# Electric Potential and Electric Field

## Electrical Potential Due to a Point Charge


### Learning Objectives

By the end of this section, you will be able to:
1. Explain point charges and express the equation for electric potential of a point charge.
2. Distinguish between electric potential and electric field.
3. Determine the electric potential of a point charge given charge and distance.
Point charges, such as electrons, are among the fundamental building blocks of matter. Furthermore, spherical charge distributions (like on a metal sphere) create external electric fields exactly like a point charge. The electric potential due to a point charge is, thus, a case we need to consider. Using calculus to find the work needed to move a test charge  from a large distance away to a distance of  from a point charge , and noting the connection between work and potential , it can be shown that the electric potential  is 
where k is a constant equal to

 
  .
The potential at infinity is chosen to be zero. Thus  for a point charge decreases with distance, whereas  for a point charge decreases with distance squared:
Recall that the electric potential  is a scalar and has no direction, whereas the electric field  is a vector. To find the voltage due to a combination of point charges, you add the individual voltages as numbers. To find the total electric field, you must add the individual fields as vectors, taking magnitude and direction into account. This is consistent with the fact that  is closely associated with energy, a scalar, whereas  is closely associated with force, a vector.
The voltages in both of these examples could be measured with a meter that compares the measured potential with ground potential. Ground potential is often taken to be zero (instead of taking the potential at infinity to be zero). It is the potential difference between two points that is of importance, and very often there is a tacit assumption that some reference point, such as Earth or a very distant point, is at zero potential. As noted in Electric Potential Energy: Potential Difference, this is analogous to taking sea level as  when considering gravitational potential energy, .

### Section Summary

1.  Electric potential of a point charge is .
2. Electric potential is a scalar, and electric field is a vector. Addition of voltages as numbers gives the voltage due to a combination of point charges, whereas addition of individual fields as vectors gives the total electric field.

### Conceptual Questions


### Problems & Exercises
