Electric Potential

This section introduces the scalar electric potential, giving you a way to talk about the potential energy of an electric field without that value being affected by a test charge. Potential difference, a related quantity, is also introduced.

Taking the simple case of a uniform electric field, this section examines more carefully the behavior of electric potential and potential difference. The concept of an equipotential surface - an imaginary surface with a continuous distribution of points with the same electric potential - is also introduced.

This section explores the electric potential and potential energy of the electric field caused by distributions of point charges. You learn that it is often easier to calculate the potential than to calculate the electric field, setting the field for the next section.

The qualitative relationship between the electric field and electric potential are studied in this section. These explorations lead to a better understanding of both quantities as well as a new way to calculate the electric field using partial dimensional derivatives of the electric potential.

This section takes the explorations a step further and derives an equation to calculate the electric potential created by a continuous charge distribution. It also provides a number of helpful problem solving hints and important reminders before giving a collection of examples with which to practice them.

Using tools and information from earlier in this text, this section shows how the electric potential of a charged conductor behaves, explains the behavior of field lines caused by non-spherical objects, and considers the special case of a cavity within a conductor. Finally, corona discharge is examined as a related phenomenon which has practical uses.

In 1923, Millikan won the Nobel Prize for a series of experiments he conducted calculating the electron's charge. The experiment, amazingly, used a combination of situations and techniques which you have now learned from this text. This section describes and explains the experiment.

Now that you have learned about electrostatics and seen them applied in a few examples, this section shows you three different applications of electrostatics which are still widely used today.

When a positive test charge q₀ is moved between points A and B in an electric field E, the change in the potential energy of the charge-field system is

The electric potential V = U / q₀ is a scalar quantity and has the units of J/C, where 1 J/C = 1 V.

The potential difference between points A and B in an electric field E is defined as

The potential difference between two points A and B in a uniform electric field E, where s is a vector that points from A to B and is parallel to E is

where d = | s |.

An equipotential surface is one on which all points are at the same electric potential. Equipotential surfaces are perpendicular to electric field lines.

If we define V = 0 at r_A = infinity, the electric potential due to a point charge at any distance r from the charge is

We can obtain the electric potential associated with a group of point charges by summing the potentials due to the individual charges.

The potential energy associated with a pair of point charges separated by a distance r₁₂ is

This energy represents the work done by an external agent when the charges are brought from an infinite separation to the separation r₁₂. We obtain the potential energy of a distribution of point charges by summing terms like Equation 25.13 over all pairs of particles.

If we know the electric potential as a function of coordinates x, y, z, we can obtain the components of the electric field by taking the negative derivative of the electric potential with respect to the coordinates. For example, the x component of the electric field is

The electric potential due to a continuous charge distribution is

Every point on the surface of a charged conductor in electrostatic equilibrium is at the same electric potential. The potential is constant everywhere inside the conductor and equal to its value at the surface.

Table lists electric potentials due to several charge distributions.

Potential and Potential Energy

The potential is characteristic of the field only, independent of a charged test particle that may be placed in the field. Potential energy is characteristic of the charge-field system due to an interaction between the field and a charged particle placed in the field.

Voltage

A variety of phrases are used to describe the potential difference between two points, the most common being voltage, arising from the unit for potential. A voltage applied to a device, such as a television, or across a device is the same as the potential difference across the device. If we say that the voltage applied to a lightbulb is 120 volts, we mean that the potential difference between the two electrical contacts on the lightbulb is 120 volts.

The Electron Volt

The electron volt is a unit of energy, NOT of potential. The energy of any system may be expressed in eV, but this unit is most convenient for describing the emission and absorption of visible light from atoms. Energies of nuclear processes are often expressed in MeV.

Similar Equation Warning

Do not confuse the electric potential of a point charge with the electric field of a point charge. Potential is proportional to 1/r, while the field is proportional to 1/r². The effect of a charge on the space surrounding it can be described in two ways. The charge sets up a vector electric field E, which is related to the force experienced by a test charge placed in the field. It also sets up a scalar potential V, which is related to the potential energy of the two-charge system when a test charge is placed in the field.

Which Work?

There is a difference between work done by one member of a system on another member and work done on a system by an external agent. In the present discussion, we are considering the group of charges to be the system and an external agent is doing work on the system to move the charges from an infinite separation to a small separation.

Potential May Not Be Zero

The electric potential inside the conductor is not necessarily zero, even though the electric field is zero. We see that a zero value of the field results in no change in the potential from one point to another inside the conductor. Thus, the potential everywhere inside the conductor, including the surface, has the same value, which may or may not be zero, depending on where the zero of potential is defined.