Rearranging these units (1 joule = 1 coulomb x 1 volt) shows us that the amount of work done on a charge by an external agent as it is moved around an electric field is expressed as

 

 

For a point charge the absolute potential of any position in its electric field can be calculated using the equation

 

 

EPE represents a charge's electric potential energy which is calculates as the magnitude of the charge times the absolute potential of its position in the electric field produced by the dentral charge.

 

 

When the charge creating the field is positive, the voltage is positive; when the central charge is negative, the voltage is negative. As the distance from the central charge, r, grows larger and larger; that is, as r approaches infinity, the absolute potential is defined to be zero. You can almost think of the "voltage" as being an indicator of the "elevation of the terrain" surrounding a central point charge. The steeper the terrain, the faster the voltage changes from one location to another. The work done by an external agent can be envisioned as "pushing or pulling" a second charge up or down these changes in elevation. The EPE that the second charge gains or loses can anticipated by the difference in the elevations of its starting and ending positions.

 

 

Surfaces which connect points that are at the same absolute potential, or voltage, are called  equipotential surfaces. In the diagram of the point charge shown in the previous example, two equipotential surfaces were labeled, A and B. Notice that equipotential surfaces meet field lines at right angles. The closer together two equipotential surfaces are to each other, the more rapid the change in voltage. This indicates a stronger electric field which is shown in the second diagram below by the fact that the field lines are grouped more closely together on the left side where the equipotential surfaces are also more closely clustered together.

 

 

Note that the electric field strength, E, can be measured in either the units V/m, or equivalently, in the unit N/C.

 

N/C = V / d
      = (J/C) / m
      = [(Nm)/C] / m

      = N/C

 

Shown below are two more illustrations showing the relationships between field lines and equipotential surfaces. The image on the left shows two positive point charges of equal magnitude; while the image on the right shows two charges of equal magnitude but differing in sign. Note that the field lines (represented by vectors) meet every equipotential surface at right angles.

 

These images were provided by the The Wolfram Demonstrations Project

 

The following two graphs compare the voltage around a positively charged conducting sphere and the electric field for the same positively charged conducting sphere under electrostatic conditions. Note that the electric field strength (E ∝ 1/r²) drops off more rapidly than does the voltage (V ∝ 1/r). Also notice that within a conducting sphere, the voltage remains constant in contrast to the fact that no electric field exists.

 

For a conducting sphere, V = kQ/r	For a conducting sphere, E = kQ/r2

 

Remember that the electric field strength, E, is a vector quantity. You are required to state both its magnitude and its direction to completely describe it at any given location. If you are ever asked to calculate the net electric field in 2-dimensions, you should first take the x- and y-components of each field, add the components to determine the net E_x and net Ey, and then calculate the resultant field and its direction. Voltage, on the other hand, is a scalar quantity and can be added directly without considering components or directions.