Download Gauss` Law (Field mapping)

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Electric Potential Mapping
Equipment
Fluke Multimeter, Anatek power supply, field probe, field mapping apparatus, concentric
ring field board, parallel plate field board, connecting cables, field mapping graph paper, carbon
paper, tape, calipers.
Preparation
Review the concept of electric field and potential.
Purpose
To study and understand electric fields and potentials.
Theory
Gauss' Law states that the electric flux ϕE out of any closed surface is proportional to the
total electric charge qtot inside the surface. The charges inside determine the electric flux out of
this surface. The electric flux out of the closed surface is given by
ϕ E = ∫ E ⋅ dA .
(1)
Gauss' Law then says
ϕE =
qtot
ε0
,
(2)
where εo = 8.854×10-12 C2/N⋅m2 is the permittivity of free space.
The experimental arrangement is depicted in Figure 3. A power supply is used to set the
potential of various connectors on the field mapping apparatus. A field mapping board is selected
and installed onto the mapping apparatus. Different boards with differing field configurations are
available. So the electric field is determined by the type of field mapping board that is installed.
This electric field is "mapped" with a voltmeter.
The field mapping board with two concentric rings corresponds to a small solid
conducting cylinder of radius ro surrounded by a larger concentric
cylinder of radius r1 as shown in Figure 1. In the field mapping board the
cylinders consist of thin pieces of metal plated onto the surface of the
r
board. The rest of the board is coated with a mildly conducting material.
r
This conducting material confines the electric field to the surface of the
0
field mapping board. The effect is to produce the same behaviour as
would be observed in the cross-section of two tall concentric cylinders.
r
In the present situation, we select a Gaussian surface that is a cylinder
1
concentric with the other two cylinders and note that the ends make no
contribution because the electric field is confined to the field mapping
Figure 1
board surface. We immediately obtain
Er =
λ
,
2 πε 0 r
(3)
where Er is the radial electrical field and λ is the charge per unit length. Letting Vo = 10V be the
potential on the inner cylinder, we then have
r
Vr − V0 = − ∫ Er dr = −
r0
r 
λ
ln   .
2πε 0  r0 
(4)
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Similarly, the parallel plate field board
behaves identically to the cross-section of two tall
conducting parallel plates, as shown in Figure 2. Here,
the electric field, E, between the plates is given by
E=
−σ
ε0
,
(5)
where σ is the charge per unit area of the plate. Let
the left plate have potential V0=0. Let the points of the
field on the dotted line a distance, x, from the left plate
have potential Vx. Then
x
Vx − V0 = Vx = − ∫ E dx =
0
σ
x
ε0
-σ
+σ
-
+
+
+
+
+
+
+
+
+
+
x=0
(6)
x=d
x
Figure 2
is the potential at x. Here, the potential varies linearly with distance as opposed to logarithmically
with distance as in the cylindrical case. Substituting x=d into Formula 6 gives the potential
between the plates, which is
Vd =
σ
d.
ε0
(7)
Voltmeter
+
-
Field Mapping Probe
Equipotential Reference Connectors
Field Mapping Board
Power Supply
-
+
+
-
10.0 V
Figure 3
Experimental Procedure
1) Set the power supply to 10.0V. Connect the power supply to the field mapping apparatus using
Figure 1, you do not need to connect the field probe yet. Measure the voltage of each of the
seven equipotential reference connectors at the top of the field mapping apparatus with respect to
the minus terminal of the power supply. Also, measure the voltage of each of the two electrodes
underneath with respect to the minus terminal of the power supply. Then turn off the power
supply.
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2) Connect the concentric ring field board to the bottom of the field mapping apparatus using the
supplied thumbscrews. The board should be oriented so that the exposed metal electrodes are
facing down towards the tabletop.
3) Fix graph paper to the top of the field mapping apparatus using the alignment pegs. Fasten
the corners with tape. Make sure the electrodes pictured on the graph paper are the same ones
that are on the field board on the other side. You can make multiple copies of the field by using
several sheets of graph paper separated by carbon paper.
4) Connect the field probe as shown in Figure 3, with the negative terminal of the voltmeter
connected to reference connector #1, and turn on the power supply. The voltmeter will read zero
voltage whenever the probe is on an equipotential line of reference connector #1. Using the probe
and meter, pencil in dots at distances not greater than one centimeter along the equipotential line.
When finished, connect the dots with arcs to obtain the equipotential circle. Repeat this procedure
for at least the first five reference connectors.
5) Now connect the negative of the meter to the negative of the power supply so that it reads
absolute voltage and not equipotential voltage. On the graph paper draw a line from the center of
the inner circle to the outer circle. Take measurements of the field voltage at various positions
along this line. Measure the postions on the line using calipers.
6) Repeat steps 2 through 4 for the parallel plate field mapping board to obtain the equipotential
lines. Use all of the reference connectors this time. Draw a line between the parallel plates at their
center. Measure voltages along this line and their corresponding positions.
Error Analysis
The uncertainty in measuring the potential is the instrument error of the Fluke multimeter
as described on the bottom of the instrument. Distance and position measurements will have an
error determined by the instrument error of the calipers. The error in positioning of the
equipotential line depends on the registration accuracy of the field probe underneath with the
pencil tip on top, the care with which you mark points, the resolution of the voltmeter near zero
potential, and the error of the volt meter. In general it can be assumed that this error is specified
by a circle centered at the pencil mark with a given radius. You will need to decide on a
reasonable value for the radius of this circle.
Assume that the error in r0 is negligible. This means that the propagated error in the
logarithmic function is given by
 r  ∆r
∆ ln   =
.
r
 r0 
(8)
To be handed in to the laboratory instructor
Prelab
1. A complete derivation of Equations 4 and 7.
2. The name of this lab is somewhat of a misnomer, since you will not actually be mapping
electric fields lines, but equipotential lines, i.e. lines joining points that are at the same potential.
However, the two are closely related. The figure below shows two metal electrodes, one circular
and the other planar, which are held at different potentials. The circular electrode is at the higher
potential. Copy the figure into your notebook and sketch some electric field lines between the two
(indicating the direction of the electric field) and sketch also some equipotential lines in the region
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between the two electrodes. Try to reproduce as accurately as you can the geometrical
relationship between the field lines and the equipotentials.
Data Requirements
3. A sheet of graph paper for the concentric ring field with measured equipotential lines and some
error bars. It is sufficient to only put one or two error bars on an equipotential line so that the
figure does not get cluttered. Label the potential of each equipotential line together with the error.
Show the electric field lines and indicate the electrodes and their potentials.
4. A table below with values for r, Vr, and ln(r/r0) together with their errors for the concentric ring
board from procedure step 5.
5. A graph of Vr versus [ln(r/r0)] with error bars if these are large enough to show up.
6. A sheet of graph paper for the parallel plate field with measured equipotential lines and their
error bars. Again, only one or two error bars are needed on each equipotential line. Label the
potential of each equipotential line together with the error. Show the electric field lines and
indicate the electrodes and their potentials.
7. A table with values for x and Vx and their errors for the parallel plate field board.
8. A graph of Vx versus x on graph paper.
9. Equation for the line of best fit for the graphs drawn in steps 5 and 8. A numerical value for λ
and σ in correct SI units.
Discussion
10. Discuss whether the results support or refute the theory of electric fields.