Unit 7. Sources of magnetic field

Transcription

1 Unit 7 ouces of magnetic field 7.1 ntoduction 7. Oested s expeiment 7.3 iot and avat s law 7.4 Magnetic flux 7.5 Ampèe s law 7.6 Magnetism in matte 7.7 Poblems Objectives Use the iot and avat s law to calculate the magnetic field ceated in the cente of a cicula loop flowed by a cuent Define the flux of a magnetic field. Use Ampèe s law to calculate the magnetic field ceated by staight conductos, cicula loops and coils. Explain the feomagnetism by mean of theoy of Weiss domains. Know the hysteesis cuve. 7.1 ntoduction n unit 6 we have studied the foces poduced by magnetic fields on moving chages and electic cuents. n this unit, we ae going to study how ceate such magnetic fields, and we ll see that thei oigin ae the moving chages and the electic cuents. Geneation of magnetic fields has an essential technological impotance, since a lot of applications ae based in this phenomenon: geneatos, ecoding heads, magnetic disks, devices fo magnetic esonance, etc 7. Oested s expeiment This expeiment was caied out by Hans Chistian Oested (pofesso of Electicity, Galvanism and Magnetism) in 18. Until then, phenomena of Electostatics, Electic cuents and Magnetism wee undestood having diffeent bases, but this expeiment was the fist demonstation that these thee phenomena wee elated. The expeiment was involved in poduce an electic cu- 1

2 ent nea a compass needle, poving that this cuent could tun the needle pependiculaly to the cuent, as can be seen on Figue 7-1. N N N O E O E O E a) b) c) Figue 7-1. Oested s expeiment. n a) the compass needle is tuned accoding the teestial magnetic field. n b) a cuent in a lowe plane to the plane of the compass needle flows. n c) a cuent in an uppe plane to the plane of the compass needle flows. The needle is tuned in opposite diection than in b) The expeiment was useful to state that the electic cuents ae the souces of magnetic fields. t poved that, although the esting electic chages lack magnetic effects, the electic cuents, that is the moving chages, poduce magnetic fields, in a simila way than magnets. The main advantage ove magnets is that contolling the intensity of cuent, magnetic fields with diffeent magnitudes can be ceated, something impossible by using a magnet. This expeiment was the announcement of an age whee electicity and magnetism went to be two demonstations of a same inteaction: the electomagnetism. 7.3 iot and avat s law The magnetic field ceated by a moving electic chage on any point P on vacuum comes fom expession: q q v v = Equation π P n this equation, q is the magnitude of the chage, v its velocity and the vecto going fom point whee the chage is placed to point P whee magnetic field wants to be calculated. esides, a new constant called magnetic pemittivity of vacuum ( ) appeas. t s a univesal constant whose magnitude is: = 4π 1-7 NA - Electic cuents ae usually handled instead electic chages, and it s inteesting to expess the magnetic field ceated by an electic cuent. Let s take a diffeential element of a conducto with length d l, enclosing a diffeential volume dv. The chage dq contained in this volume comes fom definition of intensity of cuent and the time dt that a chage takes to cove dl: dq= dt. dl On the othe hand, dl is elated to dift speed v a = dt

3 o, applying supeposition pinciple using Equation 7-1 (see Figue 7-), magnetic field ceated by on a point P will be: v dl dq va dt dt dl d = = = π 4π 4π A d d dl d l = 3 α P 4π Equation 7- Figue 7-. Diffeential magnetic field poduced by an element of cuent in the point P Equation 7- is known as iot and avat s law. Two details equie we be caeful when applying this law. On fist, we must note that the diection and sense of magnetic field comes fom a coss poduct between d l and. Then, applying the scew o ight hand ule, diection and sense of magnetic field can be stated; note that is a vecto going fom the diffeential element of conducto to point P. On the othe hand, it s necessay to undeline that iot and avat s law is not integated; it s useful to get the diffeential magnetic field ceated by a diffeential element of conducto, but not fo a long conducto. n ode to get the magnetic field ceated by a long conducto flowed by an intensity of cuent, supeposition pinciple must be applied using Equation 7-: dl = 3 4π Conducto Equation 7-3 This geneal equation will be applied on following section to calculate the magnetic field poduced by a cicula loop flowed by a cuent on its cente. t can be also applied to some othe geomety, but mathematical computations become moe difficult. Magnetic field poduced by a cicula loop with a cuent in its cente Let s suppose a cicula loop (adius R) flowed by a cuent as can be seen on pictue. The objective is calculating the magnetic field ceated on its cente. n this case, the solution can be simplified due to some aspects: the intensity of cuent is the same (constant) fo all diffeential elements of conducto, and then can be taken out of integal; coss poduct R dl u Figue 7-3. Cicula loop flowed by a cuent 3

4 dl R fo all diffeential elements of conducto point in the same diection and so, we can integate only the modulus of magnetic field ( d l is the tangent vecto to cicumfeence on each point); the distance between each diffeential element of conducto and the cente of loop is also a constant. Fo any diffeential element d l = d l R = dlr u, whee u is the unit vecto pependicula to plane of loop. Note that sense of u is also given by applying the scew ule to the sense of intensity of cuent. Applying Equation 7-3: cente πr πr d R d R l l = u πr u u 3 3 4π = R 4π = = R 4πR R When a point diffeent than cente of loop is consideed (but on the flat suface of loop), modulus of magnetic field is difficult to calculate due to lack of symmety. n addition, if a point out of this flat suface is consideed, magnetic field isn t pependicula to the plane of loop. t can be seen that a cicula loop behaves in the same way that a magnet, with its noth pole o pole whee the lines of field go out, and its south pole, o pole whee the lines of field go in. N Figue 7-3. Field lines fo a loop and fo a magnet Example 7-1 The magnetic field in the cente of a loop of 5 cm of adius flowed by a 3 A cuent is: 7 4π 1 3 = = 37,7 T,1 7.4 Magnetic flux f we have a diffeential element of aea d in a magnetic field, it s defined the diffeential flux of magnetic field though such diffeential suface as inne poduct between and d d Φ = d 4

5 Magnetic flux is measued in Tm, in.. This unit is called webe (Wb). f we conside not a diffeential aea, but any aea, magnetic flux of magnetic field though aea is the addition (integal) of all the diffeential fluxes: Φ = d Equation 7-4 f magnetic flux is unifom along suface, then can be witten out of integal, and magnetic flux is: Φ = d = d = = sinα Equation 7-5 α d d Figue 7-4. Magnetic flux though a suface Related to the flux, thee is an impotant diffeence between magnetic flux and electic flux. The electic flux though a suface depends on the net chage enclosed inside the suface (Gauss s law), but magnetic flux though a closed suface is always zeo d = Equation 7-6 closed suface This fact is due to the non-existence of magnetic monopoles. As a monopole can t exist isolated, inside a closed suface we ll always have the same quantity of noth and south magnetic poles, and so, the same numbe of magnetic field lines going in on the suface and going out, and net magnetic flux will be zeo. Fo this eason, it s said that the magnetic field is a solenoidal vecto field. ut as positive and negative electic chages can exist isolated, a net electic flux can exist though a closed suface, and so Gauss s law can be stated. A simila law to Gauss s law will be stated fo magnetic fields (Ampeè s law), as we ll see below, but instead conside the magnetic o electic flux, we ll conside the magnetic integal along a line L (ciculation), and instead the chages being inside the suface, we ll take the intensities of cuent cossing the suface whose bode is the line L. 5

6 E Figue 7-5. Electic dipole. t s possible to suound a chage with an enclosed suface; all the field lines will coss it in the same sense and theefoe thee will be a net flux Example 7- Figue 7-6. Magnetic dipole. t isn t possible to suound a magnetic pole with an enclosed suface and obtain a net flux. The field lines will always coss the suface twice, enteing and exiting Calculate the magnetic flux though a coil of 1 tuns as that shown on pictue, placed inside a unifom magnetic field =,3 j T. olution Magnetic flux though the coil is Φ = = N cos θ = 1, cos6º =,6 Wb x z θ = 6º 5 cm =,3 T 8 cm y 7.5 Ampèe s law As magnetic poles always can be found by couples, magnetic field lines ae closed lines. The integal of magnetic field along any closed line C = d l (ciculation of magnetic field) won t be zeo. This magnitude can be easily computed using Ampèe s law, that we ae only to state (we don t demonstate it): The ciculation of magnetic field vecto along any enclosed cuve equals the poduct of the constant by the addition of the intensities of cuent cossing any suface bodeed by the cuve. The sign of the intensity will be positive when it was in accodance with the scew o the ight hand ule with the sense of the ciculation, and negative in anothe case. = dl = C Equation 7-7 t means that the ciculation only depends on the cuents cossing a suface bodeed by the cuve, but doesn t depends on those intensities not cossing this suface. 6

7 1 3 dl Figue 7-7. The closed cuve (black line) suounds a net cuent, and so C = dl Figue 7-8. The closed cuve (black line) suounds a net cuent , and so C = ( ) 4 n the same way that happened with Gauss s law of Electostatics, Ampèe s law can be used to calculate magnetic fields poduced by cuents, but only in those cases whee high symmety makes possible the calculation of the ciculation and solving fo the magnetic field. esides, if we choose a field line as line to integate along, at any point of this line, d l and will be paallel, and inne poduct is simplified to poduct of modulus: dl = dl We ae now going to apply Ampèe s law in seveal cases whee these conditions of symmety ae veified. Magnetic field ceated by an infinite staight caying-cuent wie Let s take an infinite conducto pependicula to the plane of pape flowed by a cuent enteing on pape; we want to compute the magnetic field poduced by this cuent on any point placed at a distance x of conducto (Figue 7-9). dl x n ode to apply Ampèe s law, we choose a cicumfeence of adius x, pependicula to conducto and centeed on a point of such conducto. This cicumfeence is a field line of magnetic field, being the magnetic field vecto tangent to this line at any point. On the othe hand, as distance to conducto is the same fo all the points of line, modulus of magnetic field will also be the same. o, ciculation of magnetic field along this cicumfeence is C = dl = dl = πx Figue 7-1. Ciculation of magnetic field along a cicumfe- On the othe hand, consideing the suface of a ence. n the figue the cuent is disk bodeed by this cicumfeence, the intensity pependicula to the plane and cossing this disk is (positive because its sense is in enteing on pape. accodance with the sense of ciculation chosen). Then, applying Ampèe s law becomes C = d l = πx = = πx πx = Equation 7-8 This equation is useful to calculate the modulus of magnetic field poduced by an infinite staight caying-cuent wie on a point at a distance x fom wie. 7

8 Magnetic field inside and outside a conducto Let s take a cylindical conducto with adius a flowed by an homogeneous cuent whose density of cuent is J = = ; we want to calculate the πa magnetic field on points placed on two diffeent aeas of the space: points outside conducto (>a) and points inside conducto (<a). J a C dl J C1 a applying Ampèe s law to C : a) Outside conducto f we think the conducto as a set of infinite staight wies, magnetic field on any point will be tangent to a cicumfeence pependicula to conducto and centeed on a point of axis of conducto, being these cicumfeences field lines (modulus of magnetic field is constant along these cicumfeences). Choosing a cicumfeence (C ) with adius >a and Jπa Ja dl = l = π = = Jπa = = π C b) nside conducto We can epeat calculations and apply Ampèe s law to a cicumfeence C 1 with adius <a. ut in this case we must be caeful because the intensity cossing a disk bodeed by C 1 isn t the whole intensity, but only a pat coesponding to suface Jπ. Then: C1 dl = l = Jπ J π = = Jπ = = π ½ Ja Magnetic field in both aeas can be epesented on pictue. Magnetic field is zeo on points of the axis and a maximum is eached on points of suface of conducto. a Figue Magnetic field inside and outside a conducto flowed by a cuent 8

9 Example 7-3 Calculate the magnetic field on point P(,d), poduced by two infinite staight caying-cuent wies of opposite senses and intensity, placed at a distance d. y P (,d) olution d x Magnetic field poduced on P is the addition of magnetic field poduced by each cuent: = 1 + Applying Equation 7-8 and the sense of the field accoding Figue 7.1: y u 45 d α 1 d 1 x 1 = i = u πd πd 1 u = sen45i cos45 j = ( i + j) = ( i + j) Finally: = + 1 = i + πd πd 1 ( i + j ) = ( i j ) 4πd Definition of ampee When two conductos with cuent ae nea, each conducto ceates a magnetic field on second conducto, and then a magnetic foce appeas on second conducto. ut the same happens on fist conducto, and magnetic foces appea on conductos. Fom this phenomenon, a definition fo ampee (the unit of intensity of cuent) can be given. Let s imagine two staight caying-cuent wies, paallel, infinite, placed at a distance d each othe and flowed by cuents 1 and in the same senses; magnetic field poduced by 1 on points of is: 1 1 = πd and magnetic field poduced by on points of 1 is: = πd 9

10 n both cases, magnetic field is pependicula to the plane made up by both cuents. As a consequence, both foces will appea on 1 and. These foces, fo a length l of conducto, will be: 1 F 1 d F1 F = 1 = πd l( ) 1 1l u l F = 1 = πd l( ) 1 l 1 u 1 oth foces ae equal in modulus and of opposite senses. Then, both conductos attact themselves. f cuents would have opposite senses, then the foce between them would be ejecting. o, a definition fo ampee is: One ampee is the intensity of cuent having two infinite, staight and paallel conductos when, placed at a distance of one mete, a foce of 1-7 N by mete of length appeas between themselves. Magnetic field inside a tooidal coil A tooidal coil is a coil (set of loops) whose loops ae aanged as if they wee a ing (see the pictue). f tooid is naow ( a b <<< R ), magnetic field inside the tooid can be taken as unifom (constant). esides, magnetic field on points of axis of tooid (the middle line) is paallel to the cicumfeence of middle line. t s easy to justify consideing the tooid as a set of loops and applying supeposition N i pinciple; if we conside a point on cente of a loop (1), the magnetic field due to this loop ( 1 ) is tangent to the cicumfeence of middle line of tooid (adius R). Magnetic fields due to two symmetical loops ( and 3) ae and 3. Addition of and 3 is also tangent to cicumfeence, because thei pependicula components ae cancelled. o, the total magnetic field on points of cicumfeence epesenting the middle line of tooid is tangent to this cicumfeence. esides, this magnetic field is constant along this line, due to the symmety of tooid. Then, applying Ampèe s law to this cicumfeence epesenting the R 3 b a 1

11 middle line of tooid: C = d l = πr C and consideing the disk bodeed by this cicumfeence, this disk is cossed once by each loop in the same sense, and then the total cuent cossing the disk will be = N Then, fom Ampèe s law, we can get the magnetic field on points inside the tooid: N πr N = πr = Equation 7-9 The sense of magnetic field points accoding scew o ight hand ule fo intensity of cuent. On points outside the tooid, if we conside a cicumfeence (adius ) passing by this point (both fo <R o >R), the intensity of cuent cossing the disk bodeed by this cicumfeence is zeo, and magnetic field on points of such cicumfeences is zeo. Magnetic field inside a staight caying-cuent coil Let s conside a staight caying-cuent coil having a length L, aea, N tuns and flowed by an intensity of cuent. f coil is naow, magnetic field inside coil can be consideed as unifom, and if coil is vey long, magnetic field outside the coil can be neglected. N tuns L Figue 7-1. Coil with length L, N tuns and flowed by a cuent. f coil is naow (little adius compaed with length) magnetic field inside can be consideed as unifom f we conside this staight caying-cuent coil as a paticula case of tooidal coil with adius infinite, we can apply equation 7-8, witing L instead πr : N = n L = Equation 7-1 N being n = the numbe of tuns by unit of length on coil. L ense of magnetic field points accoding scew o ight hand ule fo intensity of cuent. On points outside coil, magnetic field can be neglected if coil is naow enough. Magnetic moment of a coil is m = N. Magnetic moment by unit of volume is called magnetization vecto (M ). This vecto is pointing in diection of 11

12 axis of coil and its sense is given by the scew o ight hand ule fo intensity of m N N cuent on coil. ts modulus is M = = = = n V L L o, magnetic field inside a coil and magnetization of a coil ae elated though equation = M Equation 7-11 Example 7-4 Calculate the magnetic flux though a coil of length 5 cm, adius 1 cm and 4 tuns; a 4 A cuent flows along coil. olution As length of coil is 5 times its adius, magnetic field inside coil can be taken as unifom: = N L And magnetic flux cossing the coil will be N times the magnetic flux cossing a loop: 7 N N π 4π 1 4 4π,1 Φ = N = = = =,11 Wb L L,5 7.6 Magnetism in matte. Magnetization. Feomagnetism. Hysteesis cuve. Matte is made up by atoms whee electons ae moving. Using a simple atomic model, electonic obitals aound the nucleus of an atom can be consideed like cicula electical cuents and, theefoe, equivalents to loops flowed by cuents. Then, an electon tuning in a cicula obital of adius with peiod T and speed v, can be consideed equivalent to a loop having the same adius and flowed by an intensity = e/t (e is the electic chage of an electon); magnetic moment of such loop will be m = = π u = π u = u e e ev T π v m m i Figue The electonic movement is equivalent to a cicula cuent v e 1 being u a unit vecto in the diection of aea vecto. n this way, matte can be seen as made up by a high quantity of loops flowed by cuents, that is, magnetic moments, equivalent to magnets as it was explained on section 7.3. esides this magnetic moment associated to the movement of electons, it s necessay to add the intinsic magnetic moment of each electon associated to its spin. n this way, each obital has a magnetic moment associated to it. ometimes, these magnetic moments ae andomly oiented on an atom, cancelling thei effects and esulting in a zeo magnetic moment fo the atom.

13 ut it can also occu that atoms of some substances have magnetic moment not zeo. n these cases, magnetization (M ) on a diffeential volume can be computed as: dm M =, being measued in Am -1 dv Theefoe, a mateial is not m magnetized if the addition of all magnetic moments in a given volume is zeo. M = M Figue 7-1. Magnetization in a mateial is the addition of magnetic moments by unit of volume n substances whee magnetic moments ae cancelled on a volume (magnetization zeo), if an extenal magnetic field is applied ( ap ), magnetic moments of atoms can be oiented, magnetizing the substance. uch magnetization is linealy elated to applied magnetic field, though a dimensionless numbe (chaacteistic fo each substance) called magnetic susceptibility χ (χ m ), in such way that M = m ap. Magnetization will einfoce o will decease the applied magnetic field, in accodance with the intenal stuctue of substance. ubstances einfocing extenal magnetic field ae called paamagnetic substances, and those deceasing the extenal magnetic field ae called diamagnetic substances; but in both cases, magnetization is vey little and also the einfocing o deceasing effects, being vey difficult to detect them. Having in mind Equation 7-11, magnetization of such substances poduces an additional magnetic field M = χ m = Positive values of χ m coespond to paamagnetic substances, and negative values to diamagnetic substances. On Table 7-1, values fo seveal substances ae summaized. Mateial χ m To ºC ap χ m < (a) M ap χ m > (b) M Figue Diamagnetic (a) and paamagnetic (b) substances m ap Aluminium,3 1-5 Coppe -, Diamante -, 1-5 Gold -3,6 1-5 Mecuio -3, 1-5 ilve -,6 1-5 odium -,4 1-5 Titanium 7,6 1-5 Volfamio 6,8 1-5 Oxygen (to 1 atm) Table 7-1. Magnetic susceptibilities of seveal substances 13

14 o, esulting magnetic field can be got afte add magnetic field due to magnetization of mateial to applied magnetic field: = + = + χ = ( 1 + χ ) = ap m ap m = 1 + χ m is called elative magnetic pemittivity, also being dimensionless. ap m ap ap ap = M = ap M Figue n diamagnetic substances, esulting magnetic field is deceased due to magnetization of mateial ap = M = ap M Figue n paamagnetic substances, esulting magnetic field is einfoced due to magnetization of mateial As changes on applied magnetic field ae vey little fo both types of mateials, paamagnetic and diamagnetic mateials aen t technologically inteesting mateials. nstead, feomagnetic mateials ae technologically vey inteesting mateials, with a lot of applications. Feomagnetism. Weiss domains. Mateials having a net (not zeo) magnetic moment on each atom ae called feomagnetic mateials. When a magnetic field is applied to these mateials, the esulting magnetic field is inceased up to some thousand times, having consequently a lot of technological applications. ehavio of these mateials can be explained by the theoy of Weiss domains set up by Piee Weiss in 197. Accoding this theoy, magnetic moments of atoms on a feomagnetic mateial aen t andomly oiented, but they ae oiented along some specific diections called diections of easy magnetization. These diections ae elated with the histoy of mateial when cystals wee made up. Mateial is divided in aeas (domains), having each aea a diection of easy magnetization, being all magnetic moments oiented in the same diection inside each domain. 14

15 Figue Diagam of magnetic domains in a cubic model and in a hexagonal model, with two and thee diections of easy magnetization. The pocess to magnetize a mateial fo fist time can be seen on Figue When no magnetic field is applied to a mateial that has neve been magnetized, mateial stats on situation a) of Figue 7-19: magnetic domains ae oiented on diections of easy magnetization. When an extenal magnetic field is applied, as on b), those domains having a component of its magnetization vecto in the same sense that ap gow against the othe domains. n this way, some domains gow (the ones of favoable magnetization) and some domains educe. The situation continues in c) with the inceasing of ap until the favoed domains can t incease thei size, and finally in d) all magnetic moments ae tuned in the same diection than the applied magnetic field, aiving to magnetization of satuation. n 1919, akhausen checked that "clicks" could be listened when the extenal caies to the Weiss domains to be aligned. At this stage, even inceasing the applied magnetic field, magnetization won t incease. This last stage of oientation of moments out of diections of easy magnetization of domains is evesible, disappeaing when field is emoved; but the gowth of favoed domains at the expense of the no favoed domains will keep and will be esponsible of the emnant magnetization (M R ) of the mateial, elated with the phenomenon of hysteesis. a) b) ap = M = ap M c) d) ap M ap M Figue Fou stages of gowing of magnetic domains 15

16 This pocess can be epesented dawing magnetization (M) against applied magnetic field ( ap ), as can be seen on Figue 7- on called fist magnetization cuve. Each piece of cuve is elated to a stage of figue Figue 7-. Fist magnetization cuve When a value of tempeatue is eached, phenomenon of feomagnetism disappeas (Cuie s tempeatue), due to themal agitation of atoms. Each feomagnetic substance has its own Cuie s tempeatue (77 ºC fo ion); above this tempeatue, mateial behaves as paamagnetic. On feomagnetic mateials, magnetization (and so, the magnetic field) can incease even thousand times of applied magnetic field. Magnitude of magnetic susceptibility (χ m ) can ise up to, but not being linealy elated to the applied magnetic field and depending on the pevious magnetic histoy of mateial. ubstance ubstance Cobalt 5 Nickel 6 Commecial ion 6 High ion puity 1 5 upemalloy (79% Ni, 5 % Mo) 1 6 Table 7-. Relative magnetic pemittivity of some feomagnetic substances. Fo this eason, it s said that such feomagnetic mateials have memoy. This fact can be explained studying the hysteesis cuve. Hysteesis cuve. When a feomagnetic mateial has been magnetized fo fist time aiving to satuation magnetization, if applied magnetic field is emoved, only stage d) is evesible, eaching the emnant magnetization when applied magnetic field is completely emoved. n ode to emove the emnant magnetization, a magnetic field opposite to fist one must be applied, the Coecive field ( c ). nceasing this opposite magnetic field, we aive to satuation magnetization, but in opposite sense than the befoe satuation magnetization. Removing this field and inceasing in opposite sense, we can each a new emnant magnetization and a new coecive field, completing the hysteesis cuve (Figue 7-1). Afte the fist magnetization cuve, mateial moves along the hysteesis cuve, being able to emembe its magnetic histoy. Aea of this hysteesis cuve is popotional to lost enegy as heat in the pocess of magnetization demagnetization. 16

17 Figue 7-1. Hysteesis cuve Accoding the shape of hysteesis cuve, two types of magnetic mateials can be stated: soft magnetic mateials, that ae those with a low emnant magnetization, and had magnetic mateials, with a high emnant magnetization, such as it is shown on figue 7-. Figue 7-. Left: oft magnetic mateial, with low emnant magnetization. Right: Had magnetic mateial, with high emnant magnetization. oft mateials ae useful as coes of tansfomes (whee low enegy must be lost) o as electomagnets (magnets contolled by an electic cuent). Had mateials ae useful to build pemanent magnets (high emnant magnetization) o to build magnetic memoies to stoe infomation (ecoding infomation by oienting the domains of a feomagnetic mateial). 17

18 Poblems 1. Figue epesents thee paallel staight cayingcuent wies, of infinite length, flowed by intensities, and 3, in the same sense. Compute magnetic field ceated on point P. ol: = (-13i - ) 1πa j Y P a 3 a a X. Two paallel, staight caying-cuent wies, sepaated a distance d, cay the same intensity in opposite senses. Compute modulus of magnetic field on point P. ol: = d/π(4r + d ) R d 3. A chaged annulus (adii a and b) with suface density of chage σ, tuns with angula speed ω aound its axis. Compute the magnetic field ceated in its cente O. ol: ( σω/ )( b a) k 4. An infinite staight conducto is flowed by an intensity 1 = 3 A. A ectangle ACD, whose sides C and DA ae paallel to the staight conducto, is placed in the same plane that the conducto, and it s flowed by an intensity =1 A. Compute the foce acting on each side of ectangle due to magnetic field ceated by the staight conducto. ol: F AD F C F A = (- i ) N = j N = 4, j N = - F CD 5. A unifom magnetic field = i T acts on space, in positive sense of X axis. a) Which is the magnetic flux though suface abcd on figue? 4 cm b) Which is the magnetic flux though suface befc? c) And though suface aefd? a ol: a) Φ = -,4 Wb 3 cm b) Φ = c) Φ =,4 Wb Z d 1 X D A Z O 1 cm 1 cm b c Y P ω a b 5 cm C σ Y cm y x e f X 18

19 6. An infinite staight conducto z'z is flowed by a cuent of intensity. The ectangula suface of figue with sides a and b can tun aound its axis X'X paallel to z'z; distance between both axes is c. nitially, the plane of suface and conducto is the same. Compute the change of magnetic flux ceated by though ectangula suface when it tuns a π/ angle aound x'x. b c + a ol: Φ Φ1 = ln π c a Z Z c X a X b 7. A vey long coaxial wie is made up by two concentic conductos with adii a, b and c. Along conductos flow the same intensity of cuent but in opposite senses. Compute the magnetic field on points placed at a distance fom axis of cylinde: a) < a (inside the inne conducto), b) a < < b (between both conductos) c) b < < c (inside oute conducto), and d) > c (outside oute conducto). ol: a) /πto b) /π c) ((c - )/(c -b ))/π b c a d) n all cases, is pependicula to the axis of wie, and sense given by the ight hand ule. 19

20 GLOARY Magnetic pemittivity of vacuum: univesal constant whose value is = 4π 1-7 NA - iot and avat s law: The diffeential magnetic field d due to an element of cuent d l on any point comes fom (being the position vecto) d d l = 3 4π Magnetic field ceated by an infinite staight caying-cuent wie at a distance : = π Pependicula diection to the plane made up by cuent and point, and sense given by the scew ule to the sense of intensity of cuent. Magnetic field inside a coil of N tuns and length L flowed by a cuent = N L Diection of axis of coil and sense given by the scew ule fo intensity. Flux of magnetic field though a suface d = Webe (Wb): Unit of magnetic flux in the, equivalent to Tm Ampèe s law: Line integal (ciculation) of magnetic field vecto along a closed cuve equals the poduct of constant by the addition of intensities cossing any suface bodeed by the cuve. The sign of intensity will be positive if it points accoding the ule of the ight hand with espect to sense of ciculation, and negative in contay case. dl =