# Electric Potential and Electric Field

## Electric Potential in a Uniform Electric Field


### Learning Objectives

By the end of this section, you will be able to:
1. Describe the relationship between voltage and electric field.
2. Derive an expression for the electric potential and electric field.
3. Calculate electric field strength given distance and voltage.
In the previous section, we explored the relationship between voltage and energy. In this section, we will explore the relationship between voltage and electric field. For example, a uniform electric field  is produced by placing a potential difference (or voltage)  across two parallel metal plates, labeled A and B. (See .) Examining this will tell us what voltage is needed to produce a certain electric field strength; it will also reveal a more fundamental relationship between electric potential and electric field. From a physicist’s point of view, either  or  can be used to describe any charge distribution.  is most closely tied to energy, whereas  is most closely related to force.  is a scalar quantity and has no direction, while  is a vector quantity, having both magnitude and direction. (Note that the magnitude of the electric field strength, a scalar quantity, is represented by  below.) The relationship between  and  is revealed by calculating the work done by the force in moving a charge from point A to point B. But, as noted in Electric Potential Energy: Potential Difference, this is complex for arbitrary charge distributions, requiring calculus. We therefore look at a uniform electric field as an interesting special case.
The work done by the electric field in  to move a positive charge  from A, the positive plate, higher potential, to B, the negative plate, lower potential, is
 The potential difference between points A and B is
Entering this into the expression for work yields
Work is ; here , since the path is parallel to the field, and so . Since 
, we see that . Substituting this expression for work into the previous equation gives
The charge cancels, and so the voltage between points A and B is seen to be
where  is the distance from A to B, or the distance between the plates in . Note that the above equation implies the units for electric field are volts per meter. We already know the units for electric field are newtons per coulomb; thus the following relation among units is valid:
In more general situations, regardless of whether the electric field is uniform, it points in the direction of decreasing potential, because the force on a positive charge is in the direction of  and also in the direction of lower potential . Furthermore, the magnitude of  equals the rate of decrease of  with distance. The faster  decreases over distance, the greater the electric field. In equation form, the general relationship between voltage and electric field is
where  is the distance over which the change in potential, , takes place. The minus sign tells us that  points in the direction of decreasing potential. The electric field is said to be the gradient (as in grade or slope) of the electric potential. 
For continually changing potentials,  and  become infinitesimals and differential calculus must be employed to determine the electric field.

### Test Prep for AP Courses


### Section Summary

1. The voltage between points A and B is
      

where  is the distance from A to B, or the distance between the plates.
2. In equation form, the general relationship between voltage and electric field is
      

where  is the distance over which the change in potential, , takes place. The minus sign tells us that  points in the direction of decreasing potential.) The electric field is said to be the gradient (as in grade or slope) of the electric potential. 

### Conceptual Questions


### Problems & Exercises
