24.2: Electric Potential Energy in a Uniform Electric Field
When an electric field accelerates a free positive charge, it acquires kinetic energy. This process is analogous to an object being accelerated by a gravitational field as if the charge were going down an electrical hill where its electric potential energy is converted into kinetic energy, although, of course, the sources of the forces are very different. The electrostatic or Coulomb force acting on the positive test charge is conservative, which means that the work done on a test charge is independent of the path taken. This is exactly analogous to the gravitational force. When a force is conservative, it is possible to define the potential energy associated with the force. It is usually easier to work with the potential energy (because it depends only on position) than to calculate the work directly.
Consider a positive test charge placed in a uniform electric field produced by the oppositely charged parallel plate. The electric field direction is from the positively charged plate to the negatively charged plate. If the test charge is moved from a positively charged plate towards the negatively charged plate, then the force acting on the test charge is positive. The electric field in this example is one-dimensional, and the work done by moving the test charge is given as integral to the force times displacement. Here since the force is positive, the work done is positive. This work can be expressed as a change in the electric potential energy of the test charge. As the test charge moves towards the negatively charged plate, its electrical potential energy is reduced. If the positive test charge had moved opposite to the direction of the electric field, then the electric potential energy would have increased, suggesting negative work. Here, the assumption is made that the test charge electric field does not alter the electric field in which it is moving.
Suppose a negative test charge is used in the above scenario. In that case, moving the negative charge in the same direction as the electric field will increase the electric potential energy of the negative test charge. On the other hand, if the negative charge was moved in the opposite direction to that of the electric field, then the electric potential energy of the negative test charge will decrease.
