Experiment E15b – Induction Law IIName: Chow Ka HoSID: 52591419

Objective:I. To measure the dependence of induction voltage on

a. The frequency of the exciting fieldb. The amplitude of the exciting voltagec. The number of turns of the coils

II. To verify the voltage induced in a coil is proportional to the rate of change of the magnetic field

Equipment:A pair of SolenoidsFunction Generator1-Ω ResistorAutomatic data loggerMicrovoltmeterCoils (250 &1000 turns)

Metal rodStringStandsConnecting wiresIron coreTransformer

Theory:Faraday’s law of electromagnetic induction gives that the induced e.m.f (ε) is

ε=−d∅dt

Where Φ is the flux passing through conduction loop.

With a solenoid with n turns and area A, the relationship should become

ε=−nA dBdt

The magnetic field is produced by current I in a solenoid. Since the magnetic field is proportional to I, the expected induced voltage will be proportional to the rate of current change.

ε∝−dIdt

Moreover, the magnetic field is proportional to the number of turns of primary and secondary coil.

ε∝−N1∧−N2

These relationships will be verified in this experiment.Procedure:Part IA. Induced voltage as a function of frequency

a. A circuit was connected by following fig. 1b. Offset voltage on the Microvoltmeter was compensated.c. Function generator was set to saw-tooth output with frequency 0.1 Hz.d. It was adjusted to slowly increase the amplitude through the field coil.e. Microvoltmeter was set to amplify the signal 104 times.f. A clearly induced voltage could be read on the pc through the data logger.g. The induced voltage was recorded.h. Frequency was increase step-by-step while the field-exciting amplitude

keeping constant.i. Steps g and h were repeated for frequency 0.1 – 1 Hz.j. A graph of induced voltage against frequency was plotted.

Part IB. Induction voltage as a function of field-exciting current amplitudea. Experiment in Part IA was repeated with fixed frequency 0.3 Hz.b. Field-exciting amplitude UX was varied step-by-step.c. A graph of induced voltage against field-exciting amplitude was plotted.

Part IC. Phase difference between the induced voltage and the field-exciting currenta. Experiment was repeated with sinusoidal input voltage.b. Frequency of 0.2 Hz was chosen.c. A graph with input and output voltage was plotted.d. Phase difference between them was measured.

Part IIA. Induction with metal roda. The primary coil and function generator were removed.b. A metal rod was hanged on the stands by strings.c. The metal rod was adjusted to point to the inside of the solenoid.d. It was pulled outward slightly and released to perform a simple pendulum.e. The induced voltage was recorded.

Part IIIA. Induction with coils and iron corea. Experiment IA and IB were repeated with different coil.b. Primary coil with 1000 turns and Secondary coil with 125 turns was used.c. A bar of iron core was inserted into the two coils to amplify the signal.

Part IIIB. Induced voltage with difference numbers of turnsa. Experiment IA was repeated with fixed frequency and voltage.b. Numbers of turns of the primary and secondary coils were changed.c. Several combinations were obtained in this part.d. Graph of induced voltage and number of turns was plotted.

Result and Data analysis:Part IA. Induced voltage as a function of frequency

Frequency (Hz) Induced Voltage (ε/ μV)0.11 ± 0.0011 2.037 ± 0.020

0.201 ± 0.0020 3.429 ± 0.0340.292 ± 0.0029 4.792 ± 0.0480.426 ± 0.0043 6.802 ± 0.0680.497 ± 0.0050 7.908 ± 0.0790.598 ± 0.0060 9.409 ± 0.0940.706 ± 0.0071 10.92 ± 0.110.807 ± 0.0081 12.36 ± 0.120.905 ± 0.0091 13.70 ± 0.141.019 ± 0.010 15.17 ± 0.15

* Voltage and Frequency measured by electronic devices which have 1% random error.* The induced

0 0.2 0.4 0.6 0.8 1 1.20

20406080

100120140160

Induced Voltage against frequency

Frequency (Hz)

Indu

cced

Vol

tage

(ε/ μ

V)

Part IB. Induction voltage as a function of field-exciting current amplitude

Field amplitude (Ux/V) Induced Voltage (ε/ μV)8.900 ± 0.089 4.430 ± 0.04410.90 ± 0.11 5.335 ± 0.05313.00 ± 0.13 6.284 ± 0.06315.00 ± 0.15 7.355 ± 0.07417.00 ± 0.17 8.311 ± 0.08319.00 ± 0.19 9.362 ± 0.094

8 10 12 14 16 18 200

20

40

60

80

100

Induced Voltage against frequency

Field amplitude (Ux/V)

Indu

cced

Vol

tage

(ε/ μ

V)

Part IC. Phase difference between the induced voltage and the field-exciting current

0 2 4 6 8 10 12 14 16 18 20

Phase Difference between Inducced Voltage and Field-exciting Amplitude

Field-excit-ing Ampli-tude

Inducced Voltage

Time (T/s)

Vol

tage

Part IIA. Induction with metal rod

0 1 2 3 4 5 6 7 8 9 10

Induced Voltage using Metal Rod

Time (T/s)

Vol

tage

(V)

Part IIIA. Induction with coils and iron core

Frequency (Hz) Induced Voltage (ε/ mV)0.1020 ± 0.0010 0.8211 ± 0.00820.2000 ± 0.0020 1.460 ± 0.0150.2870 ± 0.0029 2.056 ± 0.0210.4090 ± 0.0041 2.879 ± 0.0290.5250 ± 0.0053 3.640 ± 0.0360.6920 ± 0.0069 4.708 ± 0.0470.8000 ± 0.0080 5.417 ± 0.0540.9100 ± 0.0091 6.075 ± 0.0611.012 ± 0.010 6.666 ± 0.067

0.0 0.2 0.4 0.6 0.8 1.0 1.20

1

2

3

4

5

6

7

Induced Voltage against frequency

Frequency (Hz)

Indu

cced

Vol

tage

(ε/ m

V)

Field amplitude (Ux/V) Induced Voltage (ε/ mV)5.000 ± 0.050 0.5530 ± 0.00557.000 ± 0.070 0.77 ± 0.00779.000 ± 0.090 0.981 ± 0.01011.00 ± 0.11 1.198 ± 0.01213.00 ± 0.13 1.404 ± 0.01415.00 ± 0.15 1.638 ± 0.01617.00 ± 0.17 1.858 ± 0.01919.00 ± 0.19 2.098 ± 0.021

0 2 4 6 8 10 12 14 16 18 200.0

0.5

1.0

1.5

2.0

2.5

Induced Voltage against frequency

Field amplitude (Ux/V)

Indu

cced

Vol

tage

(ε/ m

V)

Part IIIB. Induced voltage with difference numbers of turns

No. of turns of Primary Coil

No. of turns of Secondary Coil Induced Voltage (ε/ mV)

125 125 0.5580 ± 0.0056125 250 1.110 ± 0.011125 500 2.232 ± 0.022125 1000 4.373 ± 0.044

0 200 400 600 800 1000 12000

1

2

3

4

5

Induced Voltage against Number of Turns

Number of Turns

Indu

cced

Vol

tage

(ε/ m

V)

No. of turns of Primary Coil

No. of turns of Secondary Coil Induced Voltage (ε/ mV)

125 250 1.110 ± 0.011250 250 2.220 ± 0.022500 250 4.162 ± 0.0421000 250 7.653 ± 0.077

0 200 400 600 800 1000 12000

2

4

6

8

10

Induced Voltage against Number of Turns

Number of Turns

Indu

cced

Vol

tage

(ε/ m

V)

Discussion:In Part IA, a straight was observed on the graph of induced voltage against the

frequency. This shows that the induced voltage (ε) is liner proportional to the frequency of the field-exciting current. However, the graph did not pass through the origin, and this is due the uncertainty of the experiment.

In Part IB, a linear realtionship between the induced voltage (ε) and the field-exciting voltage UX are also found. These two relatioship verify that the expectation that the induced voltage should be proportional to the rate of current change.

ε∝−dIdt

In Part IC, the graph with field-exciting voltage and induced voltage was plotted. From the graph, the phase difference between two voltages was found to be about 90o. The induced voltage leads the field-exciting voltage because it is proportional to the rate of current change.

In Part IIA, we started to try something else. A metal rod was set to perform a simple pendulum, and it moved in and out periodically. From the graph, the period of the induced voltage was about 0.8sec. The string used in experiment was about 13cm long. The theoretical period can be calculated.

T=2π √ Lg=447 √ 0.13

9.78=0.72 sec

Which is similar to the experimental result, and the difference may be due to the repulsion and attractive forces from the solenoid while the metal rod moving.

In Part IIIA, two coils with an iron core inserted have been used. Despite that the used coil had a little change in turns and area. The induced voltage was increased about 1000 times. This can be some evidence which verify that permeability can affect induction.

In Part IIIB, graphs of induced voltage against number of turns were plotted. Linear relationship between the number of turns of the coils and induced voltage was

obtained. Induced voltage is linear proportional to the primary coil because of the

magnetic field is proportional to the number of turns. From equation,ε=−nA dBdt

This shows that the induced voltage is proportional to the secondary coil.

From the above relationship, we verified most of the dependence of induction. These can be separated into two types of dependence. Once depends on the input magnetic field, and other depends on the induce coil. The induced voltage mostly depends on the input magnetic field, for example, field-exciting voltage, frequency and turns of primary coil. Only few depend on induce coil, e.g. turns of secondary coil and area. Permeability of the area between two coils also affects the induction.

Error analysis:There are several errors occurred in this experiment. First, the electronic devices

in the laboratory can be greatly distorted the result. In a testing case, slightly moving a magnet placed 1m around the equipment still can cause a largely distortion.

Secondary, the function generator was not very stable. Even thought, it only caused about 0.01Hz change in the experiment, the result can still be affected since low frequency was used.

Finally, there is a 1% uncertainty on all the electronic devices.

Conclusion:The dependences of the induction are verified in this experiment. By

understanding more about the induction, we can even calculate the theoretical induced voltage and the efficiency of the setup. This could be a great help in the improvement of some devices such as transformer.