DEMYSTIFING MAGNETISM

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1 PHYS2012 ag02doc DEMYSTIFING MAGNETISM ELECTRICAL PROPERTIES OF MATERIALS What is an electic field? What ceates an electic field? fee chages electic displaceent D electic dipoles bound suface chages polaization P electic field E 1 E ( D P) 0 MAGNETIC PROPERTIES OF MATERIALS What is a agnetic field? What ceates a agnetic field? fee cuents H-field, agnetic field, agnetic field intensity H [A -1 ] agnetic dipoles bound suface cuents agnetization M [A -1 ] -field, agnetic induction, agnetic flux density [T] H M M 0 1 H H H 0 0 H M You have to be caeful in intepeting these equations as the agnetic susceptibility - not a siple nube can depend upon histoy of saple < 0 sall diaagnetic ateials > 0 sall paaagnetic ateials > 0 lage feoagnetic ateials Isotopic ateials: H M sae diection & scalas Non-isotopic ateials: H M aely in sae diection & tensos peeability of fee space µ 0 = TA -1 (NA -2 H -1 ) ag02doc Octobe 13,

2 What is a agnetic field? A oving chage expeiences a foce in a agnetic field + q F qvsin I I + I F I Lsin F qv df i dl I F out of page ight hand pal ule I What ceates a agnetic field? Moving chages cuents agnetic fields Obital otion and spin of electons in atos peanent agnets iot-savat Law 0 i dl d 3 4 agnetic fields & cuents use ight hand ules (scew & pal) dl I F ag02doc Octobe 13,

3 FUNDAMENTAL LAWS GOVERNING MAGNETISM Apee s Law line integal dl i 0 total i total depends on fee cuents and ediu (not siply the cuent though a wie); total cuent passing though the loop defined by the integation) i Fo fee cuents i total = N i nube of tuns N (agnetic devices have any tuns) J da H dl N i f i f fee cuents does not depend upon ediu i fee dl H Faaday s Law and Magnetic flux Magnetic flux da [T 2 ] Faaday s Law (geneation of electicity by tie vaying agnetic fields) ef E dl d dt Gauss s Law fo Magnetis Total agnetic flux though any closed suface is zeo da 0 No agnetic poles 2 poles of a agnet -field lines fo continuous loops -field lines ae closed ag02doc Octobe 13,

MAGNETIC FIELD SURROUNDING A LONG STRAIGHT WIRE Apply Apee s Law to the cicufeence of a cicle of adius dl 0 i 2 i 0 MAGNETIC FIELD CENTRE OF A CIRCULAR LOOP A cuent i is aintained in a thin, tightly

4 MAGNETIC FIELD SURROUNDING A LONG STRAIGHT WIRE Apply Apee s Law to the cicufeence of a cicle of adius dl 0 i 2 i 0 MAGNETIC FIELD CENTRE OF A CIRCULAR LOOP A cuent i is aintained in a thin, tightly would coil of N tuns, with a adius, R What is the agnetic field, at the cente of the coil? Z Y X Can t use Apee s Law (no syety) need to use iot-savat Law The agnetic field at the cente of a coil of adius R with one tun is found fo the iot-savat Law: i d i dl R i dl 4 R 4 R o o d 3 2 since dl and R ae at ight angles to each othe Fo N tuns and integating d i i o Ni 4R 4R 2 R o o N d N dl N 2 R 2 2 Diection of is in Z-diection, if cuent anticlockwise in XY plane (Right Hand Scew Rule) ag02doc Octobe 13,

MAGNETIC FIELD ALONG AXIS LONG SOLENOID Conside the ectangula

is the cuent though the wie and N is the nube of tuns y Apee s Law

o n I 0 N tuns: Cuent i Magnetic field ~ confined within coils dl 0

5 MAGNETIC FIELD ALONG AXIS LONG SOLENOID Conside the ectangula contou C shown in the figue It encloses a total cuent of N i whee i is the cuent though the wie and N is the nube of tuns y Apee s Law C ds L N i o Magnetic field at cente of a long solenoid Ni L o n I 0 N tuns: Cuent i Magnetic field ~ confined within coils dl 0 stat Diection of fo Right Hand Scew Rule C = 0 L ag02doc Octobe 13,

at the ean adius of the tous, if the tous width d, is uch less than its aveage cicufeence 2 The diection of is clockwise (Right

6 MAGNETIC FIELD - AIR FILLED TOROID What is the agnetic field inside an ai-coe tooid of adius and coss-sectional aea A with N tuns Cuent in Z diection Cuent in Z diection Y X R d ds The agnetic field thoughout the coil will not deviate appeciably fo its value at the ean adius of the tous, if the tous width d, is uch less than its aveage cicufeence 2 The diection of is clockwise (Right Hand Scew Rule) Applying Apee s Law aound the cicle with adius, we get Ni o dl o N i and A 2 What is the aea A? ag02doc Octobe 13,

7 MAGNETIC FIELD ALONG THE Z-AXIS FOR A CURRENT LOOP IN THE XY-PLANE The iot-savat law can be used to calculate the agnetic field along the Z axis fo a cuent loop in the xy plane i dl o d 3 4 z d dcos d dsin z x x coponents cancel d z d a z a cos z >> a a a x d z 0 i dl 0 i dl a 0 i dl a 0 i a cos z a z a 2z a ia 0 a i 0 i A z dz (2 a) A a 3/ 2 3/ z z >> a 0 i In the cente of the loop (z = 0) 2a 3/ 2 dl 2 ag02doc Octobe 13,

8 MAGNETIC DIPOLE MAGNETIC DIPOLE MOMENT The cicula cuent loop is a vey ipotant exaple because it leads to the concept of the agnetic dipole and its agnetic dipole oent ight hand scew ule p agnetic dipole oent p i A At a point along the axis z >> a cuent loop in xy-plane i z 0 2 z 3 i A Toque on a agnetic dipole in an extenal agnetic field p p sin Fo = z what is the toque when the loop is in an xz plane and in the xy plane? Daw diagas to illustate the answes Potential enegy of a agnetic dipole U p p cos Take the zeo of the potential enegy function to be at = 90 o Daw a diaga showing the dipole at angles, = 0, = 90 o, = 180 o Sketch a gaph of the potential function fo fo 0 o to 180 o How is the dipole aligned in its lowest enegy state? in its highest enegy state? M470 M872 ag02doc Octobe 13,

MATLA MAGNETIC FIELD FROM A CURRENT LOOP The iot-savat law can be used to calculate the agnetic field fo any cuent configuation i dl o d 3 4 Although this equation looks difficult to pefo

the agnitude of the agnetic field suounding the cuent loop in the xz plane To siplify the pogaing, the agnetic field is calculated in abitay units with a = 1 The liits of the egion whee the

9 MATLA MAGNETIC FIELD FROM A CURRENT LOOP The iot-savat law can be used to calculate the agnetic field fo any cuent configuation i dl o d 3 4 Although this equation looks difficult to pefo calculations with, using Matlab it is elatively easy We will conside a vey ipotant type of cuent configuation: a cuent loop of adius a, centeed at the oigin and in the xy plane We will calculate the agnitude of the agnetic field suounding the cuent loop in the xz plane To siplify the pogaing, the agnetic field is calculated in abitay units with a = 1 The liits of the egion whee the agnetic field (detecto space) is calculated is given in tes of the adius of the cuent loop a We su the contibution fo each cuent eleent d at each detecto point (x P, y P, z P ) dl d 3 ag02doc Octobe 13,

10 Steps in the calculation using cuent_loop Divide the cicufeence into N eleents of length L The cente of each eleent (x C, y C, z C ) and its (L x, L y, L z = 0) coponents ae specified by the adius of the cicle a, and an angle which is easued anticlockwise with espect to the x axis Set up a two diension gid fo the detecto points (x P, y P = 0, z P ) at which the agnetic field is calculated Fo each eleent, calculate: The displaceent ( x, y, z ) fo the cente of each eleent (x C, y C, z C ) to each detecto point P(x P, y P, z P ) The coss poduct i j k i j k dl Lx Ly Lz Lx Ly 0 x y z x 0 z y z x z x y dl i L j L k L dl Su the contibution of the agnetic field fo each eleent 3 Plot the agnetic field Fo ou agnetic field calculated in the xz plane, we can plot the agnetic field along the z axis (x P = 0) An expession fo the agnetic field along the Z axis when >> a is 2 o 2ia 1 o in ou abitay units zp We can copae the two esults, one fo the using the iot-savat Law and one using the appoxiation How well do the esults agee with the default values? Incease the ange fo the z values so that >> a by inceasing both zpin and zpax How do the appoxiation now copae? Note how the agnetic field dops off vey apidly with distance What is the y coponent of the agnetic field? Is this what you expected? You can copy and edit this poga to plot the agnetic field in an xy plane just above the cuent loop ag02doc Octobe 13,

11 MATLA SCRIPT % cuent_loop clea all close all clc % cuent loop ============================================ a = 1 ; % adius of cuent loop N = 115; % nube of eleents in cuent loop theta = zeos(1,n); % angle of cuent loop eleent xc = zeos(1,n); % xyz coodinates fo point cuent lop eleent yc = zeos(1,n); zc = zeos(1,n); dtheta = 360/N; theta(1) = dtheta/2; theta(end) = 360-dtheta/2; fo c = 2 : N-1 theta(c) = (c-1)*dtheta+theta(1); end xc = a*cosd(theta); yc = a*sind(theta); L = 2*pi*a/N; % length of each cuent loop eleent Lx = L*cosd(90+theta); Ly = L*sind(90+theta); clea theta % Detecto space (xp, yp, zp) whee is calculated =================== NP = 217; % Detecto points NP x N xpax = 8*a; % Diensions of detecto space zpin = 1*a/4; zpax = 8*a; xp = linspace(-xpax,xpax,np); zp = linspace(zpin,zpax,np); [xxp zzp] = eshgid(xp,zp); x = zeos(np,np);y = x; z = x; % Calculation of agnetic field : su ove each cuent eleent fo c = 1 : N x= xxp - xc(c); z = zzp - zc(c); y = yc(c); = sqt(x^2 + y^2 + z^2); 3 = ^3; x = x + Ly(c)*z/3; y = y - Lx(c)*z/3; z = z + Ly(c)*x/3; end = sqt(x^2 + y^2 + z^2); = /ax(ax()); % noalize to 1 ag02doc Octobe 13,

12 % GRAPHICS ===================================================== figue(1) pcolo(xxp,zzp,^02); coloap(hot) shading intep; axis equal; axis([-xpax xpax 0 zpax]); xlabel('xp');ylabel('zp'); set(gca,'xtick',[-6:2:6]); set(gca,'ytick',[0:2:6]); ectangle('position',[ ],'FaceColo','k'); title('magnetic field fo cuent loop') coloba figue(2); suf(xxp,zzp,,'facecolo','intep', 'EdgeColo','none', 'FaceLighting','phong') daspect([1 1 1]) axis tight view(-122,36) calight left coloap(jet) gid off axis off coloba title('magnetic field fo cuent loop') % along z-axis: iot-savat & appox _theoy = abs(1/zp^3); _theoy = _theoy/ax(_theoy); figue(3) index=find((1,:)==1); plot(zp,(:,index),'b'); hold on plot(zp,_theoy,''); xlabel('zp'); ylabel(''); legend('iot-savat','appox'); title('magnetic field fo cuent loop: xp = 0') ag02doc Octobe 13,

13 (au) Magnetic field fo cuent loop 2a x a z Magnetic field fo cuent loop: xp = 0 iot-savat Appox zp/a ag02doc Octobe 13,

ELECTROMAGNETS ROWLAND RINGS A Rowland ing is a tooidal ing with any windings aound its cicufeence Fo an ion Rowland ing with N windings and a ean adius, what ae the -field and the agnetic flux

14 ELECTROMAGNETS ROWLAND RINGS A Rowland ing is a tooidal ing with any windings aound its cicufeence Fo an ion Rowland ing with N windings and a ean adius, what ae the -field and the agnetic flux inside the ing? Apply Apee s Law about the cicufeence of length L= 2 N i f N i f H dl N i f H L N i f H L 2 Assue that the ion in the Rowland ing is opeated in the linea egion so that N i N i H 0 H 0 L 0 2 The agnetic flux is 0 A N i 0 A N i da A L 2 whee A is the coss-sectional aea of the ing What ae the diections of the fields, H and M? What is eant by the linea egion? ag02doc Octobe 13,

15 How would the esults be diffeent if a sall gap of length d was in the Rowland ing? i f i f d Apply Apee s Law about the cicufeence of length L H dl Fe N i f H ( L d) H d N i gap f Assue the -field is confined to the gap, then by Gauss s Law da 0 H H Fe gap o gap o Fe Hgap H Fe o o Ni f -field not liited to the axiu value of the agnetization o L d d o A N i f L d d What ae the diections of the fields, H and M? ag02doc Octobe 13,

16 ELECTROMAGNETS INFINITE SOLENOID Assue the agnetic field is totally enclosed within the coil Apply Apee s Ciculation law to the thee loops dl N I I H G L K E F D C I J A Loop ACD = 0 Loop EFGH EF L+0 - GH L+0 = 0 unifo inside coil N Loop IJKL L + 0 = NI 0 I 0n I L ELECTROMAGNETS od inside a coil Assue that the electoagnet is vey long The elative peeability of its ion coe is µ The electoagnet coil cuent is i and the nube of winding pe ete is n Give expessions fo, H and M in the ai, in the gap egion between the coil windings and the ion coe and inside the ion coe gap egion gap ai H gap H ai ion coe Fe H Fe i coil windings ag02doc Octobe 13,

17 Magnetic field of electoagnet confined to egion inside the solenoid s coil ai = 0 H ai = 0 M ai = 0 The H-field is siply deteined by the cuent i in the coil windings H Fe = H gap = H M gap = 0 1 Apply Apee s Law to a loop 1234 M H H Fe Cuent i out of page H M 2 1 X X X X X X 3 4 Cuent i into page Coss-section though electoagnet H dl n Li H L n Li H ni gap H o H ni Fe gap M H H ni Fe What akes a stong electoagnet? Why is the ion coe ipotant (what is a typical value fo µ )? Ciculation loop: squae of length L M269 M342 M415 M439 M562 M670 M919 M961 ag02doc Octobe 13,

SHORT MAGNETIC LENS Magnetic lenses ae integal pats of ost electon icoscopes A shot cuent caying coil

Conside the foce on an electon at point A taveling in the YZ plane with velocity v (0, v,0) A The

is diected in +X diection (out of page) giving ise to an X-coponent fo the velocity at a point which is

18 SHORT MAGNETIC LENS Magnetic lenses ae integal pats of ost electon icoscopes A shot cuent caying coil (solenoid) can act like a lens to focus an electon bea Z object (0,, ) A y z A electon bea v A iage Y X Conside the foce on an electon at point A taveling in the YZ plane with velocity v (0, v,0) A The agnetic field at A is (0,, ) y A y z The agnetic foce on the electon is F qv F ( v,0,0) A y z This foce is diected in +X diection (out of page) giving ise to an X-coponent fo the velocity at a point which is likewise acted upon by the agnetic field The electon how has a velocity v ( v, v,0) x y ag02doc Octobe 13,

19 And the foce now acting on the electon is F q v i j k F e vx vy 0 0 y z F ev iˆ ev ˆj ev kˆ y z x z x y electon bea deflected towad axis of otion +Z Electon at A oving paallel to +Y-axis v y F x due to z Electon acted upon by the adial coponent of the agnetic field foce on electon in +X diection +Xcoponent to the velocity y +Y axis fo the otion of the electon bea +X z adial coponent of agnetic field +Z Electon at has a velocity coponent in the +X diection F y due to z v x F z due to y Electon acted upon by the axial coponent of the agnetic field y foce on electon in -Z diection ie towads to axis focusing action y +Y axis fo the otion of the electon bea +X z adial coponent of agnetic field Thee is a coponent of the foce diected towads the Y axis in this way electons appoaching the lens in a diection paallel to the Y-axis but displaced adially fo it ae deflected towads the Y-axis The adial coponent ( z ) of the agnetic field is esponsible fo a lateal coponent of velocity which unde the action of the axial coponent ( y ) of the agnetic field gives ise to a coponent of velocity towads the axis M378 M507 ag02doc Octobe 13,

20 M269 M342 M378 M415 M439 M470 M507 M562 M670 M872 M919 M961 ag02doc Octobe 13,

FARADAY'S LAW. dates : No. of lectures allocated. Actual No. of lectures 3 9/5/09-14 /5/09

FARADAY'S LAW. dates : No. of lectures allocated. Actual No. of lectures 3 9/5/09-14 /5/09 FARADAY'S LAW No. of lectues allocated Actual No. of lectues dates : 3 9/5/09-14 /5/09 31.1 Faaday's Law of Induction In the pevious chapte we leaned that electic cuent poduces agnetic field. Afte this