lambda \left( \vec{r}^{\prime} \right) dl}{\left(\vec{r} - \vec{r}^{\prime}_{i}\right)^{2}} \hat{r}$$
electric field over the axis of a finite linear charge from a reference point \(x_{P}\) $$E_{x} = \frac{kq}{x_{P}^{2} - \left( \frac{1}{2} L \right)^{2}},\quad x_{p} > \frac{1}{2} L$$
\(E_{y}\) component due to a segment with uniform linear charge density $$\begin{split} E_{y} &= \frac{k \lambda}{y}\left( \sin \theta_{2} - \sin \theta_{1} \right) \\ &= \frac{k q}{L y}\left( \sin \theta_{2} - \sin \theta_{1} \right) \end{split}$$
\(E_{x}\) component due to a segment with uniform linear charge density $$E_{y} = \frac{k \lambda}{y}\left( \cos \theta_{2} - \cos \theta_{1} \right)$$
electric field \(\vec{E}\) at a distance R from an infinite linear charge $$E_{R} = 2k \frac{\lambda}{R}$$
electric field for a surface charge $$\vec{E} \left( \vec{r} \right) = \frac{1}{4\pi \epsilon_{0}} \int_{\mathcal{A}} \frac{\sigma \left( \vec{r}^{\prime} \right) ds}{\left(\vec{r} - \vec{r}^{\prime}_{i}\right)^{2}} \hat{r}$$
electric field for a volume charge $$\vec{E} \left( \vec{r} \right) = \frac{1}{4\pi \epsilon_{0}} \int_{\mathcal{V}} \frac{\rho \left( \vec{r}^{\prime} \right) dv}{\left(\vec{r} - \vec{r}^{\prime}_{i}\right)^{2}} \hat{r}$$
electric dipole moment $$\vec{p} = q \vec{L}$$
dipole torque $$\vec{\tau} = \vec{p} \times \vec{E}$$
dipole potential energy $$U = -\vec{p} \cdot \vec{E}$$
electric field flux through a surface \(\mathcal{S}\) $$\Phi_{E} \equiv \int_{\mathcal{S}}\vec{E} \cdot d\vec{s}$$