Unit 7: Sources of magnetic field

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Unit 7: Souces of magnetic field Oested s expeiment.

Magnetic field ceated by a: Staight cuent-caying wie Coil

1 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 wie Coil Magnetic flux tough a suface. Maxwell s equations fo Magnetostatics. Magnetism in matte. Feomagnetism. Jean-aptiste iot Félix Savat Andé Maie Ampèe

Oested s expeiment. 1820 1. If switch is off, thee isn t cuent and compass needle is aligned along noth-south axis F 2.

2 Oested s expeiment If switch is off, thee isn t cuent and compass needle is aligned along noth-south axis F 2. If switch is on, cuent aligns compass needle pependicula to cuent. 3. If cuent flows in opposite diection, compass needle is aligned in opposite diection. F An electic cuent ceates a magnetic field Tiple, chapte 27,2

3 iot and Savat s law Magnetic field ceated by a cuent is pependicula to cuent, and depends on the intensity of cuent and distance fom cuent. dl i d P Magnetic field ceated by a cuent element (Idl ) at a point P is: d µ idl µ idl = 0 = 0 π 3 π u u = dl diection is the same as i µ 0 (vacuum magnetic pemittivity)=4π10-7 Tm/A Tiple, chapte 27-2

4 iot and Savat s law Magnetic field ceated by a finite piece of wie is the sum (integal) of each cuent element at P: = µ 4 0 i π A dl 3 dl This equation can be applied to diffeent conducto shapes, staight conductos, cicula conductos,. A i P

5 Magnetic field lines Magnetic field lines ceated by a staight cuentcaying wie ae cicula in shape aound conducto: i Diection of magnetic field comes fom ight-hand o scew ule Tiple, chapte 27.2

6 Magnetic field lines Magnetic field lines ceated by a cicula loop: i

7 Magnetic field lines As magnetic poles cannot exist isolated (noth pole o south pole), any field line exiting fom a noth pole must go to a south pole, and all magnetic field lines ae closed lines.

8 Ampèe s law. Ampèe s law elates the integal of magnetic field along a closed line and the intensity passing though a suface enclosed by this line. Closed line C must be chosen by us (if possible, should be a magnetic field line): I I I i c d l = µ I 0 i Ampèe s law i c Tiple, chapte 27.4

9 Ampèe s law. Each intensity has it own sign, accoding to the ght-hand o scew ule. I>0 I 1 >0 I 2>0... I i <0 c Ampèe s law is equivalent (in Electomagnetism), to Gauss s law in Electostatics.

10 Ampèe s law. It s used to compute magnetic fields whee symmety exists. In ode to easily compute the integal of line, the chosen closed line C should have two featues: a) Modulus of magnetic field should be equal at evey points on closed line C. b) Magnetic field vecto () should be paallel to closed line C at evey points along C. = dl = In this way: dl = dl c c c L

11 Application of A.L: staight cuent-caying wie. Let s take a staight cuent-caying wie. Field lines of this conducto ae cicumfeences. Choosing one of such lines of adius R, suface enclosed by such line and applying A.L: L R d l = dl = 2π R = µ 0I L L µ = 0 I 2π R i Tiple, chapte 27.4

12 Foce between two staight cuent-caying wies A conducto ceates a magnetic field on second conducto and a foce appeas on this conducto. The same happens on fist conducto. 1 F 21 d F12 I 2 l I Tiple, chapte 27.4

13 Magnetic flux Given a suface element ds, magnetic flux though such suface element is defined as (inne poduct): dφ= ds If suface is finite (suface S): ds φ Unit: Webe Wb = T m 2 = S ds Tiple, chapte 28.1

14 Magnetic flux On a closed suface, as magnetic field lines ae closed lines, an enteing line must always exit fom volume, and magnetic flux though a closed suface is always zeo: ds Closed Suface = 0 enteing flux (-) must be equal to exiting flux (+).

15 Maxwell s equations fo Magnetostatics On 1865, J.C. Maxwell stated his fou famous Maxwell s equations, a summay of electomagnetic field. Fo steady magnetic fields (magnetic fields not changing on time), these equations can be witten as: E dl = 0 E is consevative dl = µ 0 Ii E ds = Closed Suface Gauss s law ds = Closed Suface Q ε 0 0 i Ampee s law Monopoles don t exist Fo Magnetodynamics, it s necessay modify these equations, and a no consevative electic field appeas and a new tem must be added on Ampee s law.

16 Application of A.L: tooid (cicula solenoid). Applying A.L. to middle line of tooid and to cicle enclosed by this line: N tuns i R d l = dl = 2π R = µ 0NI L L = µ 2 0 π Ni R i =0 y applying A.L. at points outside of tooid, esult is that magnetic field is zeo at any point outside tooid. =0 Tiple, chapte 27.4

17 Application of A.L: solenoid. On a solenoid, if L>>>, the magnetic field can be taken as unifom inside solenoid and null outside solenoid. Fom tooid (L=2πR): N tuns L µ Ni = L = 0 µ 0 ni Magnetic moment of a solenoid is: n = N L Numbe of tuns by unit of lenght =Ni S µ Magnetic moment by unit of volume inside a solenoid is called magnetization: µ = V NiS SL If we put a feomagnetic mateial inside solenoid, magnetic field is multiplied by thousands (with the same intensity of cuent flowing along solenoid). M = = µ 0

18 Magnetism in matte. Feomagnetism. Magnetic popeties of feomagnetic mateials can be explained by thei atomic stuctue. An electon in its atomic obit can be consideed as an electic cuent flowing though a loop. So, the electon poduces a magnetic field, and the magnetic moment (µ) of the electon can be computed. = µ In the atoms of many mateials, such magnetic moments ae cancelled, but in feomagnetic mateials, a esulting magnetic moment is not zeo: =0 Atom of Non feomagnetic mateial µ µ 0 is e - Atom of Feomagnetic mateial Tiple, chapte 27.5

19 Magnetism in matte. Feomagnetism. In feomagnetic mateials, thee ae egions (magnetic domains) with thei magnetic moments all pointing in the same diection. Magnetic moment by unit of volume is called dµ Magnetization (it is a vecto): M = dv In a domain: M 0 but in all domains M = 0 one domain alldomains Diections on wich domains ae oiented, ae called easy magnetization diections, and ae elated to the cystalline stuctue of the mateial.

20 Magnetism in matte. Feomagnetism. When we apply an extenal magnetic field ( app ), the magnetic moments at edge of domains change thei diection accoding to app. Magnetization is not zeo and a magnetic field m appeas (due to magnetization) einfocing the applied magnetic field. m depends on app though a chaacteistic of mateial called magnetic susceptibility (χ m ): m = χ So, the esulting magnetic field will be: m app = ( app + m = 1+ χm) app app m = app + m > app 1

21 Magnetism in matte. Feomagnetism. y inceasing the applied magnetic field app, some domains ae pointing in the diection of easy magnetization (akhausen effect). Such magnetization inceases m and total field still moe m app = app + m > app 2

22 Magnetism in matte. Feomagnetism. Until all domains ae pointing in easy magnetization diections close to the applied magnetic field app. m app = app + m > app 3

23 Magnetism in matte. Feomagnetism. The last step occus when all domains ae pointing in the diection of the extenal applied field. We have got the highe magnetization (satuation magnetization). Resulting field () is thousand times the applied field: = ( app + m = 1+ χm) Pemalloy: χ m = app m app = app + m > app 4

24 Feomagnetism. Fist magnetization cuve. Magnetization pocess is non linea (but only in one egion (3) of cuve), and dawing vs app we get fist magnetization cuve of a feomagnetic mateial: Fist imantation cuve of a feomagnetic mateial M 4 3 Satuation magnetization 2 1 app

25 Feomagnetism. Removing magnetization. Removing the extenal applied field, the magnetic moments etun to thei easy magnetization diections, but not those they had initially, and some magnetization emains (emnant magnetization): M 0 To cancel emnant magnetization, an opposite magnetic field must be applied (coecive field, c ): c M =0

26 Feomagnetism. Hysteesis cuve. If the coecive field gows, satuation magnetization can be eached in opposite diection to the fist. In an altenating field, magnetization and demagnetization lose enegy by fiction, and this pocess can be epesented by a hysteesis cuve (cycle): Remnant magnetization M Satuation magnetization Fist magnetization cuve Coecive field app Satuation magnetization

27 Feomagnetism. Had and soft magnetically mateials. Aea enclosed by hysteesis cuve is elated to quantity of enegy lost by fiction. Had mateials (high emnant magnetization) ae suitable to make magnets o data stoage devices. Soft mateials ae suitable to make electomagnets Had mateial Soft mateial

28 Witing and eading magnetic devices A solenoid wound aound a soft magnetic mateial can be used to oganize (wite as 0 o 1) a had magnetic mateial. Stoed infomation can be ead by electomagnetic induction. I I x 2400

Magnetostatics. Magnetic Forces. = qu. Biot-Savart Law H = Gauss s Law for Magnetism. Ampere s Law. Magnetic Properties of Materials. Inductance M.

Magnetostatics. Magnetic Forces. = qu. Biot-Savart Law H = Gauss s Law for Magnetism. Ampere s Law. Magnetic Properties of Materials. Inductance M. Magnetic Foces Biot-Savat Law Gauss s Law fo Magnetism Ampee s Law Magnetic Popeties of Mateials nductance F m qu d B d R 4 R B B µ 0 J Magnetostatics M. Magnetic Foces The electic field E at a point in