Version 2.2 NE03 - Faraday's Law of Induction

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1 Definition Version. Laboratory Manua Department of Physics he University of Hong Kong Aims o demonstrate various properties of Faraday s Law such as: 1. Verify the aw.. Demonstrate the ighty damped osciation of the ha probe as a simpe penduum. 3. Find the amount of energy ost due to ighty damped osciation. Sef-earning materia: heory - Background Information Magnetic fied strength, magnetic fux density, magnetic fux and magnetic fux inkage Magnetic Magnetic fied strength (B-fied) Magnetic fux fux density SI Unit B Where F F magnetic magnetic (1) v Q i i F magnetic Ha effect and Ha probe effect. Qv is the magnetic force acting on a ong straight wire or on a charge; is veocity of the charge in the B-fied; is no. of charge; is current of the ong straight wire; is ength of the wire perpendicuar to B-fied B A Magnetic fux inkage () Bd A BA (3) inkage N (4) Where Bd A BA for a magnetic fied ( B ) which is constant over the cross-sectiona area ( A ) and perpendicuar to the area; is the number of turns of wire in the coi. N esa 1 ( ) Weber ( Wb ) Ha probe is a device to detect a constant magnetic fied strength, B by making use of Ha he Ha votage, B V Ha, deveoped by a Ha probe is given by V Ha Bi (5) nqt Where is the magnetic fied perpendicuar to the ha probe; i is amount of current passing through the ha probe n is number of charge-carrier per unit voume in the ha probe Q is amount of charge in charge-carrier 1 1 esa ( ) = 10,000 Gauss ( G ) which is a non-si unit and it s commony used 1 Weber ( Wb ) = 1 tesa meter square meter ( m ) Page 1 of 1

2 Version. Faraday's Law of Induction (a) Statement Faraday s aw states that he induced e.m.f. is directy proportiona to the rate of change of fux-inkage or rate of fux cutting. (b) Mathematica treatment According to Faraday's Law of Induction, a changing magnetic fux through a coi induces an d eectromotive force (e.m.f.) given by induced N (6) dt where Bd A BA for a magnetic fied ( B ) which is constant over the cross-sectiona area ( A ) and perpendicuar to the area. N is the number of turns of wire in the coi. For finding average e.m.f., the equation (6) coud approximatey equa to d BA induced N N induced N BA and A is a constant and dt t t B independent of time ( ). induced NA t t d As a resut, for this experiment, the area of the coi A r 4 is a constant where r is the radius of coi and d is the diameter of the coi. and as the coi passes into or out of the magnetic fied, there is an average e.m.f. given by induced N d 4 B t. (7) d, diameter of the coi N, Number v of turns Uniform magnetic fied pointing into the paper i f Figure 1. Position of center of mass Center of mass of a body (a) Definition: he terms "center of mass" and "center of gravity" are used synonymousy in a uniform gravity fied to represent the unique point in an object or system which can be used to describe the system's response to externa forces and torques. he concept of the center of mass is that of an average of the masses factored Page of 1

Version. by their distances from a reference point. In one pane, that is ike the baancing of a seesaw about a pivot point with respect to the torques produced.

3 Version. by their distances from a reference point. In one pane, that is ike the baancing of a seesaw about a pivot point with respect to the torques produced. x cm x x 1 (b) Experimenta method m 1 Centre of mass m m xcm x x m m m m m Figure. Definition center of mass Figure 3. Experimenta method to find the position of center of mass Once the coi wand is baanced by the index finger simiar to the situation in Figure 3, the position of the finger is the ocation of the center of mass. Simpe harmonic motion (S.H.M.) A. Definition of S.H.M. If x is dispacement, and a x is the acceeration (in the x direction), this can be expressed mathematicay: a x (8) where is the anguar frequency. he minus sign indicates that a is aways in the opposite direction x. As the force is aways in the direction towards the center, the force is caed the restoring force. B. Properties of S.H.M. Dispacement x Acost Veocity v Asint Acceeration Period Frequency Anguar frequency x a Acost x 1 f f Page 3 of 1

4 Version. m k g mass-spring system simpe penduum For a uniform circuar motion, the period is independent of radius. As a resut, for a simpe harmonic motion, the period is independent of ampitude. (i.e. caed isochronous motion) Q O Dispacement, x R Ampitude, A Ampitude, A Extreme point Equiibrium position Extreme point he partice changes its veocity direction at this point F net = 0 Acceeration = 0 ms -1 he partice changes its veocity direction at this point Veocity = 0 ms -1 Veocity = 0 ms -1 Figure 4. Properties of S.H.M. at extreme point and equiibrium position C. Simpe penduum as an exampe of S.H.M. i) he restoring force ( For very sma, F ma mg sin ma net Since the force is in opposite direction to the dispacement) mg ma x sin (9) x mg ma g a x S.H.M. Comparing with a f (10), x g (11) or g (1) g 1 f g x m Figure 5. Simpe penduum ii) he period depends on the ength of string and not the ampitude. he ength of penduum can Page 4 of 1

5 Version. be adjusted so that the period is 1 second. (Cock)his is one of the way to find the gravitationa acceeration g by the measuring the period and the ength of the string. D. Damping of S.H.M. or Damped harmonic motion i) Now, consider the case of simpe penduum. he ii) ampitude of the osciation of a penduum graduay decreases to zero owing to the air resistance. (Energy is ost) Suppose the air resistance is directy proportiona to the speed v (i.e., where b is a f air bv constant and the negative sign means air resistance is aways in the opposite direction to the motion)) hen equation a x v a x becomes (13) where constant and m is the mass of object a x v 0 (14) b is a m A o -A o x ( t ) +A o 0 Direction of motion Figure 6. free osciation Air resistance Air resistance Direction of motion Figure 7. Direction of motion and air resistance iii) Equation (14) is a differentia equation of damped osciation and x wi depend on and iv) According to the soution of equation(14), 3 cases coud be resuted ( k is arge, great damping) Heavy damped motion. 0 (critica vaue) Critica damped motion 3. 0 ( k is sma, sma damping) sighty damped motion t xt () x(0) Free osciation Lighty damped motion Critica damped motion Heaviy damped motion Page 5 of 1

6 Version. Figure 8. Ampitude ratio of free osciation and different damped motion v) In this experiment, ony the sighty damped motion is concerned Sighty damped motion (i.e. 0 ) a. t cos sin t x e A qt B qt Ce cos qt (15) where A 1, b. he period 1 1 B 1,C are constants and q is constant ( q ) and is sighty smaer. c. he ampitude is decreasing at a constant rate x ( t ) A o +A o e i.e. A o A1 A n1... e (16) A A A 1 he ampitude decreases as e n t 0 A 1 A t Sighty damped motion -A o Figure 9. Ampitude of sighty damped motion he bue dotted curves in the graph above are the pair of functions e, e often referred to as the enveope of the osciation curve, as they enveope it from above and beow. Eddy current (a) Definition Any piece of meta or coi moving in a magnetic fied, or exposed to a changing one, has e.m.fs induced in it, as we might expect. hese can cause currents, caed eddy currents, to fow inside the meta and they may be quite arge because of the ow resistance of the paths they foow. (b) Magnetic effect of eddy current According to Lent s aw, eddy currents wi circuate in directions such that the magnetic fieds they create oppose the motion (of fux change) producing them. his acts as a brake on the moving body. i induced B magnet B induced N v B magnet Meta pate Bar magnet S Page 6 of 1

7 Version. Energy and Power Figure 10. Eddy current in a piece of meta pate If the center of mass of the penduum starts from rest at an initia height h i, its potentia energy is U i = mgh i (17) As the penduum swings and passes through the magnet, some energy is ost to mechanica frictiona heat and some energy is converted to eectrica energy and then to therma energy in the resistor. hus the center of mass of the penduum does not rise to the same height but rather to a ower fina height, h f. See Figure 11. he tota energy ost by the penduum is equa to its change in potentia energy: ota Energy Lost = ΔU = mg(h i h f ) (18) or ota Energy Lost U mg 1 cos (19) where is the distance from the pivot point to the center of mass of the coi wand. and is the anguar position of the coi wand. e.m.f. h i i h b f h f Centre of mass r R i V Coi Figure 11: Coi Height Decreases Figure 1: Coi and Resistor Circuit he therma energy dissipated in the resistor ( R ) is given by E P dt P t oss oss oss Eoss Poss t = Area Under Power against ime graph (0) where P oss is the power dissipated and P is time. Figure 13: Coi and Resistor Circuit Page 7 of 1

8 Version. Consider the circuit as shown in Figure 1, V r is the potentia difference (p.d.) across the resistor ( r ),i is the current through the coi, and R is the resistance of the coi. According to Kirchhoff oop rue reads: V V (1) e. m. f. R r Because of energy conservation, P... P P he p.d. across the resistor ( r ), V r e m f R r Pe. m. f. Poutput i R i r, is measured by votmeter; according to Ohm s aw, Vr Vr ir i r he power output of the circuit is given by P i R r In this experiment, R 1.9 output Vr Poutput ( R r) r and r 4.7 () Apparatus 1 PASCO Scientific EM-8099 Induction Wand 1 PASCO Scientific EM-8641 Variabe Gap Lab Magnet 1 PASCO Scientific CI-6503 Votage Sensor 1 PASCO Scientific CI-650A Magnetic Fied Sensor 1 PASCO Scientific CI-6538 Rotary Motion Sensor 1 PASCO Scientific CI-6400 Science Workshop 500 data ogger PASCO DataStudio computer interface 1 arge rod stand Misceaneous wirings (3) Experiment 1: Average induced e.m.f. Setup Procedures: 1. Put a rod in the stand and camp the cross-rod to it as shown in Figure 14. Put the Rotary Motion Sensor at the end of the cross-rod.. Attach the coi wand to the Rotary Motion Sensor with the tabs on the 3-step puey just to the sides of the wand as shown in Figure Pug the Votage Sensor into Channe A of the ScienceWorkshop 500 interface. 4. Pug the Rotary Motion Sensor into Channes 1 and. 5. Pug the Magnetic Fied Sensor into Channe B. 6. Pug the Votage Sensor banana pugs into the banana jacks on the end of the coi wand. 7. Put the poe pates on the magnet as shown in Figure 16. (Caution: Hod the pates tight and approach sowy to the strong magnet; be carefu of the pates edges and watch out your fingers). 8. Drape the Votage Sensor wires over the rods as shown in Figure 14 so the wires wi not exert a torque on the coi as it swings. It heps to hod the wires up whie recording data. Page 8 of 1

magnetic poes? 1. Using caiper to measure the diameter of the coi and record it in the abe 1.3 of the worksheet (Page4).

Cick Save Activity As to create your group which UID shoud be your/your partner university no.(e.g. NE03_EXP1_1345678.ds). 4.

Press ARE button on the sensor to set zero of the sensor. 7.

9 Version. Figure 14: Rod Stand Figure 15: abs Figure 16: Magnet Poe Pates Experimenta Procedure: How to measure magnetic fied strength between the magnetic poes? 1. Using caiper to measure the diameter of the coi and record it in the abe 1.3 of the worksheet (Page4).. Cick the DataStudio working fie NE03_EXP1_UID.d 3. Cick Save Activity As to create your group which UID shoud be your/your partner university no.(e.g. NE03_EXP1_ ds). 4. Keep the Magnetic Fied Sensor away from the magnetic pates. 5. Cick SAR to start data measurement. 6. Press ARE button on the sensor to set zero of the sensor. 7. Put the Magnetic Fied Sensor inside the poe pates on the magnet to measure the constant magnetic fied strength between the magnet poes. 8. Hod the Magnetic Fied Sensor steady about 5 seconds. 9. Cick SOP to stop data measurement. 10. Read the magnetic fied strength between the magnet poes in esa ( ) from the magnetic fied strength vs. time graph. 11. Find out the mean vaue of the magnetic fied strength by using show seected statistic, seect the mean function as shown in Figure 18. Figure 17: Coi Passes through Magnet Page 9 of 1

10 Version. Figure 18: Functions in. 1. Repeat the measurement of the magnetic strength fied twice. Record the reading in abe 1.1 in worksheet. How to measure the ampitude and the time duration of induced e.m.f.? 1. Adjust the gap between the magnet poes so the coi wand wi be abe to pass through but put the magnet poes as cose together as possibe.. Adjust the height of the coi so it is in the midde of the magnet. 3. Aign the wand from side-to-side so it wi swing through the magnet without hitting it. (Beware: the case of coi wand is pastic and fragie. Pease avoid any coision.) 4. Keep the induction wand in the steady position. 5. Cick SAR and pu the coi wand back about 40 degree and et it swing through the magnet. 6. When the coi wand stops swinging and stays at equiibrium position, cick SOP to end the measurement. 7. Use the Zoom seect to enarge the portion of the votage vs. time graph where the coi passed through the magnet. 8. Use the Smart oo to determine the ampitude of first peak 9. Use the Smart oo again to determine the time difference from the beginning to the end of the first peak. 10. Repeat steps (1) to step (9) twice and record the data in the abe 1. in the worksheet. 11. Save the working fie and then exit the program. Page 10 of 1

Version. Experiment : Lighty damped osciation Lighty damped osciation due to friction and air resistance, eddy current and power oss of the resistance in the coi Experimenta Procedure: 1.

Now, the amount of energy ost to friction, air resistance and eddy current wi be measured by etting the penduum swing with the coi connected in a compete circuit and presence of magnetic fied. 4.

11 Version. Experiment : Lighty damped osciation Lighty damped osciation due to friction and air resistance, eddy current and power oss of the resistance in the coi Experimenta Procedure: 1. Open the DataStudio fie caed "NE03_EXP_UID.ds".. Cick Save Activity As to create your group which UID shoud be your/your partner university no.(e.g. NE03_EXP1_ ds). 3. Now, the amount of energy ost to friction, air resistance and eddy current wi be measured by etting the penduum swing with the coi connected in a compete circuit and presence of magnetic fied. 4. Connect the resistor with both pugs in the coi wand. his competes the series circuit of the resistor and coi as shown in the Figure Cick SAR with the coi at rest in its equiibrium position between the cois. hen rotate the wand to an initia ange of 5 degrees and et it go. he ange is shown in digita meter 6. Cick SOP after the coi stops swinging. 7. Measure the initia anguar position and anguar position to which the penduum firsty rises to another side after it passes once through the magnet. Record it in abe.3 in the worksheet for the 1 st tria measurement. 8. Cacuate the initia and fina heights by using the distance from the center of mass to the pivot and the initia and fina anguar position in abe. in the worksheet. Record it in abe.3 in the worksheet for the 1 st tria measurement. 9. Cacuate the tota energy ost using equation (18). 10. Use the Zoom seect to enarge the portion of the anguar position vs. time graph where at east five periods of curve is observed. 11. Use the Smart oo to determine the periods and ampitudes from the Anguar position vs. time graph. Record the data in abe.4. Figure 3 Potentia Energy vs. ime graph 1. Highight both peaks on the power vs. time graph and find the area by using show seected statistic, ick the Area as shown in Figure 18. his area is the energy dissipated by the resistor. Figure 4 Power vs. ime graph 13. Save the working fie and then exit the program. Page 11 of 1

12 Version. References: Physica penduum and Center of mass 1. Chapter 10, Physics for Scientists and Engineers with Modern Physics 8 th Edition, John Jewett and Raymond Serway S.H.M.and Damped motions Chapter 15, Physics for Scientists and Engineers with Modern Physics 8 th Edition, John Jewett and Raymond Serway Magnetic fied and Ha effect 5. Chapter 9, Physics for Scientists and Engineers with Modern Physics 8 th Edition, John Jewett and Raymond Serway Faraday s aw and Eddy Current 6. Chapter 31, Physics for Scientists and Engineers with Modern Physics 8 th Edition, John Jewett and Raymond Serway Page 1 of 1

Candidate Number. General Certificate of Education Advanced Level Examination January 2012

Candidate Number. General Certificate of Education Advanced Level Examination January 2012 entre Number andidate Number Surname Other Names andidate Signature Genera ertificate of Education dvanced Leve Examination January 212 Physics PHY4/1 Unit 4 Fieds and Further Mechanics Section Tuesday