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

 

 

When the charge creating the field is positive, the voltage is positive; when the central charge is negative, the voltage is negative. As 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  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.

 

 

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 closer together on the left side than on the right.

 

 

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

 

The following two graphs compare the voltage around a positively charged conducting sphere and the electric field for a positively charged conducting sphere. 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.