PHYS 1441 Section 001 Lecture #14

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1 PHYS 1441 Section 001 Lectue #14 Tuesday, June 29, Chapte 28:Souces of Magnetic Field Souces of Magnetic Field Magnetic Field Due to Staight Wie Foces Between Two Paallel Wies Ampée s Law and Its Veification Solenoid and Tooidal Magnetic Field Biot-Savat Law Magnetic Mateials Today s homewok Hysteesis #8, due 11pm, Satuday, July 2!! PHYS , Summe 1

2 Tem 2 Announcements In class, tomoow, Thusday, June 30 Coves fom CH 26.1 though what we cove today BYOF Reading Assignment CH27.8, CH28.8, 9 and 10 PHYS , Summe 2

3 Special Poject #5 B due to cuent I in a staight wie. Fo the field nea a long staight wie caying a cuent I, show that (a) the Ampee s law gives the same esult as the simple long staight wie, B=µ 0 I/2πR. (10 points) (b) That Biot-Savaat law gives the same esult as the simple long staight wie, B=µ 0 I/2πR. (10 points) Must be you OWN wok. No cedit will be given fo fo copying staight out of the book, lectue notes o fom you fiends wok. Due is at the beginning of the exam on Tuesday, July 5 PHYS , Summe 3

4 Souces of Magnetic Field We have leaned so fa about the effects of magnetic field on electic cuents and moving chage We will now lean about the dynamics of magnetism How do we detemine magnetic field stengths in cetain situations? How do two wies with electic cuent inteact? What is the geneal appoach to finding the connection between cuent and magnetic field? PHYS , Summe 4

5 Magnetic Field due to a Staight Wie The magnetic field due to the cuent flowing though a staight wie foms a cicula patten aound the wie What do you imagine the stength of the field is as a function of the distance fom the wie? It must be weake as the distance inceases How about as a function of cuent? Diectly popotional to the cuent Indeed, the above ae expeimentally veified This is valid as long as << the length of the wie The popotionality constant is µ 0 /2π, thus the field stength becomes µ 0 is the pemeability of fee space B = µ I 0 2π PHYS , Summe B 7 µ 0 = 4π 10 T m A I 5

6 Example 28 1 Calculation of B nea wie. A vetical electic wie in the wall of a building caies a DC cuent of 25A upwad. What is the magnetic field at a point 10cm due noth of this wie? Using the fomula fo the magnetic field nea a staight wie B = µ I So we can obtain the magnetic field at 10cm away as 0 2π 7 B = 0 I 2 µπ ( ) = 4π 10 T m A ( 25A) 2π 0.01m ( ) ( ) = T PHYS , Summe 6

7 Foce Between Two Paallel Wies We have leaned that a wie caying the electic cuent poduces magnetic field Now what do you think will happen if we place two cuent caying wies next to each othe? They will exet foce onto each othe. Repel o attact? Depending on the diection of the cuents This was fist pointed out by Ampée. Let s conside two long paallel conductos sepaated by a distance d, caying cuents I 1 and I 2. At the location of the second conducto, the magnitude of the magnetic field poduced by I 1 is µ 0I1 B1 = 2π d PHYS , Summe 7

8 Foce Between Two Paallel Wies The foce F by a magnetic field B 1 on a wie of length l, caying the cuent I 2 when the field and the cuent ae pependicula to each othe is: F = IBl 2 1 So the foce pe unit length is F µ 0 I1 = I2B1 = I2 l 2π d This foce is only due to the magnetic field geneated by the wie caying the cuent I 1 Thee is the foce exeted on the wie caying the cuent I 1 by the wie caying cuent I 2 of the same magnitude but in opposite diection F So the foce pe unit length is = l 2 How about the diection of the foce? Wednesday, If the cuents June 29, ae in the same PHYS diection, , the Summe attactive foce. If opposite, epulsive. 8 µ π II d

9 Example 28 5 Suspending a wie with cuent. A hoizontal wie caies a cuent I 1 =80A DC. A second paallel wie 20cm below it must cay how much cuent I 2 so that it doesn t fall due to the gavity? The lowe has a mass of 0.12g pe mete of length. Which diection is the gavitational foce? This foce must be balanced by the magnetic foce exeted on the wie by the fist wie. F g mg F = = M µ = 0 II 1 2 l l l 2π d mg 2π d I 2 = = l µ I PHYS , Summe Down to the cente of the Eath ) ( ) ( ) 0 1 ( 2 2π 9.8ms kg 0.20m Solving fo I 2 ( 7 4π 10 T m A) ( 80A) = 15A 9

10 Opeational Definition of Ampee and Coulomb The pemeability of fee space is defined to be exactly 7 µ 0 = 4π 10 T m A The unit of cuent, ampee, is defined using the definition of the foce between two wies each caying 1A of cuent and sepaated by 1m F µ 7 0 II 1 2 4π 10 T m A1A 1A 7 = = = 2 10 Nm l 2π d 2π 1m So 1A is defined as: the cuent flowing each of two long paallel conductos 1m apat, which esults in a foce of exactly 2x10-7 N/m. Coulomb is then defined as exactly 1C=1A s. We do it this way since the electic cuent is measued moe accuately and contolled moe easily than the chage. PHYS , Summe 10

11 Ampée s Law What is the elationship between the magnetic field stength and the cuent? µ 0I Does this wok in all cases? Nope! OK, then when? B = Only valid fo a long staight wie 2π Then what would be the moe genealized elationship between the cuent and the magnetic field fo any shapes of the wie? Fench scientist Andé Maie Ampée poposed such a elationship soon afte Oested s discovey PHYS , Summe 11

12 Ampée s Law Let s conside an abitay closed path aound the cuent as shown in the figue. Let s split this path with small segments each of Δl long. The sum of all the poducts of the length of each segment and the component of B paallel to that segment is equal to µ 0 times the net cuent I encl that passes though the suface enclosed by the path In the limit Δl à0, this elation becomes BP Δ=µ l 0Iencl B dl =µ I 0 encl Ampée s Law Looks vey simila to a law in the electicity. Which law is it? PHYS , Summe Gauss Law 12

Veification of Ampée s Law Let s find the magnitude of B at a distance away fom a long staight wie w/ cuent I This is a veification of Ampee s Law We can apply Ampee s law to a cicula path of adius.

13 Veification of Ampée s Law Let s find the magnitude of B at a distance away fom a long staight wie w/ cuent I This is a veification of Ampee s Law We can apply Ampee s law to a cicula path of adius. µ I = 0 encl B dl = Bdl = B dl = 2π B µ 0I Solving fo B B = encl µ = 0 I 2π 2π We just veified that Ampee s law woks in a simple case Expeiments veified that it woks fo othe cases too The impotance, howeve, is that it povides means to elate magnetic field to cuent PHYS , Summe 13

14 Veification of Ampée s Law Since Ampee s law is valid in geneal, B in Ampee s law is not just due to the cuent I encl. B is the field at each point in space along the chosen path due to all souces Including the cuent I enclosed by the path but also due to any othe souces How do you obtain B in the figue at any point? Vecto sum of the field by the two cuents The esult of the closed path integal in Ampee s law fo geen dashed path is still µ 0 I 1. Why? While B in each point along the path vaies, the integal ove the closed path still comes out the same whethe thee is the second wie o not. PHYS , Summe 14

15 Example 28 6 Field inside and outside a wie. A long staight cylindical wie conducto of adius R caies cuent I of unifom density in the conducto. Detemine the magnetic field at (a) points outside the conducto (>R) and (b) points inside the conducto (<R). Assume that, the adial distance fom the axis, is much less than the length of the wie. (c) If R=2.0mm and I=60A, what is B at =1.0mm, =2.0mm and =3.0mm? Since the wie is long, staight and symmetic, the field should be the same at any point the same distance fom the cente of the wie. Since B must be tangential to cicles aound the wie, let s choose a cicula path of the closed-path integal outside the wie (>R). What is I encl? So using Ampee s law µ 0 I = B dl = 2 B π 0 Solving fo B PHYS , Summe B = µ I 2π I encl 15 = I

16 So using Ampee s law 2 µ 0I R = B dl = 2 B What does this mean? Example 28 6 cont d Fo <R, the cuent inside the closed path is less than I. How much is it? I encl = I π π R 2 2 = I π Solving fo B B = R 2 µ 0 I 2π 2 = R µ 0 I 2π R 2 The field is 0 at =0 and inceases linealy as a function of the distance fom the cente of the wie up to =R then deceases as 1/ beyond the adius of the conducto. B = µ 0 2π I µ I B = 2π R 2 0 PHYS , Summe 16

Example 28 7 Coaxial cable. A coaxial cable is a single wie suounded by a cylindical metallic baid, as shown in the figue. The two conductos ae sepaated by an insulato.

17 Example 28 7 Coaxial cable. A coaxial cable is a single wie suounded by a cylindical metallic baid, as shown in the figue. The two conductos ae sepaated by an insulato. The cental wie caies cuent to the othe end of the cable, and the oute baid caies the etun cuent and is usually consideed gound. Descibe the magnetic field (a) in the space between the conductos and (b) outside the cable. (a) The magnetic field between the conductos is the same as the long, staight wie case since the cuent in the oute conducto does not impact the enclosed cuent. (b) Outside the cable, we can daw a simila cicula path, since we expect the field to have a cicula symmety. What is the sum of the total cuent inside the closed path? I encl = I I = 0. So thee is no magnetic field outside a coaxial cable. In othe wods, the coaxial cable self-shields. The oute conducto also shields against an extenal electic field. Cleane signal and less noise. PHYS , Summe B = µ 0 2π 17 I

18 Solenoid and Its Magnetic Field What is a solenoid? A long coil of wie consisting of many loops If the space between loops ae wide The field nea the wies ae nealy cicula Between any two wies, the fields due to each loop cancel Towad the cente of the solenoid, the fields add up to give a field that can be faily lage and unifom Fo a long, densely packed loops Solenoid Axis The field is nealy unifom and paallel to the solenoid axes within the entie coss section The field outside the solenoid is vey small compaed to the field inside, except the ends The same numbe of field lines spead out to an open space PHYS , Summe 18

Solenoid Magnetic Field Now let s use Ampee s law to detemine the magnetic field inside a vey long, densely packed solenoid Let s choose the path abcd, fa away fom the ends We can conside fou

19 Solenoid Magnetic Field Now let s use Ampee s law to detemine the magnetic field inside a vey long, densely packed solenoid Let s choose the path abcd, fa away fom the ends We can conside fou segments of the loop fo integal b B dl = B dl + c B dl + d B dl a b c The field outside the solenoid is negligible. So the integal on aàb is 0. Now the field B is pependicula to the bc and da segments. So these integals become 0, also. PHYS , Summe 19

20 Solenoid Magnetic Field d So the sum becomes: B dl = B dl = Bl c If the cuent I flows in the wie of the solenoid, the total cuent enclosed by the closed path is NI Whee N is the numbe of loops (o tuns of the coil) enclosed Thus Ampee s law gives us If we let n=n/l be the numbe of loops pe unit length, the magnitude of the magnetic field within the solenoid becomes B = µ ni 0 B depends on the numbe of loops pe unit length, n, and the cuent I Does not depend on the position within the solenoid but unifom inside it, like a ba magnet PHYS , Summe Bl = µ 0 NI 20

21 Example Tooid. Use Ampee s law to detemine the magnetic field (a) inside and (b) outside a tooid, which is like a solenoid bent into the shape of a cicle. (a) How do you think the magnetic field lines inside the tooid look? Since it is a bent solenoid, it should be a cicle concentic with the tooid. If we choose path of integation one of these field lines of adius inside the tooid, path 1, to use the symmety of the situation, making B the same at all points on the path, we obtain fom Ampee s law µ 0 NI B dl = B( 2π ) = µ 0 I encl = µ 0 NI Solving fo B B = 2π So the magnetic field inside a tooid is not unifom. It is lage on the inne edge. Howeve, the field will be unifom if the adius is lage and the tooid is thin. The filed in this case is B = µ 0 ni. (b) Outside the solenoid, the field is 0 since the net enclosed cuent is 0. PHYS , Summe 21

Biot-Savat Law Ampee s law is useful in detemining magnetic field utilizing symmety But sometimes it is useful to have anothe method of using infinitesimal cuent segments fo B field Jean Baptiste

22 Biot-Savat Law Ampee s law is useful in detemining magnetic field utilizing symmety But sometimes it is useful to have anothe method of using infinitesimal cuent segments fo B field Jean Baptiste Biot and Feilx Savat developed a law that a cuent I flowing in any path can be consideed as many infinitesimal cuent elements The infinitesimal magnetic field db caused by the infinitesimal length dl that caies cuent I is Biot-Savat Law is the displacement vecto fom the element dl to the point P Biot-Savat law is the magnetic equivalent to Coulomb s law B field in Biot-Savat law is only that by the cuent, nothing else. PHYS , Summe 22

23 Example B due to cuent I in a staight wie. Fo the field nea a long staight wie caying a cuent I, show that the Biot-Savaat law gives the same esult as the simple long staight wie, B=µ 0 I/2πR. What is the diection of the field B at point P? PHYS , Summe Going into the page. All db at point P has the same diection based on ight-hand ule. The magnitude of B using Biot-Savat law is B= db= µ 0I + dl ˆ µ 0I + dy sinθ 2 = 4π 2 4π y= Whee dy=dl and 2 =R 2 +y 2 and since y= Rcotθ we obtain 2 2 dy = Rdθ + Rcsc θdθ = 2 sin θ = Rdθ dθ = ( R) 2 R Integal becomes µ 0I + dy sinθ B = 2 4π = y= µ I π 0 π R θ = 0 The same as the simple, long staight wie!! It woks!! 4 1 µ 0I 1 π µ sinθdθ = cos θ = 0 I 1 0 4π R 2π R 23

Magnetic Mateials - Feomagnetism Ion is a mateial that can tun into a stong magnet This kind of mateial is called feomagnetic mateial In micoscopic sense, feomagnetic mateials consist of many tiny

24 Magnetic Mateials - Feomagnetism Ion is a mateial that can tun into a stong magnet This kind of mateial is called feomagnetic mateial In micoscopic sense, feomagnetic mateials consist of many tiny egions called domains Domains ae like little magnets usually smalle than 1mm in length o width What do you think the alignment of domains ae like when they ae not magnetized? Randomly aanged What if they ae magnetized? The size of the domains aligned with the extenal magnetic field diection gows while those of the domains not aligned educe This gives magnetization to the mateial How do we demagnetize a ba magnet? Hit the magnet had o heat it ove the Cuie tempeatue PHYS , Summe 24

25 B in Magnetic Mateials What is the magnetic field inside a solenoid? B 0 = µ 0 ni Magnetic field in a long solenoid is diectly popotional to the cuent. This is valid only if ai is inside the coil What do you think will happen to B if we have something othe than the ai inside the solenoid? It will be inceased damatically, when the cuent flows Especially if a feomagnetic mateial such as an ion is put inside, the field could incease by seveal odes of magnitude Why? Since the domains in the ion aligns pemanently by the extenal field. The esulting magnetic field is the sum of that due to cuent and due to the ion PHYS , Summe 25

26 B in Magnetic Mateials It is sometimes convenient to wite the total field as the sum of two tems B = B 0 + B M B 0 is the field due only to the cuent in the wie, namely the extenal field The field that would be pesent without a feomagnetic mateial B M is the additional field due to the feomagnetic mateial itself; often B M >>B 0 The total field in this case can be witten by eplacing µ 0 with anothe popotionality constant µ, the magnetic pemeability of the mateial B= µ ni µ is a popety of a magnetic mateial µ is not a constant but vaies with the extenal field PHYS , Summe 26

27 Hysteesis What is a tooid? A solenoid bent into a shape Tooid can be used fo magnetic field measuement Why? Since it does not leak magnetic field outside of itself, it fully contains all the magnetic field ceated within it. Conside an un-magnetized ion coe tooid, without any cuent flowing in the wie What do you think will happen if the cuent slowly inceases? B 0 inceases linealy with the cuent. And B inceases also but follows the cuved line shown in the gaph As B 0 inceases, the domains become moe aligned until nealy all ae aligned (point b on the gaph) The ion is said to be appoaching satuation Point b is typically at 70% of the max PHYS , Summe 27

Hysteesis What do you think will happen to B if the extenal field B 0 is educed to 0 by deceasing the cuent in the coil? Of couse it goes to 0!! Wong! Wong! Wong! They do not go to 0. Why not?

28 Hysteesis What do you think will happen to B if the extenal field B 0 is educed to 0 by deceasing the cuent in the coil? Of couse it goes to 0!! Wong! Wong! Wong! They do not go to 0. Why not? The domains do not completely etun to andom alignment state Now if the cuent diection is evesed, the extenal magnetic field diection is evesed, causing the total field B pass 0, and the diection eveses to the opposite side If the cuent is evesed again, the total field B will incease but neve goes though the oigin This kind of cuve whose path does not etace themselves and does not go though the oigin is called the Hysteesis. PHYS , Summe 28

29 Magnetically Soft Mateial In a hysteesis cycle, much enegy is tansfomed to themal enegy. Why? Due to the micoscopic fiction between domains as they change diections to align with the extenal field The enegy dissipated in the hysteesis cycle is popotional to the aea of the hysteesis loop Feomagnetic mateial with lage hysteesis aea is called magnetically had while the small ones ae called soft Which one do you think ae pefeed in electomagnets o tansfomes? Soft. Why? Since the enegy loss is small and much easie to switch off the field Then how do we demagnetize a feomagnetic mateial? Keep epeating the Hysteesis loop, educing the ange of B 0. PHYS , Summe 29

30 Induced EMF It has been discoveed by Oested and company in ealy 19 th centuy that Magnetic field can be poduced by the electic cuent Magnetic field can exet foce on the electic chage So if you wee scientists at that time, what would you wonde? Yes, you ae absolutely ight! You would wonde if the magnetic field can ceate the electic cuent. An Ameican scientist Joseph Heny and an English scientist Michael Faaday independently found that it was possible Though, Faaday was given the cedit since he published his wok befoe Heny did He also did a lot of detailed studies on magnetic induction PHYS , Summe 30

Electomagnetic Induction Faaday used an appaatus below to show that magnetic field can induce cuent Despite his hope he did not see steady cuent induced on the othe side when the switch is thown But

31 Electomagnetic Induction Faaday used an appaatus below to show that magnetic field can induce cuent Despite his hope he did not see steady cuent induced on the othe side when the switch is thown But he did see that the needle on the Galvanomete tuns stongly when the switch is initially thown and is opened When the magnetic field though coil Y changes, a cuent flows as if thee wee a souce of emf Thus he concluded that an induced emf is poduced by a changing magnetic field è Electomagnetic Induction PHYS , Summe 31

32 Electomagnetic Induction Futhe studies on electomagnetic induction taught If a magnet is moved quickly into a coil of wie, a cuent is induced in the wie. If a magnet is emoved fom the coil, a cuent is induced in the wie in the opposite diection By the same token, the cuent can also be induced if the magnet stays put but the coil moves towad o away fom the magnet Cuent is also induced if the coil otates. In othe wods, it does not matte whethe the magnet o the coil moves. It is the elative motion that counts. PHYS , Summe 32

33 Magnetic Flux So what do you think is the induced emf popotional to? The ate of changes of the magnetic field? the highe the changes the highe the induction Not eally, it athe depends on the ate of change of the magnetic flux, Φ B. Magnetic flux is defined as (just like the electic flux) Φ B = B A= BAcosθ = B A θ is the angle between B and the aea vecto A whose diection is pependicula to the face of the loop based on the ight-hand ule What kind of quantity is the magnetic flux? Scala. Unit? 2 T m o webe If the aea of the loop is not simple o B is not unifom, the magnetic flux can be witten as B da 2 1Wb = 1T m PHYS , Summe Φ B = 33

34 Faaday s Law of Induction In tems of magnetic flux, we can fomulate Faaday s findings The emf induced in a cicuit is equal to the ate of change of magnetic flux though the cicuit dφb ε = dt Faaday s Law of Induction If the cicuit contains N closely wapped loops, the total induced emf is the sum of emf induced in each loop ε = N dφ B dt Why negative? Has got a lot to do with the diection of induced emf PHYS , Summe 34

35 Lenz s Law It is expeimentally found that An induced emf gives ise to a cuent whose magnetic field opposes the oiginal change in flux è This is known as Lenz s Law In othe wods, an induced emf is always in a diection that opposes the oiginal change in flux that caused it. We can use Lenz s law to explain the following cases in the figues When the magnet is moving into the coil Since the extenal flux inceases, the field inside the coil takes the opposite diection to minimize the change and causes the cuent to flow clockwise When the magnet is moving out Since the extenal flux deceases, the field inside the coil takes the opposite diection to compensate the loss, causing the cuent to flow counte-clockwise Which law is Lenz s law esult of? Enegy consevation. Why? PHYS , Summe 35

Unit 7: Sources of magnetic field

Unit 7: Sources of magnetic field Unit 7: Souces of magnetic field Oested s expeiment. iot and Savat s law. Magnetic field ceated by a cicula loop Ampèe s law (A.L.). Applications of A.L. Magnetic field ceated by a: Staight cuent-caying