Learning Objectives
By the end of this section, you will be able to do the following:
- Explain the similarities and differences between electric potential energy and gravitational potential energy
- Calculate the electric potential difference between two point charges and in a uniform electric field
electric potential | electric potential energy |
As you learned in studying gravity, a mass in a gravitational field has potential energy, which means it has the potential to accelerate and thereby increase its kinetic energy. This kinetic energy can be used to do work. For example, imagine you want to use a stone to pound a nail into a piece of wood. You first lift the stone high above the nail, which increases the potential energy of the stone-Earth system—because Earth is so large, it does not move, so we usually shorten this by saying simply that the potential energy of the stone increases. When you drop the stone, gravity converts the potential energy into kinetic energy. When the stone hits the nail, it does work by pounding the nail into the wood. The gravitational potential energy is the work that a mass can potentially do by virtue of its position in a gravitational field. Potential energy is a very useful concept, because it can be used with conservation of energy to calculate the motion of masses in a gravitational field.
Electric potential energy works much the same way, but it is based on the electric field instead of the gravitational field. By virtue of its position in an electric field, a charge has an electric potential energy. If the charge is free to move, the force due to the electric field causes it to accelerate, so its potential energy is converted to kinetic energy, just like a mass that falls in a gravitational field. This kinetic energy can be used to do work. The electric potential energy is the work that a charge can do by virtue of its position in an electric field.
The analogy between gravitational potential energy and electric potential energy is depicted in Figure 18.23. On the left, the ball-Earth system gains gravitational potential energy when the ball is higher in Earth's gravitational field. On the right, the two-charge system gains electric potential energy when the positive charge is farther from the negative charge.
Let’s use the symbol to denote gravitational potential energy. When a mass falls in a gravitational field, its gravitational potential energy decreases. Conservation of energy tells us that the work done by the gravitational field to make the mass accelerate must equal the loss of potential energy of the mass. If we use the symbol to denote this work, then
where the minus sign reflects the fact that the potential energy of the ball decreases.
The work done by gravity on the mass is
where F is the force due to gravity, and and are the initial and final positions of the ball, respectively. The negative sign is because gravity points down, which we consider to be the negative direction. For the constant gravitational field near Earth’s surface, . The change in gravitational potential energy of the mass is
Note that is just the negative of the height h from which the mass falls, so we usually just write .
We now apply the same reasoning to a charge in an electric field to find the electric potential energy. The change in electric potential energy is the work done by the electric field to move a charge q from an initial position to a final position (). The definition of work does not change, except that now the work is done by the electric field: . For a charge that falls through a constant electric field E, the force applied to the charge by the electric field is . The change in electric potential energy of the charge is thus
or
This equation gives the change in electric potential energy of a charge q when it moves from position to position in a constant electric field E.
Figure 18.24 shows how this analogy would work if we were close to Earth’s surface, where gravity is constant. The top image shows a charge accelerating due to a constant electric field. Likewise, the round mass in the bottom image accelerates due to a constant gravitation field. In both cases, the potential energy of the particle decreases, and its kinetic energy increases.
Watch Physics
Analogy between Gravity and Electricity
This video discusses the analogy between gravitational potential energy and electric potential energy. It reviews the concepts of work and potential energy and shows the connection between a mass in a uniform gravitation field, such as on Earth’s surface, and an electric charge in a uniform electric field.
If the electric field is not constant, then the equation is not valid, and deriving the electric potential energy becomes more involved. For example, consider the electric potential energy of an assembly of two point charges and of the same sign that are initially very far apart. We start by placing charge at the origin of our coordinate system. This takes no electrical energy, because there is no electric field at the origin (because charge is very far away). We then bring charge in from very far away to a distance r from the center of charge . This requires some effort, because the electric field of charge applies a repulsive force on charge . The energy it takes to assemble these two charges can be recuperated if we let them fly apart again. Thus, the charges have potential energy when they are a distance r apart. It turns out that the electric potential energy of a pair of point charges and a distance r apart is
To recap, if charges and are free to move, they can accumulate kinetic energy by flying apart, and this kinetic energy can be used to do work. The maximum amount of work the two charges can do (if they fly infinitely far from each other) is given by the equation above.
Notice that if the two charges have opposite signs, then the potential energy is negative. This means that the charges have more potential to do work when they are far apart than when they are at a distance r apart. This makes sense: Opposite charges attract, so the charges can gain more kinetic energy if they attract each other from far away than if they start at only a short distance apart. Thus, they have more potential to do work when they are far apart. Figure 18.25 summarizes how the electric potential energy depends on charge and separation.