Standard experiment report
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.
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
F[pic]r21 (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
E[pic]r (3)
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=-[pic]Fds. The force provided by an electric field however is qE, giving the potential to be -[pic]qEds. The mechanical work Wa→b therefore (the work done in moving the test charge from a to b) is given by the equation
-Wa→b [pic]Eds (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,
Va→b =[pic]Eds (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
E=-[pic]i -[pic]j - [pic]k (5)
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.
[pic]
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 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
Instrumentation
i) Rectangular Perspex dish.
ii) Graph paper.
iii) Water.
iv) Two pairs of electrodes.
v) Oscillator.
vi) Resistors.
vii) Oscilloscope.
Diagram of the experimental setup
Plan of Measurements
1) 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.
3) 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.
5) For ‘set one’ place a metal bar in the positions indicated by the dotted lines. Record the equipotential lines.
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.
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 lab-book) 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)
2] (29/11/2009)
3] (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.
[pic]
[pic]
[pic]
[pic]
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Fig. 2 The set referred to as ‘set one’ Fig. 3 The set referred to as ‘Set two’
[pic]
Fig. 1Two diagrams graphically displaying the electric fields produced by two charges[2].
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