Overview
Similar to the case of gravity, the force of the electric field can be written as the gradient of some potential function {% V %}.
{% \vec{E} = - \nabla V %}
Written this way, we can see that there is flexibility in the choice of function to use as {% V %}. That is, given a function
{% V(x,y.z) %} such that the above holds, then any function expressed as
{% V'(x,y,z) = V(x,y,z) + C %}
where {% C %} is constant, also satifies the equation for the electric field.
One way to construct a potential function is to choose some reference point {% x_0 %} as the point where the potential is equal to 0. Then, the pontential can be expressed as
{% \displaystyle V(\vec{r}) = -\int_o^r \vec{E} \cdot d\vec{l} %}
Potential of a Single Charge
{% V(r) = - \int_{-\infty}^r \vec{E} \cdot d \vec{l} = \frac{-1}{4 \pi \epsilon_0} \int_{-\infty}^r \frac{q}{r^2} dr = \frac{1}{4 \pi \epsilon_0} \frac{q}{r} %}
Superposition Principle
The potential due to a collection of charges is the sum of the potentials of each individual charge.
{% V = \sum_i V_i %}