Electric Potential

Here we describe the conservative thus scalar nature of the electric field as a conservative quantity called the electric potential,

\[ \boxed{V(\boldsymbol r) \equiv -\int_{\mathcal O}^{\boldsymbol{r}} \boldsymbol{E} \cdot \mathrm d \boldsymbol{r'}} \]

One may solve for the electric potential itself only knowing the geometry and position of the charge. First for a single point charge,

\[ V(\boldsymbol{r}) = \frac{1}{4\pi\epsilon_0}\frac{q}{|\boldsymbol{r}|} \]

For a continuous charge,

\[ \boxed{V(\boldsymbol{r}) = \frac{1}{4\pi\epsilon_0}\int {\frac{1}{|\boldsymbol{r}|}\ \mathrm dq}} \]

The Reference Point \(\mathcal O\)

The reference point is simply where we define the origin of the electric potential \(V(\mathcal{O} = 0)\). In most situations such that the electric potential converges to zero the distance away from the source charge goes to infinity (\(V(\infty) = 0\)) we choose the origin to be at infinity,

\[ V(\boldsymbol r) = - \int_{\infty}^{r} \boldsymbol{E} \cdot \mathrm d \boldsymbol{r'} \]

Electric Field to a Scalar Quantity

We can related how the electric potential is considered the scalar quantity of the electric field. First, what’s often more of a useful quantity is the difference of the electric betwteen two points \(a\) and \(b\) thus \(\Delta V\).

\[ \Delta V \equiv \int_a^b \boldsymbol{E} \cdot \ \mathrm d \boldsymbol{r'} \]

We can write \(\Delta V\) as a line integral of the gradient w.r.t. the line (fundamental theorem of calculus).

\[ \Delta V = \int_{a}^{b}{\nabla V} \cdot \ \mathrm d \boldsymbol{r'} \]

Equating the two we now form the equation to relate the electric field is related to a scalar quantity – the electric potential.

\[ \boxed{\boldsymbol{E} = - \nabla V} \]