PHYS112 : Labs

4 Electric Potential and Field Mapping

Introduction

Gravitational potential energy change is defined as the opposite of the work done in moving a mass through a gravitational field,

$$\Delta U^{grav}=-W^{grav}$$

Near the surface of the earth $U^{grav}$ is just $mgh$ (where $h$ is the height above the earth's surface and we typically define the zero of $U^{grav}$ as the value at the surface). This works because we can consider the gravitational force to be constant near the earth. A more exact expression would be $U^{grav}=-\frac{Gm_1m_2}{r}$ (where $r$ is the distance between $m_1$ and $m_2$ and we define the zero to be at infinity). Note that both of these expressions are implicitly for $\Delta U^{grav}$ even though we do not always write the $\Delta$. In both cases we have defined the potential energy to be zero at a specific point and we are always considering the difference with respect to that point.

Let's continue to consider the earth as our example. Call $m_1$ the mass of the earth and $m_2$ the ``test mass.'' Now, if we wanted to define a property of the earth's gravitational field that did not depend on the ``test mass,'' we could divide $U^{grav}$ by the magnitude of the ``test mass.'' This quantity is called the gravitational potential. Using the gravitational potential, we could find the potential energy for any test mass just by multiplying by the value of the test mass. The gravitational potential gives us another way of looking at the gravitational field. It is a scalar defined at every point in space (it has no direction!) so it is simpler to work with than the field, which has both magnitude and direction. A useful way to graph the potential is to consider the surfaces along which the potential is a constant-these are call equipotentials. A plot of the equipotentials for the gravitational potential of the earth is shown in Fig. 4. Keep in mind that this plot is a two dimensional representation of a three dimensional object. The circular lines that you see are really cross-sections of spheres. Everywhere on the surface of one of these spheres, the gravitational potential has the same value. Note that the equipotentials nearer to the earth are spaced more closely. This is because the force is stronger there and the gravitational potential changes more rapidly. This is just like an elevation map, where closely spaced lines indicate a steep grade. Note that the gravitational force points radially in toward the earth and so is always perpendicular to the equipotentials.

Figure 4: Gravitational equipotentials near the earth.

We can make an analogous definition for electric fields. The electric potential energy change is $\Delta U^{elec}=-W^{elec}$. We define the electric potential V by dividing by the value of the test charge,

$$V=\frac{U^{elec}}{q}$$

We can plot equipotentials for electric fields as well. Fig. 5 shows the electric potentials, with field lines, for positive and negative point charges. The field lines, which point in the same direction as the force felt by a positive test charge, are always perpendicular to the equipotentials.

Figure 5: The equipotentials and field lines for two point charges. Which one is negative?

This week we will map equipotential lines, lines of constant electric potential, in diagrams of charged conducting objects drawn on weakly conducting paper with conducting ink.

Experiments and Analysis

You have been supplied with a DC voltage supply, two wires, thumbtacks, a few sheets of conducting paper, conducting paint, and a multimeter, which we will use to measure differences in electric potential.

Field lines and Equipotential Lines of two Point Charges

Figure 6: Two conducting ``points'' (small dots of conducting ink) on conducting paper, charged by a voltage supply.

1. You have been given a piece of (weakly) conducting paper with two drops of conducting paint centered and about half a page apart (see Figure 6). Anchor a wire to each paint drop with a thumb tack. This makes a good electrical contact between the wire and the paper in a small region, which is a reasonable approximation to a point for our purposes. Attach the other ends of the wires to the two terminals of the DC voltage supply. Don't hesitate to ask for help in setting this up.

2. Once you have assembled the circuit shown in the diagram, turn on the power supply. It should read 5 V on its voltage scale. If it doesn't, turn it off, and ask for help. Set the multimeter on the 20 V DC scale (or 30 V, or 40 V, depending on the meter). In this mode, it will then tell you the potential difference in volts between the two leads. If things are connected correctly and well, if you place a multimeter lead near each of the two ``point charges,'' the multimeter and the voltage meter on the voltage supply should agree, roughly.

3. Now, place one lead on the ground terminal (the black one or the green one-they're connected) and use the other to find a point on the paper at a potential difference of 1 V from ground. Mark that point with chalk, and find another. You should be able to find a (possibly curved) line of points at 1 V. It may or may not run off of the page, but try to get as much of it as you can.

4. In a similar fashion, find and draw the equipotential lines at 2 V, 3 V, 4V. Where is 5 V? Does this make sense?

5. Now that you have equipotential lines, you can use them to help you sketch in the field lines fairly accurately. Explain how equipotential lines help you draw field lines.

6. Place one of the multimeter leads anywhere on the 2 V line and the other anywhere on the 4 V line. What do you expect, and what do you observe? What if you place them both on the 3 V line? Explain.

Field lines and Equipotential Lines of two Line Charges (a 2D Capacitor)

Figure 7: Two parallel lines of conducting ink on conducting paper, charged by a voltage supply.

1. You should have a second piece of conducting paper with two parallel conducting lines that approximate a parallel plate capacitor (see Figure 7). Repeat the process you used with the point charges.

2. What can you say about the field in the region between the lines?

3. What can you say about the field outside?

4. Compare your equipotential map with those of other lab groups. (Each group has a different line separation.) What happens to the electric field when the conducting lines get closer together or further apart?

Field lines and Equipotential Lines of Something Else

Repeat the process used with the point charges and parallel conducting lines with an arrangement of your own design. This is mainly for fun (woo hoo!). Why is it important to have two separate conducting shapes on the page?

Before You Leave

Discuss with your instructor preliminary answers to the questions posed above.

In Your Lab Report

Answer the the questions posed above with each each experiment. Also hand in your three pieces of conducting paper showing the equipotential lines and field lines you drew.

Copyright © 2006-2009, L.A. Riley, T. J. Carroll, J.S. Scott

Updated Sun Apr 26 23:00:14 2009

This work is licensed under a Creative Commons License.