Lab 1     Field Plots

In this lab conductive sheets of paper are used to measure voltage as a function of position. That is, the voltage field produced by various configurations of charged electrodes is measured.

Electrodes are drawn on conductive paper using conductive ink. A power supply is used to establish voltages between electrodes. Current flows through the conductive paper, setting up a voltage field between the electrodes. A voltmeter measures values of the voltage field at positions between electrodes.

Knowledge of the voltage field allows the electric field and the current field to be calculated.

Equipment

• Conductive paper
• Conductive ink
• Corkboard
• DC Power Supply
• Voltmeter
Pre-drawn electrodes Electrodes represent a coaxial cable Electrodes represent a parallel plate capacitor

Plotting equipotential lines

• Use stickpins to mount a predrawn plot of two concentric circles representing the cross section of a coaxial cable on the corkboard.
• Using stickpins, connect the positive terminal of the DC power supply to the inner electrode and the negative power supply to the outer electrode. Use the voltmeter to check the connection. Measure the voltage along the electrode. It should be nearly constant and equal to the power supply terminal voltage.
Current flows from the positive power supply terminal through the wire and contact to the drawn electrode. Current then flows through the resistive paper to the outer electrode, through the contact to the wire and back to the negative power supply terminal.
Due to resistance, there will be a voltage drop at the contacts and along the electrodes. We want to ignore these voltage drops. They should be small compared the voltage between the electrodes.
• Use the voltmeter to measure values of the potential field. Connect the voltmeter common wire to the outer electrode. Measure the voltage at the inner electrode and at centimeter intervals from the center along four radial lines.
The theoretical expression for the voltage field between the inner conductor of radius, a, and the outer conductor, radius b, of a coaxial cable is; where r is the distance from the center and VBB is the power supply voltage.
• Use the voltmeter to determine lines of constant potential. Draw constant potential lines with values of 0.25 * VBB, 0.50 * VBB, and 0.75 * VBB on the conductive paper with a pencil.

Electric field

The electric field is a vector. It is the negative gradient of the potential field. The x component of the electric field is, The units of the electric field are volts/meter.

Measure the potential at a point, then move, say one centimeter, and measure the potential again. The component of the electric field in the direction you moved is the change in voltage divided by the distance moved.

The electric field is a vector, with three components. (We are concerned with only two components since we are working in only two dimensions.) The magnitude of the electric field is the maximum directional derivative of the voltage. The electric field is always in the direction of the maximum decrease voltage. Voltage determines electrostatic potential energy and charges experience forces that move them in the direction of lower potential energy.

Determine the magnitude and direction of the Electric Field 2.5 cm, 5.0 cm and 7.5 cm from the center.

Electric field lines

A complete field plot includes electric field lines. Electric field lines cross constant potential lines at 90o. The curvilinear rectangles formed by the electric field lines and the constant potential lines have approximately the same aspect ratio everywhere. The magnitude and direction of the electric field can be read from the electric field lines. The direction of the electric field at any point is tangent to the field line at that point. The density of the field lines gives the magnitude of the electric field. The field is stronger where the field lines are closer together.

Electric field lines are perpendicular to constant potential lines because the maximum change in potential is in the direction perpendicular to a constant potential line.

Once constant potential lines have been determined, electric field lines can be drawn.

1. Draw electric field lines so that they cross constant potential lines at 90 o.

2. Draw electric field lines so that all curvilinear rectangles formed have approximately the same aspect ratio.

• Draw electric field lines on the constant potential plot.

• Obtain a field plot for the parallel plate capacitor. Draw constant potential lines and field lines. Of particular interest is the fringing field at the left and right sides of the parallel plates where the field deviates from theoretical values.

The theoretical expression for the potential field is, where VBB is the voltage of the upper plate relative to the lower plate, d is the distance between plates, and y is the distance of a field point from the lower plate.

• Experiment with other electrode configurations, as time permits.

Drawing electrodes

• Place the conductive paper on a smooth hard surface.
• Vigorously shake the conductive ink for 10-20 seconds.
• Practice on a scrap of paper. Press lightly down on the spring-loaded tip while firmly squeezing the pen barrel. Draw the pen across the paper. Drawing speed and exerted pressure determines the line width.
• Next, draw electrodes on the grid of the conductive paper. If the line is thin, draw it again. A solid line is essential for good measurements.

The line will dry in 3-5 minutes, but it will take 20 minutes to reach maximum conductivity.

Mount the conductive paper on the corkboard using stickpins at each corner.

Question to keep in mind

1. What is your experimental error in each situation?
2. What factors contribute to error?
3. How do measurements compare with theoretical values?