Presentation by Robert Mines

PH 202 Lab

VERIFICATION OF FARADAY’S LAW OF INDUCTION USING CONCENTRIC SOLENOIDS

• 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

• 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

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.

• 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.

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

• 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

EXPERIMENTAL SETUP

• 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

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.

DETERMINING THE THEORETICAL EMF• The slope of the linear fit is equal to , and from this and its uncertainty can be calculated:

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:

DETERMINING THE THEORETICAL EMF• Now the area of the secondary solenoid must be calculated:

DETERMINING THE THEORETICAL EMF• Now, we can calculate the theoretical EMF form the data presented:

\S

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:

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.

CONCLUSION

• These values were consistent at 0.577 standard deviations.

• Accordingly, this result verifies that:

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.

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.

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.