PX157 - B7c - electric potential

\vec{E} (\vec{r}) = lim_{q_{0} \to 0} \frac{{\vec{F}}_{q_{0}}}{q_{0}}

V (\vec{r}) = lim_{q_{0} \to 0} \frac{U_{q_{0}}}{q_{0}}

units: volt ( $V = J C^{- 1}$ )
scalar field defined at all points in space, even if no charge is present at that point

\frac{W_{a \to b}}{q_{0}} = \frac{U_{a}}{q_{0}} - \frac{U_{b}}{q_{0}} = V_{a} - V_{b} \equiv V_{a b}

$V_{a b}$ is the potential difference, which represents work done by the electric field when unit charge moves from $a$ to $b$ , or work that must be done to move a unit charge slowly from $b$ to $a$
potential for a point charge, $q$ , at position, ${\vec{r}}^{'}$ , measured at position, $\vec{r}$ :

V (\vec{r}) = \frac{q}{4 π ϵ_{0} | \vec{r} - {\vec{r}}^{'} |}