Main variables:

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*}