Electric Fields and Potentials

Introduction

Physicists use the concept of a field to explain the interaction of particles or bodies through space, i.e., the "action-at-a-distance" force between two bodies that are not in physical contact. For example, the earth modifies the surrounding space such that any object with mass, such as the moon, is attracted. We have seen previously in our study of the "Universal Law of Gravitation" that the gravitational field gets weaker as you go farther away from the source but never completely disappears. An electron similarly modifies the space around it in such a way that other particles with like charge are repelled, while particles with the opposite charge are attracted. Like the gravitational field, the electric field gets weaker with distance from the source but is never completely gone. Any charge placed in an electric field will experience a force, as will any mass placed in a gravitational field. Just as mass in a gravitational field has some potential energy, so does a charge in an electric field.

Discussion of Principles

Electric Field

A charged body experiences a force, F, whenever it is placed in an electric field, E. The vector relationship between the force and the electric field is given by The magnitude of the force divided by the magnitude of the charge, q, on the body is numerically equal to the magnitude of the electric field. From Eq. (1), we see that the direction of the electric field vector at any given point is the same as the direction of the force that the field exerts on a positive test charge located at that point. A positive test charge will be repelled by a positive charge and attracted by a negative charge. Therefore, electric field lines start on positive charges and end on negative charges. The number of lines starting from a positive charge or ending on a negative charge is proportional to the magnitude of the charge. The electric field at any point is tangential to the electric field lines. The magnitude E of the electric field strength is proportional to the number of electric field lines per unit area perpendicular to the lines. A positive charge placed in an electric field will tend to move in the direction of the electric field lines and a negative charge will tend to move opposite to the direction of the electric field lines.

Work, Potential Energy, and Electrostatic Potential

Work is done in moving a charged body through an electric field. The amount of work, W, done depends on the electric field, the magnitude of the charge and the displacement, d, that the charge makes through the field. When the displacement is so small that the electric field can be assumed uniform in the region through which the charge moves, the work, W, is given by the dot product of the force and displacement vectors. The negative of the work done by the electric force is defined as the change in electric potential energy, U, of the body. Put another way, it is the difference in the potential energies ΔU associated with the starting and ending positions. The electrostatic potential, V, is defined as the electric potential energy of the body divided by its charge:

V = U/q

. Hence, in terms of electrostatic potential, the work done by the electric field is where

ΔV = V_final − V_initial

is the potential difference. The magnitude of the electric field can also be determined from Eqs. (3) and (5) where θ is the angle between the field and displacement vectors. When a charged particle moves in an electric field, if its electric potential energy decreases, its kinetic energy will increase. In other words, the total energy is conserved. A positive charge placed in an electric field will tend to move in the direction of the field. The work done by the electric field in this case will be positive as the field and the displacement vectors are in the same direction and the dot product of the two vectors will be positive as shown in Figure 1(a). Note that the velocity vector indicates the direction of the displacement vector for the charge. From Eq. (4)

−W = ΔU = (U_final − U_initial)

, we see that the change in electric potential energy will be negative in this case. It will, therefore, lose electric potential energy and gain kinetic energy. This tells us that electric potential decreases in the direction of the electric field lines. A positive charge, if free to move in an electric field, will move from a high potential point to a low potential point. Now consider a negative charge placed in an electric field as shown in Figure 1(b). It will tend to move in a direction opposite to the electric field and accelerate as it does so. The work done by the electric field will be Again, note that the work done by the electric field is positive and the negative charge will lose electric potential energy and gain kinetic energy as it moves against the field. A negative charge, if free to move in an electric field, will move from a low potential point to a high potential point. To move a positive charge against the electric field, work has to be done by you or a force external to the field. The charge is forced to move from a low potential point to a high potential point and the work done by the external force is negative. The reverse will be true for a negative charge. As in the case of the gravitational field, the zero point of the electric potential is chosen arbitrarily. In general, the zero point is at infinity; however, often in circuits, the zero point for the potential is the ground or a conductor that is directly connected to the ground.

Equipotential Lines and Electric Field Lines

Consider the field due to a single point charge. A point in this space near the source of the field (i.e., near the point charge), and another point far from the source of the field are at different potentials. This is true even if no charges reside at the two points. In Fig. 2, points A and B are at different potentials due to the electric field of the positive charge. All fields have certain points that are at the same potential. For example, when a point charge is the source of the field, then any two points that are the same distance from the point charge (points A and C in Figure 2) will be at the same potential. There are an infinite number of points—all lying on the same sphere—that are at the same distance, all of which are at the same potential. In three dimensions, these points form a surface called an equipotential surface. In two dimensions—say, the equatorial plane of the sphere—the circle (equator) where the sphere intersects the plane is an equipotential line; the potential difference between any two points on this line is zero. It follows from Eq. (4)

−W = ΔU = (U_final − U_initial)

that no work is done in moving a charge along an "equipotential" (i.e., an equipotential line or surface). Eq. (4) also tells us that no work is done when the displacement and the field are perpendicular to one another. Figure 3 shows a few of the electric field lines (blue lines with arrows) for a positive charge. The dashed red circles are two of the many equipotential circles. At any given point on one of these circles, the tangent to the circle will be perpendicular to the electric field line direction at that point. Because it is not simple to find the electric field lines directly, in this experiment, you will first locate the equipotential lines, and then draw the field lines by knowing that they are perpendicular to the equipotential lines.

Mapping of the Electric Field

We employ a special conducting paper (not too conducting and not too insulating) and a voltmeter. Some shapes have been drawn on the paper, and to these we will connect a power supply. Because the paper is conducting, a current can flow between the electrodes. By Ohm's law, the current flows along the electric field. Thus, if we can measure either the current or the electric field, we can determine the other. With our equipment, we cannot directly measure the electric field; however, what we can do is measure voltage differences using the voltmeter setting on the multimeter. From these voltage differences we can learn about the electric field lines because the voltage difference between two points is a measure of the integrated electric field between the electrodes (see Eq. (6)

E =

ΔV

d cos θ

). If the electric field is along the path between the leads, then

|ΔV|

is a maximum; if the electric field is normal to this path, then

|ΔV| = 0.

Thus, when the voltmeter probes are separated by a fixed distance, the direction of maximum effect gives the direction of the electric field. The electric field points from high to low electrical potential (voltage). Recall that on the earth, the gravitational field points from high to low gravitational potential. As noted above, the electric field lines are perpendicular to surfaces called equipotential surfaces. In two dimensions, these are lines. Along an equipotential, the electric field is a minimum—that is, it is zero. A voltmeter will yield zero voltage difference between two points that are on the same equipotential line. See Figure 4.

Apparatus

Procedure

Please print the worksheet for this lab. You will need this sheet to record your data.

Setting Up the Experiment

The graphite sheets provided have equally spaced division marks imprinted on them. The division marks may be used as reference when transferring data collected from the graphite sheets to a template that you will be supplied later in the lab. The graphite sheets also have various shapes (configurations) drawn on the surface with conducting nickel ink (similar to Figure 6). Graphite paper has been chosen for two reasons. First, the graphite paper has an electrical resistance that is much less than the voltmeter, so that the voltmeter will not draw much current. Second, it has a large enough electrical resistance that it does not draw more current than the power supply can provide.

Equipotentials for a Parallel Plate Configuration

Equipotentials for a Dipole Configuration

Equipotentials for the Surface and Point Configuration

Copyright © 2013-2014 Advanced Instructional Systems Inc. and Texas A&M University. Portions from North Carolina State University. | Credits