A Lab Experience with Deriving Faraday's Laws
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Transcript of A Lab Experience with Deriving Faraday's Laws
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Presentation by Robert Mines
PH 202 Lab
VERIFICATION OF FARADAY’S LAW OF INDUCTION USING CONCENTRIC SOLENOIDS
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• In this experiment, the induced electromotive force (E) and the time rate of change in current ( were measured directly.
• Using equations derived from Faraday’s law, was used to calculate E.
• Then, using propagation of errors the measured and theoretical values were tested for consistency to verify Faraday’s Law.
INTRODUCTION
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• In 1831, Michael Faraday noticed unusual behavior between magnets and coils of wire:
• If a magnetic field passed through a loop of wire, an EMF would be induced.
• If the field passed through the wire in the opposite direction, an EMF of equal magnitude but opposite sign would be produced.
• If the current in a coil of wire was changed, an EMF could be induced in another wire.
QUALITATIVE EXPLANATION OF FARADAY’S LAW
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QUANTITATIVE DESCRIPTION OF FARADAY’S LAW• Based off his qualitative experiments, Michael Faraday derived the following result:
• N = The number of turns in wire coil.
• = The time derivative of magnetic flux.
• B = The Magnetic Field Vector Magnitude
• A = Area through which B passes.
• = The angle between the area and field vectors.
• In other words, a magnetic flux induces an EMF and a current that serve as an electromagnetic inertia to resist changes in the circuit’s environment.
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• In this experiment, a current was applied to a solenoid, and this induced a magnetic field in a smaller coaxial solenoid.
• The magnetic field of any current can be determined using Ampere’s Law:
DETERMINING THE MAGNETIC FIELD OF A SOLENOID
+ + + = =
where = , I is the applied current, and n is the number of wire coils per unit length.
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INDUCED EMF DUE TO A CHANGING CURRENT
• Inside of the solenoid the area vectors and field vectors are essentially parallel at every point, so
• Setting the number of coils in the secondary solenoid as N2 and applying Faraday’s Law, we find
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• Science Workshop 750 Interface (CI-6565 A)
• Voltage Sensor (CI-6503)
• Primary and Secondary Coil (SE-8653)
• Patch Cords with Banana Plugs
• Personal Computer
• Power Amplifier II (CI-6552A)
• Digital Multi-Meter (1 Ω)
• Digital Caliper (0.01 mm)
REQUIRED EQUIPMENT
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EXPERIMENTAL SETUP
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• Using the digital multimeter, the resistance across the outer solenoid was measured.
• Uncertainty for this value was taken to be 1%.
• Using the digital caliper, the length of the outer solenoid was measured.
• Uncertainty was estimated since electrical tape obscured the end of the solenoid.
• The inner and outer diameter of the secondary solenoid were measured using the digital caliper.
• A separate value for uncertainty was calculated later.
• Number of turns was specified by the manufacturer.
Quantity Measurement
Resistance (R)
Length (L)
Outer Diameter (Dout)
Inner Diameter (Din)
N1
N2
PHYSICAL PROPERTIES OF THE SOLENOIDS
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SOFTWARE SETUP AND DATA COLLECTION• Secondary solenoid was inserted into the primary.
• Voltage sensors and circuit connected to data studio and power amplifier.
• In Data Studio, a voltage ramp up wave was generated with an amplitude of 9.60 V and a frequency of 260 Hz.
• The resulting induced EMF in the secondary coil was measured by the voltage sensor.
• Using the oscilloscope tool, Data Studio plotted the applied voltage and induced voltage.
• The data was separated into two plots.
• On the applied voltage plot, a linear fit of voltage vs time was generated.
• The measured voltage/EMF was taken as the average of the points that asymptotically approached the maximum possible induced voltage. Statistical uncertainty was calculated.
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DETERMINING THE THEORETICAL EMF• The slope of the linear fit is equal to , and from this and its uncertainty can be calculated:
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DETERMINING THE THEORETICAL EMF• Now, the number of turns per unit length of the primary solenoid “n” and its uncertainty must be
calculated:
• Next, the average diameter and uncertainty of the secondary solenoid must be calculated:
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DETERMINING THE THEORETICAL EMF• Now the area of the secondary solenoid must be calculated:
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DETERMINING THE THEORETICAL EMF• Now, we can calculate the theoretical EMF form the data presented:
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\S
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MEASURING EMF FROM DATA• The last graph shows the induced EMF compared to time.
• Using the data selection tool, a series of points asymptotically approaching the maximum induced voltage was selected.
• The statistical package in Data Studio determined that this constituted 27 data points with a mean E = 0.060 V and a standard deviation of 3.970 X 10 -3 V.
• Statistical uncertainty was calculated for the measured E:
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COMPARISON OF ERRORS
• Now, the theoretical and measured values was tested for consistency using comparison of errors:
• Since the values agreed within 3 standard deviations, the values are consistent.
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CONCLUSION
• These values were consistent at 0.577 standard deviations.
• Accordingly, this result verifies that:
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SOURCES OF ERROR: LIMITATIONS OF MEASURING EQUIPMENT
• The caliper could not be used to measure the exact length of the solenoid since electrical tape used as insulation obscured the location of the end of the wire.
• The Digital Multimeter fluctuated substantially when measuring the resistance across the solenoid depending on how much force was applied and where the contact occurred.
• Also, measuring the inner diameter was hindered by the support structure.
• Last, we assumed that there was no uncertainty in the number of coils provided by the manufacturer.
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SOURCES OF ERROR: THEORETICAL ISSUES• First, we assumed that all of the magnetic field lines were parallel to the area vector:
• The field actually has a slight curvature in the solenoid, so it may have actually been less than we theoretically predicted as is consistent with the measured value being less than the theoretical value.
• Second, we assumed that there was no external magnetic field.
• In all reality, the solenoid produces an external magnetic field, and we cannot go infinitely far from it when in a real situation.
• This external magnetism would be opposite in sign to the first portion of the path decreasing the observed field vector accordingly decreasing the observed induced EMF.
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REFERENCES
• “Experiment 6: Faraday’s Law.” Physics Experiments for PH 201 and 202. 4th ed. University of South Alabama Department of Physics. Mobile, AL: Department of Physics, 2010. 152-159. Print.