Potential

$$V(r)\equiv -\int \_o^{r}E(r^{\prime })\cdot d{l}^{\prime }$$

With $o$ being a chosen reference point, that can be changed.

E = - \nabla V \hspace{5mm} \left\[ \frac{Nm}{C} = \frac{J}{C} = V \right]

A simple scalar $V$ will therefore constitute the definition of three interdependent functions inscribing an entire field.

Potential Difference

$$V(b)-V(A)=\int \_a^{b}E\cdot dl$$

Identities

• $\nabla V^{\prime }=\nabla V$ where $\nabla V^{\prime }$ is a potential from a different reference point.

Properties

• Obeys superposition principle $V=V\_1+V\_2+\dots$ so that it is the sum of the potentials due to all the source charges separately. Much easier to work with because $V$ is only an ordinary value.
• A reference point $O=\infty$ at infinity is applicable only when the charge distribution does not extend to infinity. (Which it only ever does in theoretical textbook problems, never in real life)