Equipotentials
EM06
(C. Davies, 29/11/2009)
This aim of this experiment was to identify the electric field patterns caused by different
electrode configurations. This was successfully done, by deductions from the positions of
the equipotential lines, and the maximum electric field strengths for each respective
configuration were found to be 0.46Vm-1, 1.2Vm-1, 0.34Vm-1, and 0.40Vm-1. It was
further discovered that the electric field strength is strongest near sharp metallic rods, and
charged particles can be deflected by the presence of sharp metallic rods.
Introduction
The aim of this investigation is to measure the equipotential lines associated with
different electrode configurations, in order to study electrostatic field patterns. It is
intended that the link connecting electric potential and the electric field vector will be
identified: this will have important implications in the study of electromagnetism.
Theory
The relationship between the electrostatic force acting on a point charge is
F = q.E
(1)
where the F, q, and E define the electrostatic force, the positive charge of the test
charge, and the electric field respectively. Note a bold font denotes a vector. This
relationship itself raises the question of why the electric field is a vector: the answer
can be deduced from equation (1). Force is a vector quantity, as it has a magnitude
and an associated direction. The charge however is a scalar quantity, for only one
value is needed to describe it completely. For the product ab to give a vector c,
therefore, when a is a scalar, it is necessary for b to be a vector. By direct comparison
with equation (1), the electric field must be a vector quantity. If the charge q is
positive, the electric field acts in the same direction as the electrostatic force: if the
charge is negative, F and E act in opposite directions.
The electric field pattern can be shown graphically.
Fig. 1Two diagrams graphically displaying the electric fields produced by two charges[2].
Notice in the above diagrams the arrows represent the direction of the electric field
lines. It is a convention that electric field lines point away from positive charges, and
towards negative charges. Field lines never cross each other, they are always
perpendicular to the surface of a charged particle, and the spacing between the lines
indicates the strength of the field: the closer the lines are, the stronger the field is[2].
Consider two particles of charge q1C and q2C: Coulomb’s Law states the
electric force created by this charge is
1
q1 q 2
r21
(2)
40 r 2
Where r denotes the distance between the two charges, and r21 represents a unit
vector. By equating equations (1) and (2), the following equation is derived
Q
E
r
(3)
4 0 r 2
Notice the elimination of a charge from this equation: Q in equation (3) represents the
charge responsible for the electric field produced.
An electric field causes there to exist a force on a test charge. Note the
definition of a force is F=-dU/ds where ‘s’ denotes the displacement across which the
potential energy spans, and ‘U’ the potential energy of the test particle. Rearranged,
this give U=-  Fds. The force provided by an electric field however is qE, giving the
F
potential to be -  qEds. The mechanical work Wa→b therefore (the work done in
moving the test charge from a to b) is given by the equation
b
-Wa→b   q Eds
a
(4)
The electric potential is defined as the potential energy divided by the charge. This
can be incorporated in to equation (4) by dividing both sides by q. If the electric
potential is denoted by V,
a
Va→b =  Eds
b
(4)
When a test charge moves between two points of different potentials, the test charge
will move towards the point with the lower potential, for as can be seen from equation
(4), a greater potential results in a greater electric field, and thus a greater force being
exerted on the test charge. For a known potential distribution, therefore, the electric
field strength (expressed in terms of the spatial co-ordinates x, y and z) can be
determined[1] by
V
V
V
E=ijk
(5)
x
z
y
The electric field is always perpendicular in direction with respect to the equipotential
lines: this can be most easily understood by recognising that the potential around a
charged particle is constant at a certain radius.
Fig. 2 A diagram showing the field lines (arrows) and equipotential lines (dashed)
If a perfect metallic conductor has an external electric field applied to it, the
electric potential anywhere in the conductor is the same {1}, and the electric field
very close to the metal’s surface is perpendicular to the metal’s surface {2}. These
effects are caused by the free charges within the metal moving so as to compensate
2
the electric field inside the material. This could be proven by placing a test charge
close to various points of the metal’s surface: if consequence {1} is true, the charge
will respond in exactly the same way wherever it is positioned, and {2} could be
proven by observing the test charge moving in a perpendicular direction with respect
to the metal’s surface.
Experimental
1)
Instrumentation
i)
ii)
iii)
iv)
v)
vi)
vii)
2)
Rectangular Perspex dish.
Graph paper.
Water.
Two pairs of electrodes.
Oscillator.
Resistors.
Oscilloscope.
Diagram of the experimental setup
Fig. 2 The set referred to as ‘set one’
Fig. 3 The set referred to as ‘Set two’
Plan of Measurements
1)
2)
3)
4)
5)
Place ‘set one’ and ‘set two’ in the tank of water, and apply a frequency of
3kHz. Note ‘set one’ should not have a bar in the middle.
Change R1 and R2 so that their sum is a constant 10kΩ.
Move the wire probe around the tank, and detect the locus of points which
provide a minimum in the 3kHz frequency displayed on the oscilloscope.
Plot this set of points, along with the position of the electrodes.
For ‘set one’ place a metal bar in the positions indicated by the dotted lines.
Record the equipotential lines.
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Error is going to be introduced by the plotting of the equipotential lines. To plot these,
reference will need to be made to the graph paper under the tank of water, and due to
the refractive properties of water, it will be difficult to measure the position of the
equipotential lines exactly. This can however be minimised by looking perpendicular
to the water surface: this removes the source of error introduced by refraction.
Another source of error will be the oscilloscope; to measure the position of the
equipotential lines, the oscilloscope will need to display a minimum. Due to the error
within the oscilloscope, the wire probe could be moved a small distance, and no
discernible effect would be observed on the oscilloscope’s display.
Safety Procedures
Care will need to be taken to ensure no loose electrical wires come in contact with the
water, as electrocution could occur.
Results
The graphs constructed for this experiment are given in the lab-report dated 31/11/09,
along with the calculations for the maximum electric field strength. For ease of
reference however, a sketch of the electric field lines determined for each
arrangement of the electrodes are provided in the appendix.
It must be noted however that the experimental procedure was modified. Due
to the type of oscillator used, the oscilloscope reading was unaffected when the
fraction in resistance was altered. For a constant frequency, therefore, the positions of
equipotentials were determined for each electrode configuration, but the fractional
resistance was not changed. This has had little impact overall on the conclusions
formed on the basis of this experiment.
Discussion
For each graph, error bars have been provided for several representative data points.
Experimental error occurs when potential, in a region, remains constant, and therefore
when the gradient of the potential equals zero (dV/ds=0). The sources of this error
have already been described, in the ‘plan of measurements’.
The equipotentials were plotted on the graphs, using the oscilloscope and
probe. It is known that the electric field lines are always perpendicular to the
equipotentials: therefore small lines perpendicular to each equipotential were
constructed, and then joined to form an electric field line. The electric field lines
always have a direction which heads towards the lower potential, as this corresponds
to a decrease in potential energy, and thus an increase in kinetic energy, for a
positively charged particle.
For the second graph, the presence of the bar distorts the equipotential lines.
This is because the bar has an electric field, and the bar is positioned vertically, so its
equipotential lines are perpendicular to the equipotential lines created by the two
original bars. The equipotential lines cannot cross though, for they are boundaries of
potential areas, and so the equipotential lines must change. This simultaneously
explains why the equipotential lines are unaffected when the bar is positioned
horizontally: the equipotential lines of the bar are parallel to those already there, and
so no ‘interference’ occurs.
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Near the bar surface, it must be noted that the equipotential lines are parallel to
the surface, and the field lines, perpendicular. This was predicted earlier, and the
verification of this indicates this experiment has been at least partially successful.
It can be seen in all the graphs that electric fields are strongest near sharp
edges. In the first and third graph, the potential gradient- the electric field- is greatest
at the base of the ‘V’ shaped electrode. In the second graph, the electric field is at its
strongest not only at the base of the ‘V’ electrode, but at the edge of the base of the
vertically orientated bar. The fourth graph has the strongest electric field between the
edges of the ‘H’ electrodes. These points all corroborate each other, in confirming
electric fields have a maximum strength at sharp edges.
The second graph also demonstrates that sharp metallic rods can focus electric
field lines. The calculation of the maximum electric field strength (given in the labbook) shows it to be around 1.2Vm-1, whereas for the first graph, the maximum field
strength is approximately 0.46Vm-1. The field strength has approximately doubled,
which emphatically demonstrates the electric field lines have been focussed. If
therefore the charge of the rod was reversed to a negative one, it can be seen the
reverse would be true, and that sharp metallic rods can similarly divert electric field
lines.
It is worth explaining why no errors have been associated with the values
calculated for the electric field strength. As can be seen on the graphs, error bars have
been constructed. This is however individual for each point on the graph. Since the
gradient of the potential constantly varies (given that an equipotential line is not being
traced), the error in the gradient of the potential will constantly vary. For example,
consider graph 1. A point very close to the base of the ‘V’ shaped electrode will have
a high potential gradient: thus in one inch, vertically down, the potential has
decreased by 2V. The error will therefore be small, for the change is concentrated in a
small region. A point close to the other electrode however has a smaller potential
gradient, which results in the condition dV/ds=0 being satisfied over a greater
distance, and thus the error will be larger. Due to such variability in the error, no error
has been associated with the values calculated.
Conclusion
This experiment has successfully shown that electric field lines, and therefore charged
particles, can be focussed or diverted by sharp metallic rods, and that electric fields
are strongest at sharp edges. These points are both expected from the theory of
electromagnetism, for sharp metallic rods have their own electric field patterns, which
will logically deflect charged particles, and similarly, electric field lines are always
perpendicular to a material’s surface, so a sharp edge will cause the field lines to be at
their closest, giving a maximum field strength. The electrode configurations studied in
this experiment could be used for chemical analysis or medical purposes: this
experiment therefore acts as a platform on which further study could be done in to
specific applications.
References
[1] Young H. D. and Freedman R. A. (2007), University Physics (with Modern
Physics) (12th edition), Addison-Wesley, ISBN 0-321-4 (UL: 530 YOU)
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[2] http://www.physicspost.com/articles.php?articleId=164&page=4&show=all
(29/11/2009)
[3] http://newton.ex.ac.uk/teaching/resources/fyo/phy1110/manuscripts/em06.pdf
(29/11/2009)
Appendix
Contained are images of the final electric field patterns determined for each electrode
configuration. Note the arrows represent electric field lines.
First graph
Emax=1.2Vm-1
Second graph
Emax=1.2Vm-1
Third graph
Emax=0.34Vm-1
6
Fourth graph
Emax=1.2Vm-1
7