13S1 FE1073 E2-Magnetic Field

NANYANG TECHNOLOGICAL UNIVERSITY First Year Engineering Course

FE1073: An Introduction to Engineering and Practices

Laboratory Manual For

Experiment E2 Magnetic Field Laboratory : Power and Clean Energy Design Location: S2-B5c-01 School of Electrical and Electronics Engineering [EEE]

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Session 2013/2014

MAGNETIC FIELD 1. OBJECTIVES When current exists in an infinitely long straight wire, a B field will exist in the region surrounding the wire. If the current is constant in time, the B field that exists will be constant in time at a given point. This presence of the constant B field can be detected by a small compass. If the current in the wire is time-varying, the B field that exists will also be time-varying. This time-varying B field can be detected by the electric field that it induces in a small inductor coil placed near the wire. In this laboratory, measurements on an apparatus with a long straight current-carrying wire will be used to accomplish the following objectives: 1.1 Determination of the direction of the B field surrounding a long straight wire using a compass. 1.2 Confirmation that the direction of the B field near the wire is consistent with the right-hand rule that relates the current direction to the direction of the B field. 1.3 Determination of the induced voltage in a small inductor coil placed near a long straight wire as a relative measurement of the B field. 1.4 Demonstration that the magnitude of the B field surrounding a long straight wire decreases with increasing r, where r is the perpendicular distance from the wire. 1.5 Determination of the induced voltage in a small inductor coil as a function of the ac current in a long straight wire. 1.6 Determination of the induced voltage in a small inductor coil as a function of the frequency of the ac current in a long straight wire.

2. EQUIPMENT LIST 2.1 Direct-current power supply 2.2 Sine-wave generator 2.3 Digital voltmeter 2.4 Digital ammeter 2.5 A 100-mH inductor coil (length ≈ 1 cm and inside diameter ≈ 0.5 cm) 2.6 Small compass; long straight wire apparatus. (Consists of a frame on which a continuous strand of wire is wrapped for 10 loops. The 10 strands are taped together over a length of approximately 40 cm to approximate a wire whose current is 10 times the current in a single strand of the wire.)

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3. THEORY When a current I exists in an infinitely long straight wire, the lines of magnetic induction B are concentric circles surrounding the wire. At a perpendicular distance r from the wire, the B field is tangent to the circle as shown in Fig. 1. The direction of the current I is perpendicular to the plane of the page and directed out of the page. The direction of the current is by definition the direction that positive charge would flow. The magnitude of the B field as a function of I and r is given by

B

0 I 2 r

(1)

where μ0 = 4 x 10-7 weber/amp-m, I is in amperes, and r is in meters. The units of B are weber/m2, named as Tesla. The direction of the B field relative to the current direction is given by following the right-hand rule. If the thumb of the right hand points in the direction of the current, the four fingers of the right hand curls in the direction of the B field. It is important to note that the B field forms circles around the conductor and at each instant they will be pointing in the direction tangent to these circles as shown in Fig. 1. Also note in Fig. 1 that the length of the B vectors are drawn shorter for the larger circles to indicate that the B field decreases with distance from the wire as given by equation 1. Ideally, the above statements apply only to an infinitely long straight wire. In this laboratory the straight portion of the wire has a finite length L. In order to satisfy the ideal condition, measurements are made at the center of the wire and within a perpendicular distance of L/4 from the wire. If the current in the long straight wire is constant in time, the B field created by that current will be constant in time. Here, the direction of the B field can be determined by observing the effect of the B field on a small compass placed in the vicinity of the long straight wire.

Fig. 1 B field near a wire carrying current perpendicular to the page and directed out of the page. If the current in the long straight wire is an alternating current produced by a sine-wave generator, the B field surrounding the wire will also be time-varying, and it will alternate in direction and magnitude. If a small inductor coil is placed next to the wire, an alternating voltage will be induced in the coil. According to Faraday’s law of induction, this induced voltage in the coil is proportional to the rate of change of the magnetic flux through the coil, and hence to the magnitude of the time-varying B field.

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Therefore, a measurement of the voltage induced in the coil, as the coil is placed at different distances from the wire, provides a relative measure of the magnitude of the B field at different distances from the wire. Note carefully that the quantity actually measured is an alternating electric voltage, but its magnitude is proportional to the B field and will be taken to be a relative measurement of the B field at a given point. 4. EXPERIMENTAL PROCEDURE – DIRECTION OF THE B FIELD 4.1 Connect the circuit shown in Fig. 2 using the direct-current power supply and the digital multimeter. Select dc current setting on the multimeter and use the 10A and common sockets for connection. Arrange the long-wire apparatus so that side A is facing you. Make sure that the direction of current flow in the bare wire is from top to bottom (Determine the direction of the current by tracing the wires from the (+) to (-) terminals of the power supply). Have the circuit checked by your instructor to ensure that the current is in the proper direction before turning on the power supply.

Fig. 2 Long wire apparatus connected to Direct Current Supply.

4.2 Turn on the power supply and increase the voltage until a current of 2.00A is read on the multimeter. Do not exceed the current beyond 2.00A. 4.3 Place the compass in the middle of the top horizontal section, directly above the wire and as close to the wire as possible. State the direction (side A, side B) that the compass needle points. Record your answer in Data Table 1. 4.4 Place the compass in the middle of the top horizontal section, directly below the wire and as close to the wire as possible. State the direction (side A, side B) that the compass needle points. Record your answer in Data Table 1. 4.5 Place the compass next to the bare wire at the four positions indicated by the open circles in Fig. 6 in the Log sheet 1. The  represents the downward current viewed from above. In the open circles representing the four compass positions, draw an arrow showing the direction that the compass needle points.

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5. EXPERIMENTAL PROCEDURE – B FIELD AS A FUNCTION OF DISTANCE 5.1 Connect the circuit shown in Fig. 3 using the long-wire apparatus and the sine-wave generator. The detailed connection diagram is given in Fig. 4.

Fig. 3 Long-wire apparatus experimental setup. 5.2 Select ac current on the digital ammeter and connect the ammeter using the 100 mA and common socket. Select the sine wave and 10 KHz buttons of the sine wave generator. Select ac voltage on the digital voltmeter. Using the leads that have been twisted about 10 to 15 times, connect them between the voltage and common sockets to the inductor coil of 100 mH selfinductance. This is extremely important because it will minimize the voltage induced in the leads themselves and ensure that the voltage induced is in the inductor coil. The inductor coil is placed on the platform as shown in Fig. 5. The axis of the inductor coil is perpendicular to an imaginary line (shown as the dotted line labeled I in Fig. 5), which is in turn perpendicular to the current-carrying wire. The inductor coil was shown in three different positions with the axis of the coil at different distances r1, r2, and r3 from the wire. At each position of the inductor coil shown, the B field will alternate in opposite directions along the axis of the coil. The coil is chosen to be short (≈ 1 cm) and of small cross section (diameter ≈ 0.5 cm) because for that choice, the B field lies approximately along the coil axis and is approximately uniform over the cross section of the coil.

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Fig. 4 Long wire apparatus connection diagram.

Fig. 5 View of the platform looking down from above. The current is perpendicular to the page alternating into and out of the page.

5.3 The amplitude of the induced voltage on the digital voltmeter will depend on the frequency of the sine-wave generator. With the inductor about 3 cm from the wire, and its axis positioned as shown in Fig. 5, turn the sine-wave generator to its maximum output amplitude by turning the amplitude knob fully clockwise. Vary the frequency of the generator by tuning the frequency dial until a maximum voltage is read on the digital voltmeter. Record the frequency in Data Table 2. Once this frequency is found, do not change the frequency. Make all measurements at this frequency. 5.4 Measure the voltage induced in the inductor coil as a function of r (Fig. 5). The quantity r is the distance from the center of the coil ( indicated by the white marker ) to the center of the wire. Take data from r = 3.0 cm to r = 9.0 cm, in increments of 1 cm. Since the B field is extremely nonuniform over the coil cross section close to the wire, data is not taken for r less than 3 cm. Record the values of the voltage in Data Table 2 under the column labeled V. If this were a true measure of the B field, the units would be in Tesla. Since the measured quantity is voltage, the units are in volt. 6. EXPERIMENTAL PROCEDURE – B FIELD AS A FUNCTION OF FREQUENCY 6.1 Use the same circuit as in the above section. 6.2 Move the inductor to a distance of 3 cm from the long wire (r = 3 cm). 6.3 Select the 1 KHz button and set the output current from the sine wave generator to 40mA.

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6.4 Vary the frequency of the sine wave from f = 5 kHz to 12 kHz at 1 kHz steps and record the voltmeter reading in Data Table 3. For each set of reading make sure the current is maintained at 40mA. The current can be adjusted by turning the amplitude knob of the sine wave generator. 7. EXPERIMENTAL PROCEDURE – B FIELD AS A FUNCTION OF CURRENT 7.1 Use the same circuit as in the above section. 7.2 Move the inductor to a distance of 3 cm from the long wire (r = 3 cm). 7.3 Set the frequency of the sine wave generator to 70 kHz. 7.4 Vary the current in the wire by turning the amplitude knob of the sine wave generator from 10 mA to 45 mA in steps of 5 mA. 7.5 Record the voltmeter reading for each current setting in Data Table 4.

8. GRAPHS 8.1 Use the data in Data Table 2 draw a graph of induced voltage V versus 1/r. 8.2 Use the data in Data Table 3 draw a graph of induced voltage V versus frequency f. 8.3 Use the data in Data Table 4 draw a graph of induced voltage V versus current I.

9. FORMAL REPORT 9.1 Derive an expression for the magnetic field B at a point of distance r, from an infinitely long wire that carries a current I. Your derivation should include the direction of the magnetic field with respect to the direction of current flow. Verify your expression by using the experimental results obtained. If your results do not show the expected relationship, explain why. 9.2 Derive and comment on the dependence of the induced voltage in the inductor coil on the (i) frequency and (ii) magnitude of the ac current flowing in the long wire. Verify your answers by using the experimental results obtained. If your results do not show the expected relationships, explain why. The report length should not be more than 15 pages.

10. REFERENCES [1] R. A. Serway & R. J. Beichner, 2004, “Physics for Scientists and Engineers with Modern Physics”, 6th Edition, Saunders College Publishing. [2] E. R. Jones & R. L. Childers, 2000, “Contemporary College Physics”, McGraw Hill.

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Experiment E2: Magnetic Field DATA SHEET 1

Name

: ______________________________________

Date

: ______________

Group : ______________________________________

Data Table 1 With compass above wire, compass direction = With compass below wire, compass direction =

Fig. 6 Indicate the compass direction at the positions shown.

Sine wave amplitude = maximum Data Table 2 r (cm)

1/r (cm-1)

r = 3cm, I = 40mA Data Table 3

V (volt)

3.00 4.00 5.00 6.00 7.00 8.00 9.00

Frequency of ac current: ________

f (KHz) 5 6 7 8 9 10 11 12

V (mvolt)

r =3 cm, Freq. = 70 KHz Data Table 4 I (mA) 10 15 20 25 30 35 40 45

V (volt)

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DATA SHEET 2

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DATA SHEET 3

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DATA SHEET 4

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DATA SHEET 5 QUESTIONS 1. Are your answers to the questions in Data Table 1 about the direction in which the compass needle points consistent with the right-hand rule for the direction of the B field?

2. State the extent to which your measurements confirm the expectation that B field is proportional to 1/r for the long wire.

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DATA SHEET 6 3. When the direct current is 2.00 A in a single wire of the bundle of 10 wires, the total current in the bundle of wire that approximates the long straight wire is 20.0 A. What is the magnitude of the B field 3.00 cm from this long straight wire carrying a current of 20.0 A? What is the magnitude of the B field 9.00 cm from the wire carrying 20.0 A?

4. A constant current flows in a long straight wire in the plane of the paper in direction shown below by the arrow. Point X is in the plane of the paper above the wire, and point Y is in the plane of the paper below the wire. What is the direction of the B field at point X ? What is the direction of the B field at point Y ?

Direction at X = _________________________ Direction at Y = _________________________



X



Y

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DATA SHEET 7 5. Based on the experimental results obtained, comment on the relationship between the induced voltage V in the inductor coil and the frequency f of the ac current flowing in the wire.

6. Based on the experimental results obtained, comment on the relationship between the induced voltage V in the inductor coil and the magnitude of the ac current I flowing in the wire.

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APPENDIX 1 Additional Theory Assuming an infinite wire, the magnetic flux density B at a distance r from a wire of M turns is

0 I M 2 r

B0 

(A1)

where I is the current flowing in the wire

Assuming that the inductor has N turns and has core. The magnetic flux density B in the core of the inductor is

B  r B0 

0  r I M 2 r

r

as its

(A2)

The magnetic flux  through the inductor is

 BA N

(A3)

where A is the cross-sectional area of the inductor.



0 r IMAN 2r

(A4)

The inductance L of the inductor is given by

L

0  r N 2 A l

(A5)

where l is the length of the inductor. From (A4) and (A5),



I M Ll 2 r N

(A6)

The induced voltage in the inductor due to a changing  is given by

E

d dt

(A7)