# F-1 ELECTRIC FIELD MAPPING

1. PURPOSE

(1) To experimentally plot equipotential lines of electric fields near several different patterns of conducting shapes.

(2) To construct, from the equipotentials, the field lines of these fields.

2. APPARATUS

A. CENCO: Electric field mapping board with probe and an assortment of "pattern-boards." Plastic pattern templates. A voltage source, 2 to 6 volts DC or AC. A zero centering galvanometer of range and type to match the voltage source. If audio frequency AC is used, headphones may replace the galvanometer as a detector. In either case, the voltage source need not be regulated. A common source may serve several student stations.

B. PASCO AND WELCH: Sheets of conductive paper, pen with conduc- tive paint. Potential divider, either (1) a string of equal resistors mounted on a terminal board, or (2) a rheostat of size 1 kilohm or greater. A voltage source and detector (see Cenco list above). Large pins ("bankers' pins", map pins, or "pushpins"), connecting wires with alligator clips, pin probe for voltmeter.

Read the discussion of electric fields and potentials, field lines and equipotentials, in any good textbook. As you study this material, test your understanding by answering these questions:

(1) Why are all parts on and within a closed conducting body at the same potential when that body is in electrostatic equilibrium? Hints: a) Why does the question specify electrostatic equilibrium? What if equilibrium had not yet been established? b) Consider what would happen if two points in the body had different potentials.

(2) What does question (1) tell you about the field strength within a conducting body in electrostatic equilibrium?

(3) Why do equipotential lines cross field lines at right angles? Hint: What would happen if they crossed at some other angle?

(4) Why is field strength sometimes expressed in the unit Volt/meter and sometimes in the unit Newton/Coulomb?

(5*) Why do you suppose the electric field strength from a point charge has a strength proportional to r-2 rather than r-1 or some other function of r?

4. GENERAL PROCEDURE

Several types of apparatus are commercially available, and these are separately described below. Whichever type you use, read the instructions for both, since some essential common description is not repeated.

In all methods you will investigate a geometric arrangement of conducting and insulating bodies. Two of the conducting bodies will be charged to equal and opposite amounts by connecting the leads of a voltage source to them (one lead to each). There may be one or more uncharged conducting bodies also; these are not connected to the voltage source.

These patterns are painted on a weakly conductive flat surface. You can measure potentials on that surface with relatively unsophisticated equipment. The field lines and equipotentials which you will map are a quite good approximation and simulation of those you would find in a similar arrangement of bodies in a vacuum.

We'll need to measure potentials at various points on the pattern surface. To do this we make use of the "potentiometric principle" in which a variable known potential is balanced against an unknown potential. A sensitive detector, in this case a galvanometer, is used as a null detector so that when the detector reads zero you'll know that the known and unknown potentials are equal. The beauty of this procedure is that when the galvanometer null is achieved, the galvanometer is drawing negligible current from the voltage sources, and therefore does not alter what we are trying to measure (as a voltmeter might).

Fig. 1 shows the circuit arrangement, with a pattern simulating unequal sized capacitor plates. In the Cenco apparatus the potential divider consists of eight equal (100 ohm) resistors. In other versions of the apparatus, it might be a slidewire rheostat or small potentiometer (pot) of size on the order of 1 kilohm. If the rheostat is used, you'll need a good voltmeter to read the potential, as shown in Fig. 2.

A. Cenco apparatus: This consists of a field-mapping board and a number of Masonite ) pattern-boards on which patterns of conducting areas have been applied over a poorly conductive graphite painted surface. Some have areas where the graphite paint has been removed to produce non-conductive areas. Along the top of the apparatus is a string of 8 equal (100 ohm) resistors in series, with accessible terminals (jacks) between them. When a voltage is applied across the resistor string, the voltage will be the same size across each resistor. If the applied voltage is V and there are 8 resistors, each will then have voltage V/8 across it.

PRECAUTION: Avoid touching the surface of the pattern board. Your perspiration can tarnish the metallic coatings. Also, fingerprints can be more conductive than the graphite paint, and may significantly alter the measured patterns of equipotentials. The pattern boards cannot be successfully cleaned, for the graphite paint tends to wipe off.

The two wires from the voltage source are connected to the terminals A and B at opposite sides of the mapping board. The chosen pattern board is bolted to the mapping board, using two knurled nuts, with its painted surface facing down. Attach an 8.5 x 11 inch paper to the top surface of the mapping board. Connect the galvanometer between one of the resistor string jacks and the binding post on the special probe.

Clear plastic templates are available which enable you to accurately trace the conductor pattern onto your drawing paper. These templates are placed on top of the apparatus, and fit on two registration pins. Remove them before taking measurements.

For consistent results the probe must be used carefully. It is like a two-tined fork. One tine has a raised metal button which must make contact with the surface of the graphite pattern board. The other tine has a small hole located directly above the button, through which you can mark points on the drawing paper. The probe must be held in such a way that the button makes contact with the pattern board. Gently slide it along until the galvanometer reads zero, indicating that you have found a point at the same potential as the jack on the resistor string. Find the locus of all such points of the same potential, marking each on the drawing paper. These points trace out an equipotential line. Map the line all the way until it closes on itself, or terminates at the edge of the pattern board.

Do the same for all seven equipotential lines.

B. Pasco and Welch conductive paper sheets: The method is the same as for the Cenco apparatus except that you can create your own design of conductive patterns with metallic ink on graphite paper. [The conductive paper has a resistance of about 1000 ohm between points 1 cm apart. A typical silver paint line has a resistance of 5 ohm/cm.]

If you make your own pattern, you'll need to know how to use the conductive paints. Pasco conductive paint comes in several forms. One is a "pen" with silver paint. Another is a graphite suspension in a squeeze bottle or can. A liquid suspension of copper particles or nickel particles is less expensive, and is sold by electronic parts suppliers (for circuit board repair.) In any case, the conductive lines need only be about 1/4 inch wide. Thickness is more important than width. Do not waste the conductive paint by painting in large solid areas. The electrons on a closed conductor reside at its surface, whether the inside is solid or hollow. So make them hollow.

The paints dry rather quickly, in 10 to 15 minutes. As they dry, the metallic particles, which are small flat flakes, settle upon each other in overlapping layers to form a good conductive strip.

Agitate the conductive ink pens before class until the agitator ball moves freely, and agitate again for at least 10 seconds before drawing each line. Stir the bottles of metallic paint before each use.

An insulating area may be created on the graphite paper sheets by cutting a hole in the sheet with scissors.

A potential divider may be made of a string of equal resistors. An alternative potential divider is a rheostat or potentiometer (pot). A high impedance voltmeter must be used to read the potential output of the divider and set it precisely. This method has a great advantage over the resistor string, for one can choose any equipotential one needs, as when one wants to measure an equipotential deep inside the Faraday ice pail pattern.

5. PATTERNS ESPECIALLY WORTHY OF INVESTIGATION:

(1) Conducting and insulating sphere. This pattern (Fig 3, II) shows two circles, one an uncharged conductor, one an insulator. Be especially careful to accurately plot the equipotentials near these circles, and take equal care in drawing the field lines there.

(2) Faraday ice pail. This pattern (Figs. 6 and 11) shows the effectiveness of metallic shielding, even when the shielding container is open at one end. The field inside is quite small.

In some cases the equal voltage increments suggested above will not show equipotentials in regions of special interest, especially where the field is weak. For example, in the Faraday Ice Pail (Fig 11), the equipotentials you draw will not go deep inside the pail. You can find equipotentials in such regions if you use smaller voltage increments, and perhaps a more sensitive detector. Do this, in order to obtain sufficient information to calculate the field strength within the pail, say at a point 1/3 of the pail's depth measured from its bottom, as shown in Fig. 11.

(3) Charged conductor within a charged shell. Fig. 12 gives you an idea of the interesting possibilities you might invent. Attach the voltage source at points A and B.

6. ANALYSIS:

(1) Trace all of the equipotential lines fully, all the way to the edges of the paper.

(2) Use a contrasting color of pen or pencil to construct a set of electric field lines on the same sheet of paper as the equipotentials. Use the physical principle which says that the field lines always cross equipotentials at right angles. Perfect conducting surfaces are equipotentials.

(3) The field strength at a point between two adjacent equipotentials may be found approximately by measuring the distance of separation of the equipotentials at that point, then taking the ratio ΔV/Δs, where ΔV is the potential difference of the two equipotentials and Δs is their distance of separation. Use this fact to find the field strength at three different points on each of your field maps. Choose `physically interesting' points, and include points of both strong and very weak field.

7. QUESTIONS

(1) What do your equipotential line maps tell you about the potential within a conducting body? Within an insulating body?

(2) What do your field line maps tell you about the field strength within a conducting body? Within an insulating body?

(3) Give a simple statement describing how the field lines and equipotentials behave near (at the boundaries of) conducting bodies. Why must this be so?

(4) You probably noticed that the equipotential lines deviate from the "textbook" patterns near the edges of the conductive paper. Suggest a physically plausible reason for this.

(5) Give a simple statement describing how the field lines and equipotentials behave near (just outside the boundaries of) insulating bodies. Why must this be so?

(6) Prove, from basic field theory, that the entire volume of a conducting body is an equipotential volume when the charges are in electrostatic equilibrium (not moving in the macroscopic sense).

Text and drawings © 1995, 2004 by Donald E. Simanek.