IPS*1510
Experiment #1: Electrostatic Field Mapping
Purpose:
-
to investigate the spatial dependence of the electric potential associated with different source
charge distributions
to investigate the spatial dependence of the electric field associated with different source
charge distributions
Equipment required:
-
two sheets of conductive paper
two pieces of cardboard with pre-drilled holes
two metal bar electrodes
one metal ring electrodes
a drafting board working surface
an electronic multimeter
a DC power supply
a ruler
sandpaper file for sharpening the probe tip
Introduction:
Electric potential is often an abstract and confusing concept. The idea of electric force is more
intuitive and its extension to electric field, force/unit charge, usually presents less conceptual
difficulty. These three quantities, however, are all inter-related, since electric potential is just a
mathematical construct that allows electric fields to be treated more easily. Whereas the field is a
vector quantity, the potential is a scalar and its mathematics is much simpler. Roughly, one can
image the potential to be like the slope of a hill — the steeper it is, the faster its value changes, the
more force is exerted. The force points in the direction of steepest descent. You may like to start
with this image of electric potential even though the analogy is not perfect. The equipotential plots
you produce in this experiment will resemble topographic contour maps.
There are several rules to keep in mind about electrical potentials:
- the value of the potential at a point is the work it takes to move a unit charge from an
arbitrary reference position to this point. Once the reference position is chosen it must remain
the same for the whole field.
- the potential on a conductor is everywhere the same if the charges are static.
- an equipotential surface is a three-dimensional surface on which the electric potential 𝑉 is the
same at every point (like the surface of a conductor). If a test charge 𝑞0 is moved from point
to point on such an equipotential surface, the electric potential energy, 𝑈 = 𝑞0 𝑉, does not
change.
Because the potential energy does not change as a test charge moves over an equipotential surface,
the electric field can do no work on the charge. It follows that 𝐸�⃗ must be perpendicular to the surface
at every point so that the electric force, 𝐹⃗ = 𝑞0 𝐸�⃗ , will always be perpendicular to the displacement
of a charge moving around on the surface. In other words:
Field lines and equipotential surfaces are always mutually perpendicular.
In this experiment, you will look at the field patterns for two different electrode configurations. In
reality, source charge distributions establish three-dimensional electric fields and potentials.
However, to simplify the procedure, we will be investigating the field patterns of electrode
configurations constrained to two dimensions, using special conductive paper. The edges of the
paper will affect the field, but if we stay several centimeters from the edge, this distortion will not be
significant. The fields you map will be very similar to their three-dimensional analogues. Note:
two-dimensional fields have equipotential lines instead of equipotential surfaces.
Uncertainty measurements
For the digital multimeter used in this experiment, the accuracy of the device is specified by the
manufacturer as 0.3% of the reading + 1 digit. Therefore, if your measurement is 0.946 V, the
uncertainty would be calculated as 0.3% * 0.946 = 0.002838 V. Then we add 0.001 as the “+1 digit”
component, to get 0.003838 V. Our voltage measurement is therefore 0.946 ± 0.004 V. If the
voltmeter reads 10.1 V, the uncertainty would instead be 0.1303 V (check this calculation).
Therefore our measurement is 10.1 ± 0.1 V.
Procedure:
Configuration #1: Parallel Plate Capacitor
1.
2.
3.
4.
Using the cardboard (the one with four holes), mark out the locations of the holes and
puncture the conductive paper with a pencil.
Place the cardboard then the conductive paper (dull side up), and finally the metal bars into
the appropriate slots on the drafting board. The bars should be aligned vertically as shown
on the next page, with the raised edge side against the paper. Secure the setup tightly with
the screws to eliminate any gap between the electrodes and the conductive paper.
Turn on the power supply. Adjust the current to 0.5 A and the voltage to 15 V. Do not
exceed these settings. Set the multimeter to the setting (this is volts DC).
Use the voltmeter ‘pen’ sensor to measure the potential at any point on the paper simply by
touching the pen to the paper at that point. To map an equipotential line, touch the paper at
some starting point and note the potential on the voltmeter. Now touch the paper at various
locations until the same potential is indicated on the voltmeter. Mark the paper at the points
of equal potential clearly with the probe tip. Continue marking the paper until the trend is
clear. Connecting the points produces an equipotential line.
5.
6.
7.
8.
9.
Produce several equipotential lines until you have a clear understanding of the nature of the
lines throughout the area between the plates. Be sure to continue the lines about a
centimeter above and below the ends of the plates. Note the potential on each plate as well.
Reproduce the equipotential lines you observed experimentally on Figure 1, indicating the
values of the potential at each electrode as well.
With a different colour, on Figure 1 draw a representative array of electric field lines
crossing the equipotential lines at right angles in the direction of decreasing potential.
Now measure the potential at 10 to 15 distances from the positive plate (the plate with
highest potential), along the centre line between the plates. Record your data in a table in
your lab book. Be sure to include calculated uncertainties.
When you have finished, turn off the power supply and remove the conductive sheet and
metal bar electrodes.
ℓ
𝑑
Figure 1: Experimentally measured equipotential and electric field lines
for the parallel plate configuration.
10. Plot the potential as a function of distance from the positive plate, along the centre line
between the plates, on the graph paper provided. Be sure to include error bars reflecting the
uncertainties of the measured values.
Questions:
Describe the plot of potential vs. distance. Along the centre line between the plates, the potential
acts as it would between infinite parallel plates. Does the graph agree with the trend predicted
for infinite plates?
What happens to the electric field at the edge of the plate region?
Configuration #2: Concentric circles
1.
2.
3.
4.
5.
6.
7.
8.
Punch out the necessary holes on the conductive paper as before, now using the cardboard
with five holes.
Fit the cardboard on to the slots on the drafting board, then the conductive paper and metal
ring on the very top. The ring should be positioned with the raised edge side against the
paper. Secure the setup with the five screws (including the slot at the centre of the
enclosed circle).
Turn on the power supply. Adjust the current to 0.5 A and the voltage to 15 V. Do not
exceed these settings. Turn on the multimeter to the setting (this is volts DC).
Measure the potential as a function of radial distance from the centre, at 8 to 10 locations.
Record your data in a table in your lab book. Be sure to include uncertainties in your table.
When you have drawn many equipotential lines, turn off the power supply and remove the
conductive sheet.
Reproduce the equipotential lines on Figure 2, indicating the values of the potential at each
electrode as well.
With a different colour, draw a representative array of electric field lines crossing the
equipotential lines at right angles in the direction of decreasing potential.
It can be shown that for this configuration the potential is given by:
𝑟
𝑉(𝑟) = 𝐶 ∙ ln � �
(1)
𝑟0
where C is a constant, and ro is the radius of the inner electrode/screw. Plot the potential on
𝑟
𝑟
a semi-log graph, as a function of �𝑟 �; remember to start the plot at �𝑟 � = 1. Be sure to
0
include error bars to reflect the uncertainties of the measured values.
0
Questions:
𝑟
Describe the plot of potential vs. �𝑟 �. Does this agree with the predicted trend in Equation (1)?
0
Based on the electric field lines drawn in Figure 2, where is the magnitude of the electric field
the strongest? Where is the magnitude of the electric field the weakest? Explain.
Figure 2: Experimentally measured equipotential and electric
field lines for the concentric circles configuration.
What your lab notebook should contain:
- title and purpose of the lab
- figures 1 and 2, taped in (do not use staples)
- data tables
- answers to the questions
- graphs
- appropriate conclusions
Voltage
(V)
𝑟
� �
𝑟0
Appendix
Graphing with Logarithmic Paper:
In science, logarithmic plots are used to visualize data with an exponential relationship, such as 𝑁 =
𝑁
𝑁0 𝑒 𝑎𝑡 . Taking the natural logarithm of both sides and plotting ln �𝑁 � versus 𝑡 yields a straight line of
slope 𝑎, as shown below.
0
𝑁
Sometimes it is a nuisance to look up a bunch of logarithms of values of 𝑁 , however semi-logarithmic
0
and logarithmic paper does this automatically. A sheet of semi-logarithmic paper is depicted in panel 2,
below.
Notice that the horizontal axis is linear but the vertical axis is logarithmic and goes from 1 to 10 (one
cycle). Hence the name “semi-logarithmic paper” or, more commonly, “semi-log” paper.
𝑁
Panel 3 provides some values of 𝑁 that obey an exponential relationship. In the right-hand column are
𝑁
the natural logarithms of 𝑁 .
0
0
A plot of the data is shown in Panel 4. Notice that the graph is on normal graph paper, not semi-log
paper. We’ll use semi-log paper in a moment. You can see that the graph is a straight line and its slope
(a) is constant and can be deduced from the graph.
Now let’s examine how the semi-log this as shown in panel 5. The same data in panel 3 is used here, and
𝑁
since out 𝑁 data is all between 1 and 10, we can use the numbers on the left-hand edge of the graph as
0
they are. All we have to do is plot the numbers as given. We do not have to find logarithms; the paper
does it for us. You have to watch out how the paper is subdivided, though. In this example, it is
subdivided in increments of 0.1, from 1 to 3, but in divisions of 0.2 from 3 to 5 and 0.5 from 5 to 10.
This time determining the slope was easier as only one logarithm needed to be calculated; the difference
of two logs is the log of the quotient. This method, yields the same value of a as before, a = 0.23 s-1.
𝑁
Panel 6 depicts a similar situation; the values of t are the same, but the values of 𝑁 are 10 times larger
0
than before. What do we do now? The answer is that the decade over which the axis with the logarithmic
scale runs is quite arbitrary. It can be 1 to 10 as in the previous example, or it can be 10 to 100 which is
now the case, or 100 to 1000, or 0.1 to 1, and so on as long as the plot starts on 10𝑛 where 𝑛 =
… , −3, −2, −1, 0, 1, 2, 3, …
It is also sometimes necessary to have multiple logarithmic cycles in order to accommodate for the range
of the data acquired. This is the case with the example shown in panel 7 below. You choose the number
of cycles in the graph paper you use to match your data; semi-log paper comes in one, two, three, …,
cycles.