Main variables:
\( \vec{F} \) = Force.
\( \epsilon_0 \) = vacuum permittivity \( \approx 8.854 \times 10^{-12} \, \text{F/m} \)
\( \vec{r} \) = Position vector.
\( q \) = Charge.
\( \vec{E} \) = Electric field.
\( \vec{A} \) = Area vector whose direction is perpendicular to the surface.
\( U \) = Electric potential energy.
\( V \) = Electric potential or voltage.
\(\rho \) = Volumetric charge density.
\(\sigma \) = Superficial charge density.
\(\lambda \) = Linear charge density.
\( \hat{\textbf{r}} \) = Unitary vector of the position.
\( \hat{\textbf{n}} \) = Unitary vector perpendicular to a surface.
\( \sum F\) = Sum over all the forces.
\( {\large\bigcirc}\kern-1.55em\iint \) = Flux integral over a closed surface.
Main Equations:
Unitary vector position that starts from the center of an object and points radially outwards:
\begin{equation*}
\hat{\textbf{r}} = \frac{ \vec{r} }{ |\vec{r}|}.
\end{equation*}
Coulomb’s law:
\begin{equation*}
\vec{F} = \frac{1}{4 \pi \epsilon_0} \frac{q_1 q_2}{r^2} \hat{\textbf{r}}.
\end{equation*}
The electric field is the force per unit charge:
\begin{equation*}
\vec{E} = \frac{\vec{F}}{q}.
\end{equation*}
Electric field for a punctual charge:
\begin{equation*}
\vec{E} = \frac{1}{4 \pi \epsilon_0} \frac{q }{r^2} \hat{\textbf{r}}.
\end{equation*}
Electric field for a wire:
\begin{equation*}
\vec{E} = \frac{1}{2 \pi \epsilon_0} \frac{\lambda}{r^2} \hat{\textbf{r}}.
\end{equation*}
Electric flux:
\begin{equation*}
\Phi = \vec{E} \cdot \vec{A}.
\end{equation*}
Gauss’ law:
\begin{equation*}
{\large\bigcirc}\kern-1.55em\iint \vec{E} \cdot d \vec{A} = \frac{Q_{\text{enclosed}}}{\epsilon_0}.
\end{equation*}
Electric field outside a sphere (for both conducting and isolating), for \(r > R\):
\begin{equation*}
\vec{E} = \frac{1}{4 \pi \epsilon_0} \frac{q }{r^2} \hat{\textbf{r}}.
\end{equation*}
Electric field inside (\(r < R\)) a conducting sphere:
\begin{equation*}
\vec{E} = 0.
\end{equation*}
Electric field inside (\(r < R\)) an isolating sphere: \begin{equation*} \vec{E} = \frac{1}{4 \pi \epsilon_0} \frac{qr}{R^3} \hat{\textbf{r}}. \end{equation*} Electric field outside a cylinder (for both conducting and isolating), for \(r > R\):
\begin{equation*}
\vec{E} = \frac{1}{2 \pi \epsilon_0} \frac{\lambda}{r^2} \hat{\textbf{r}}.
\end{equation*}
Electric field inside (\(r < R\)) a conducting cylinder:
\begin{equation*}
\vec{E} = 0.
\end{equation*}
Electric field inside (\(r < R\)) an isolating cylinder:
\begin{equation*}
\vec{E} = \frac{1}{2 \pi \epsilon_0} \frac{qr}{R^2} \hat{\textbf{r}}.
\end{equation*}
Electric field for an infinite plane at any point:
\begin{equation*}
\vec{E} = \frac{\sigma}{2 \epsilon_0 } \hat{\textbf{n}}.
\end{equation*}
Magnitude of the electric field between opposite parallel conducting plates that are charged:
\begin{equation*}
E = \frac{\sigma}{\epsilon_0 }.
\end{equation*}
Electric potential energy:
\begin{equation*}
U = \frac{1}{4 \pi \epsilon_0} \frac{q_1 q_2 }{r}.
\end{equation*}
Electric potential or voltage:
\begin{equation*}
V = \frac{U}{q}.
\end{equation*}
Electric potential for a punctual charge:
\begin{equation*}
V = \frac{1}{4 \pi \epsilon_0} \frac{q}{r}.
\end{equation*}
The relationship between the voltage and the electric field:
\begin{equation*}
\Delta V = – \vec{E} \cdot \vec{\ell}.
\end{equation*}
The relationship between the voltage and the electric field in integral form:
\begin{equation*}
\Delta V = – \int \vec{E} \cdot d \vec{\ell}.
\end{equation*}
The relationship between the electric field and the voltage:
\begin{equation*}
\vec{E} = – \left( \frac{\partial V}{\partial x} \hat{\textbf{i}} + \frac{\partial V}{\partial y} \hat{\textbf{j}} + \frac{\partial V}{\partial z} \hat{\textbf{k}} \right).
\end{equation*}
Work in terms of the voltage:
\begin{equation*}
W = q \Delta V.
\end{equation*}