Dr. Fritz Wilhelm page 1 of 19 C:\physics\230 lecture\ch29 magnetic fields.docx; S: 5/3/ :01:00 PM P:

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1 D. Fitz Wilhelm page of 9 C:\physics\23 lectue\ch29 magnetic fields.docx; S: 5/3/29 2:: PM P: Homewok: See website. Table of Contents: 29. Magnetic Fields and Foces, Motion of a Chaged Paticle in a Unifom Magnetic Field, a Moement pependicula to the magnetic field; Cycloton fequency, Applications, a Velocity Selecto, Foce on a cuent caying conducting wie in a magnetic field, Toque on a cuent loop in a magnetic field, The Hall Effect, 9 Addendum: A) Deiation of the law of iot-saat, ) Magnetic Foces on Moing Chages, in geneal, 6 C) Magnetic foces on moing chages in paallel, 6 D) Calculating the magnetic field though the ecto potential, 9

2 D. Fitz Wilhelm page 2 of 9 C:\physics\23 lectue\ch29 magnetic fields.docx; S: 5/3/29 2:: PM P: Magnetic Fields and Foces. We all know how ba magnets attact some metallic objects like ion files. Many expeiments hae eealed that magnetic field lines always stat in one pole and end up in anothe. Thee ae no magnetic monopoles out of which magnetic fields aise. This is in exact contast to electic fields fo which thee ae positie o negatie chages. (The magnetic fields geneated by magnets hae thei oigin in cicula cuents inside of the so-called paa-magnetic mateial.) Let us appoach the concept of magnetic fields by contasting them to electic fields: The mathematical desciption fo electic fields emeging fom single chages was: ρ (29.) die = ε Applying a olume integal and using Gauss theoem this leads to the Gaussian law: (29.2) diedv EdA Q = = = chage inside the olume ε ε olume suface of the olume Fo a magnetic field we hae always : (29.3) di = This is the same as saying that magnetic fields do not hae souces o sinks o monopoles. Magnetic field lines appea between the poles of a pemanent magnet. One is called the Southpole, the othe the Nothpole. y conention, we say that the magnetic fieldlines ae diected fom the Nothpole to the Southpole. No matte how many times we cut a ba magnet in half, we always end up with two poles, which attact othe magnets. The Southpole of one magnet is attacted by the Nothpole of anothe magnet, and ice esa. We hae found in the last centuy that all pemanent magnets ae due to many cicula little cuents inside of the magnetic mateial. The magnetic field of the eath is due to a huge cicula cuent of molten ion inside the eath. This cuent, and with it the magnetic field of the eath, has changed duing the geological histoy of the Eath. The magnetic poles do not coincide pefectly with the geogaphic poles. And actually, the magnetic Southpole coesponds oughly to the geogaphic Nothpole. The N-point of the compass needle points to the magnetic Southpole of the Eath, which is the geogaphic Nothpole. It is easy to obsee that any moing chage q is deflected when enteing a magnetic field accoding to: (29.4) F = q F = q Fom this we can deduce the unit fo the magnetic field, which is (29.5) q The magnetic field lines point into the page. The foce points upwads. Newtons s tesla T Gauss Coulombs m 4 [ ] = = = =

3 D. Fitz Wilhelm page 3 of 9 C:\physics\23 lectue\ch29 magnetic fields.docx; S: 5/3/29 2:: PM P: The foce is of couse pesent in addition to the foce ceated by an electic field. The total foce on a chage q is theefoe: F = q E+ Loentz Foce (29.6) ( ) The wok done by the foce of a unifom magnetic field (not time dependent) on a moing chage is always as we can easily see (A time-dependent magnetic field ceates an electic field, see late): dw ( ) (29.7) = F ds = q ds = is paallel to ds The foce is pependicula to both the elocity and the magnetic field ecto. ds ds F, simila to the gaitational foce on a planet in obit, which also does not do any wok because the foce is pependicula to the elocity. Fom the Wok-Enegy theoem we know that 2 (29.8) W = K = K2 K = m 2 As wok is, thee can be no change in the kinetic enegy of a chage in a magnetic field. The diection of the elocity of a chaged paticle in a magnetic field can change, but not its magnitude, o its kinetic enegy Motion of a Chaged Paticle in a Unifom Magnetic Field. 29.2a Moement pependicula to the magnetic field; Cycloton fequency: F F 2 m 2 q m q= = mω ; = ; = m q Assume that we hae a positiely chaged paticle, like a poton, injected into a unifom magnetic field such that the initial elocity of the paticle is pependicula to the field. Assume that the field points into the plane: The chaged paticle will expeience a foce pependicula to both the elocity and the magnetic field. It expeiences a centipetal acceleation which causes it to moe in a cicle: F = q = ma (29.9) (29.) ω = q = cycloton fequency m

4 D. Fitz Wilhelm page 4 of 9 C:\physics\23 lectue\ch29 magnetic fields.docx; S: 5/3/29 2:: PM P: Applications. 29.3a Velocity Selecto : We can use the set-up of pependiculay cossed electic and magnetic fields to select exact elocities out of a paticle beam. In the pictue below the magnetic field points into the plane. We place a paallel plate capacito inside a unifom magnetic field. The paticle beam with aying elocities points fom the left and entes the capacito. The electic field points fom top to bottom. Only those paticles will pass though the cossed field aea fo which the upwad magnetic foce equals the downwad electic foce, esulting in paticles with the speed equal to E = qx F = q Fz = x y x F = qe E We hae to coss them in such a way that the esultant Loenz foce on the paticle is. qe = q Choose : x, y q = qxyk (29.) Choose : E = Ezk qe = qezk Ez qez = qxy = x y Only paticles with this exact elocity will continue in a staight line, all othes will be deflected up o down and hit the capacito plates. If we inject these paticles with an exactly known elocity into a pependicula magnetic field, the paticles will be deflected into a cicula motion accoding to (29.9). y measuing the adius of thei cicula motion the atio between mass and chage can be exactly detemined. (Thomson s e/m expeiment). We see hee also that the atio between the electic and the magnectic field stength has the dimension of a elocity : m/s (We shall see late that in an electomagnetic field this atio is equal to the speed of light c.)

5 D. Fitz Wilhelm page 5 of 9 C:\physics\23 lectue\ch29 magnetic fields.docx; S: 5/3/29 2:: PM P: Foce on a cuent caying conducting wie in a magnetic field: If we inset a conducting wie into a magnetic field, this field will obiously excet a foce on each one of the conducting electons o chages moing with the dift elocity d though this wie. The total numbe of chages in a cylindical segment of wie with length L and cosssection A is simple to calculate: j= ρ = nq cuent I times dt: (29.2) ρ A dt = j A dt = I dt = dq q j E V Δx It is equal to the chage density times the olume of the N cylinde ρq AL = qal. We get an infinitesimal V foce df acting on a small potion of the wie by using just a small distance fo the length, i.e. Δx. When a cuent I is flowing with the speed =dx/dt the infinitesimal amount of chages affected in a small segment of wie is gien by ρq A dx = ρqadt. Now we ecognize that this expession is equal to the total In the fomula fo the foce on a single chage q we hae to tansfom this expession fo a single chage into the expession fo a small amount of chages dq passing though a segment of wie with coss section A and length dx. dq = I dt (29.3) df = dq = I dt We can wite the elocity ecto as a poduct of its magnitude and a unit ecto. We then cancel the elocity magnitude in the numeato with the denominato. ds ds (29.4) df = I = I u ds u = ds is a small segment of the cuent caying wie in the diection of the wie, which is the diection of the cuent density: That leaes us with the poduct (29.5) df = Ids u = Ids A Caution with the definitions hee: N = numbe of chages 9 q=indiidual elementay chage= ±.6 C (29.6) Q=Nq=total chage; nv = numbe density of chages dq=idt = small amount of chages flowing though a segment of wie dq nv q = cuent density=j; in ou model: Volume V=A x; I=j A= = j da dt A

6 D. Fitz Wilhelm page 6 of 9 C:\physics\23 lectue\ch29 magnetic fields.docx; S: 5/3/29 2:: PM P: Fo a small potion of the wie we get the infinitesimal foce : df = I ds (29.6) ( ) df df = Ids becomes fo a staight wie of length L: F = I L (29.7) ( ) q Ids In geneal, we must integate oe the whole length of the cuent caying wie to get the total foce exeted on this wie: df Ids (29.8) df = I ( ds ) line line Note the diection of the foce : It is pependicula to both the line-segment and to the magnetic field lines. Whenee the cuent and the magnetic field ae in the same plane, the foce is pependicula to that plane. (Reiew the ight hand ule which is alid fo all coss poducts, of couse.) Hee, to the left, the cuent is diected upwad, the magnetic field is diected into the page, the foce is diected to the left. A flexible wie will be bent to the left. L ' If the field is unifom (constant), the integation is just oe the line and is nothing but the ecto sum oe infinitesimal segments making up the line. The ecto sume connects the tail of the fist ecto segment with the head of the last ecto segment. This ecto sum is theefoe equal to the staight line ecto connecting the initial point of the wie to its endpoint. summaize: ds This sum is if the line foms a closed loop. (Like the total elocity o displacement ecto fo a closed path.) To

7 D. Fitz Wilhelm page 7 of 9 C:\physics\23 lectue\ch29 magnetic fields.docx; S: 5/3/29 2:: PM P: The magnetic foce on a cued cuent caying wie in a unifom magnetic field is equal to that on a staight wie, connecting the endpoints of this wie. b F = I ds = I ds = IL ' line a (29.9) b ds = L ' a (29.2) F = IL ' L ' is the distance ecto connecting the initial point of the wie with its final point of the potion inseted into the unifom magnetic field The net magnetic foce acting on any closed cuent loop in a unifom magnetic field is Toque on a cuent loop in a magnetic field. Een though the total foce on a cuent loop in a magnetic field is zeo, the toque is not. Conside a ectangula loop of cuent inseted in a unifom magnetic field. The magnetic field points to the ight F out τ A L upcuent F in h points to the ight in the adjacent pictue, and the ectangula loop of cuent lies in the same plane as the magnetic field. The cuents in opposite sides of the ectangle moe in opposite diections. Theefoe the foces on the hoizontal pats of the loop ae because the cuents ae paallel (o anti-paallel) to the magnetic field. All foces ae pependicula to the plane. Now, if the foce on the left etical line points outside of the plane, the foce on the ight etical line points into the plane, thus ceating a toque aound the etical axis. oth toques ceate a counte clockwise otation and add up to a esultant toque which points upwad along the axis of otation. This is a typical case of a toque ceated by a so-called ecto couple: Two equal and opposite foces ae applied at the two end-points of a ba with length 2 which is capable to otate aound its cente point. (29.2) τ = 2 F The ecto points fom the cente of otation to the endpoint at which the foce F is applied. If all eleant quantities ae at ight angles to each othe we get the magnitude fo the toque as:

8 D. Fitz Wilhelm page 8 of 9 C:\physics\23 lectue\ch29 magnetic fields.docx; S: 5/3/29 2:: PM P: (29.22) τ = 2 F = Lh I = AI = aea cuent magnetic field L Ih A= aea We define the aea of the loop with the cuent I as a suface-ecto A pependicula to the suface aea and with the magnitude of the aea. We define the mathematics by always ciculating aound an aea along its bounday with the finges culing in the diection of the positie cuent, and the thumb pointing in the diection of the suface ecto. A We define the magnetic dipole moment as the ecto of magitude I (cuent) times A and in the diection of the suface ecto A. (29.23) = IA = magnetic dipole moment h τ The diection of the magnetic moment can be L obtained in the easiest way by culing the finges of the ight hand along the loop with the diection of the cuent. (We assume a cuent of positie chages; fo negatie chages the diection is eesed.) You thumb will then point in the diection of the magnetic moment. In this way, the aea will be enclosed by the finges of you ight hand. z = IA = IAk k is a unit ecto to the suface. z y x j y j x = IA = IAk When the magnetic dipole moment is placed into a magnetic field, a toque is being ceated which e-oients the dipole. The magnetic field will line up the magnetic moment with itself, thus obtaining a position with minimum potential enegy fo the dipole-magnetic field system. Using the notation of the dipole moment the toque can be coneniently witten as: (29.24) τ = = IA The toque of the magnetic moment of a positie chage cuent is paallel to the axis of otation.

9 D. Fitz Wilhelm page 9 of 9 C:\physics\23 lectue\ch29 magnetic fields.docx; S: 5/3/29 2:: PM P: The toque is when the suface is pependicula to the magnetic field, i.e. when the suface ecto and theefoe the magnetic dipole ae paallel to. This is in analogy to what happens when we put an electic dipole into an electic field: τ E = p E ; p = qdu (29.25) whee u is a unit ecto pointing fom -q to +q. If a coil consists of N tuns of wie, the magnetic moment of the loop is N times the magnetic moment of a single loop: (29.25) = NIA Just like the potential enegy of an electic dipole immesed in an electic field was gien by Ue = p E we define the potential enegy of the magnetic loop as the wok necessay fo an outside agent to otate the loop in the magnetic field: θ θ θ (29.26) dw = τdθ = sinθdθ = ( cosθ cosθ ) θ θ θ 2 We choose ou efeence angle fo the potential enegy at 9, when the suface ecto of the dipole is pependicula to the magnetic field, i.e. when the loop lies in the plane of the magnetic field. Thus, the potential enegy of a magnetic dipole immesed in a magnetic field is gien by: (29.27) U = When the dipole is paallel to the magnetic field the potential enegy is smallest(-μ), when the dipole is anti paallel it is highest (+μ). (This is the same as with the enegy of the electic dipole.) 29.6 The Hall Effect : If a cuent caying conducto is placed into a magnetic field, chages ae deflected to one side of the conducto, thus ceating an electic field acoss the conducto, because of an excess accumulation of one kind of chages on one side of the conducto. The ensuing electic foce on the deflected (by ) chages opposes the magnetic foce. The accumulation stops when the electic foce due to the accumulated chage suplus equals the magnetic foce esponsible fo the deflection. (29.28) qd = qe y measuing the oltage diffeence acoss the conducting slap V H and the cuent I we can detemine the chage density in the conducting mateial. Let us assume it has the shape of a ectangula slap with coss-section dimensions of base d =. mm and height h=2.cm, we get: (29.29) VH = Eh = dh

10 D. Fitz Wilhelm page of 9 C:\physics\23 lectue\ch29 magnetic fields.docx; S: 5/3/29 2:: PM P: F = q d F E = qe h d Thus we can measue the dift elocity d. y measuing the Hall potential and the cuent I we hae a way to detemine the moing chage density nq. We need to expess the dift elocity by the cuent: The dift elocity is elated to the cuent density and the cuent itself. (29.3) I = j A = nv q d hd (29.3) = I I d ; VH Eh dh n qhd = = = n qhd h V A V I = n qd V (29.32) I VH = = R n qd n V I = qd V H I d y measuing the Hall oltage, the cuent, the magnetic field we can detemine the density of the conducting chages and thei sign. R H =/nq is called the Hall coefficient, which is the inese of the chage density ρ q. Example: A coppe stip 2 cm (h) wide and d=mm thick is inseted into a magnetic field pependicula to the stip width. =2T and I=2A. Calculate the Hall oltage. We find: j=i/a =I/dh = A/cm 2. n=ρ m N A /M mol =8.95 6E23/64=8.4E22/cm 3. This numbe coincides well with the concept of fee electon pe atom in coppe as the conducting electon. E H =.49V/m; V H = 2.98mV. Such measuements allow one to measue the actual chage density in any conducting o semiconducting mateial, and the sign of the chage caies. It was a big supise when physicists found that in some some semi-conductos the moing chages wee not electons, but positiely chaged holes.

11 D. Fitz Wilhelm page of 9 C:\physics\23 lectue\ch29 magnetic fields.docx; S: 5/3/29 2:: PM P: Addendum: A) Deiation of the law of iot-saat. Abstact: The same mathematical diffeential equations hae the same solutions. Undelying diffeential equations fo the gaity potential, the electic potential, and the components of the magnetic ecto potential shae the same diffeential equations. Theefoe thei solutions ae the same also. The diffeential equations in question hee ae the Poisson equations, o Laplace equations. ρ ρ die = ; E = E = gad V; di( gadv ) = ; ε ε ρ di( gadv ) V = V = Poisson equation fo the electic potential V. ε (29.33) ρ Poisson equation:,, V= + + Vxyz (,, ) = x y z x y z ε cul( gadv ) V = (29.34) di = ; cul = j thee is no scala field fom which deies. ut = cula = A ( A) = j (with the choice dia=) this leads to the simila equation fo the components of the ecto potential A as fo the scala potential V. A= A= j; j = q We get the same esults fo the potential functions caused by a single chage q (29.35) q q V( ) = if q is located at = V( ) = ε 4 πε In elatiistic physics it tuns out that V and A fom a single fou-dimensional ecto. The electic field and the magnetic field ae also components of a fou dimensional quantity, called a tenso. This dies home the fact, that the physical facts elated to electomagnetism ae elatiistic and fou dimensional in thei ey natue. q A) = ; = adius ecto fom the chage q at to point of the field. (29.36) ( ) Expeiments show that magnetic field lines hae no souces, the field lines close on themseles. Thee ae no magnetic chages fom which field lines emege. Thus, magnetic field cannot be descibed though a local elationship like the electic fields of electostatics, whee electic fields stat in a posite chage and end in a negatie chage.

12 D. Fitz Wilhelm page 2 of 9 C:\physics\23 lectue\ch29 magnetic fields.docx; S: 5/3/29 2:: PM P: (29.36) die ρ = ; cule = E = gad V ε This means that the electic field at a location x,y,z is due to the local change of a scala field V. Local change of a field means that it is the esult of the local ecto opeato. Magnetic fields nee stat in chages, theefoe : (29.36) di = always Magnetic fields cul aound wies with cuents in them o een aound single moing chages, which theefoe ceate a cuent density : (29.36) cul = j 2 7 Ns 6 = ; 2 =.26 = pemeability C A (29.36) the dimension of cul is T/m, that of cuent density j is m 2 T/m Tm j = the dimension of is = A/m 2 A We ask ouseles, just like in the case of the electostatic field, if the magnetic field at a point can be the esult of the local change of anothe field. As the cul is diffeent fom, the ecto field cannot deie ( ) fom a potential scala field, and the ealuation along a closed loop is not equal to. (The concepts of conseatie fields do not apply!!!!!) What could be the local change of a magnetic field: theoetically, thee ae only thee choices: can be the gad of a scala field, which is exluded by the fact that cul it cannot be the diegence of a ecto field because that is in itself a scala, and the magnetic field is ecto, so it could only be the cul of anothe ecto field, which tuns out that that is the case : (29.36) = cula A A is called the ecto potential. Thus, the magnetic field is the cul of a ecto potential. It must satisfy the fact that di, which is the case always because di = = di cula = A = always; ( ) ( ) (29.36) compae to cule = = cul ( gadv ) V = Poof : Fo any thee ectos we hae: Ax Ay Az (29.36) A ( C) x y z = C ( A ) = ( C A) Cx Cy Cz If any of these ectos ae paallel, the mixed poduct is. (Popeties of the deteminant.)

13 D. Fitz Wilhelm page 3 of 9 C:\physics\23 lectue\ch29 magnetic fields.docx; S: 5/3/29 2:: PM P: We get fom this that cul = j and = cula A (29.36) ( A) = j We ealuate the double coss poduct accoding to the ules of ecto opeatos : a ( b c) = b ( ac ) c ( ab ) (29.36) ( A) = j = ( A) A( ) dia A y setting the expession dia = we ae fixing a constant of integation fo the ecto potential. We ecognize the emaining opeato as the Laplace opeato which we encounteed ealie in chapte 25. The equation fo the ecto potential A is a Poisson equation just like the one fo the electic potential: ρ A= j = ρ just like V = (29.36) ε Ax = jx and Ay = jy and Az = jz As the same equations hae the same solutions we can immediately wite down the solution fo the ecto potential. Reiewing the electic potential, we found the solution fo a single chage q (29.36) q ρ V( ) = fom V = ε ε Each component of the ecto potential obeys the same Poisson equation (29.37) A = ρ ; A = ρ ; A = ρ x x y y z z Theefoe the solutions fo the ecto potential must be simila to those fo the electic potential, the only diffeence lying in the constant facto: ρ x ρ y ρz (29.38) Ax = ; Ay = ; Az = In the peious equtations we placed the chage density into the oigin. If we place the chage density into a location with the adius ecto, the distance ecto fom the chage to the point whee we calculate the field is with the distance. and now:(29.38) A ( ) = q which is the ecto potential at point. ceated by the single moing chage at the point (): dq = ρdv ρ q chage density single chage ε 4 πε cuent density j = ρ single chage with : q

14 D. Fitz Wilhelm page 4 of 9 C:\physics\23 lectue\ch29 magnetic fields.docx; S: 5/3/29 2:: PM P: To get the ecto field () we hae to calculate A.In ode calculate a deiatie of these functions we need to pay attention that we calculate the deiaties at the location of the function: (x,y,z) and not at the location of the chage (x,y, z ). To ease up on ou witing we can put the chage back into the oigin and get functions which ae easie to manage: kq e kq V() = = x + y + z and e A ( ) = q = q x + y + z V() A ( ) Single chage q ρ q die = ; V = ε ε cuent density j cul = j; A = We obtain the magnetic field ceated by this single moing chage at point, by taking the cul of A at point accoding to (29.36). (29.38) ( ) = Awith x + y + z i j k x y z x y x + y + z x + y + z x + y + z z

15 D. Fitz Wilhelm page 5 of 9 C:\physics\23 lectue\ch29 magnetic fields.docx; S: 5/3/29 2:: PM P: z y x = q ; y z x z (29.38) y = q z x y x z = q x y When we cay out the deiaties we use the Catesian foms : (29.38) = x + y + z The patial deiatie of the negatie squae-oot esults in the facto -/2 ; the deiatie of the squaes with espect to y and z esults in the factos 2y and 2z: z 2y (29.39) = z y x y z x + y + z 2 Theefoe: (29.39) ( ) 3 2y 2z z y x = q = q 2 4 z y y z 3 3 π ( x + y + z ) ( x + y + z ) We ecognize that the expession in the numeato is the x-component of the coss-poduct We get as the final esult fo the magnetic field ( ) ceated by the chage q haing the elocity at location ( ) q The components in the numeato ae the components of the coss-poduct q (29.39) ( ) = 3 The ecto which appeas in the numeato of the equation points fom the moing chage at location () (which in ou deiation we put at the oigin ) to the magnetic field at location. If we put q at location, the ecto must be eplaced by and the distance by q u ( ) = 2

16 D. Fitz Wilhelm page 6 of 9 C:\physics\23 lectue\ch29 magnetic fields.docx; S: 5/3/29 2:: PM P: This law is also known as the law of iot-saat: If we want to calculate the magnetic field at a point, ceated by a small segment of cuent Ids at point () we use see (29.3) (29.4) dq = I ds We call the magnetic field ( ) the field ceated by the moing chage at the point () q u I ds u = d = ; = x x + y y + z z (29.4) ( ) ( ) ( ) ( ) ( ) The unit ecto points fom the cuent element at point () to the point, whee the magnetic field is being calculated. If we put the cuent element into the oigin, then = x, y, z =,,. ) Magnetic Foces on Moing Chages, in geneal: If we place anothe chage q 2 moing with elocity 2 into the magnetic field ( ) at point, then this moing chage q2 2will feel the foce Foce on chage (2) ceated by the magnetic field ceated by moing chage q : (29.4) u F2 = q22 = q22 q 2 Pulling the scalas out of the equation we get : qq 2 (29.4) F2 = q22 = 2 2 ( u ) Conesely, the moing chage at location (2) ceates a magnetic field 2 at location () and theefoe exets a foce F 2 on paticle q. qq 2 (29.4) F2 = 2 ( 2 u2 ) The unit ectos point in opposite diections. In geneal, this foce is not equal and opposite to the foce F 2. C) Magnetic foces on moing chages in paallel: Let us see what we get when the elocities ae paallel to each othe: In that case, the unit ectos ae pependicula to the elocities. If we expand the double coss poducts we get: The foce on chage () by chage (2) is gien by:

17 D. Fitz Wilhelm page 7 of 9 C:\physics\23 lectue\ch29 magnetic fields.docx; S: 5/3/29 2:: PM P: qq F = ( u ) = k ( u ) u ( ) = ku (29.4) k The foce on chage (2) ceated by the moing chage at () (29.4) F = k ( u ) = k ( u ) u ( ) = ku As the unit ectos ae opposite to each othe we say that the two chages (of the same sign), moing paallel to each othe, attact each othe with the foce : qq 2 (29.42) F2 = 2 2 This is a fundamental law of physics, as fundamental as Coulomb s law. Two paallel moing chages (of the same sign) exet an attactie foce on each othe which is popotional to the poduct of the chages and speeds, and inesely popotional to the distance between them. Equal chages moing paallel attact each othe, opposite chages moing paallel to each othe epel each othe. In ode to find the foce exeted between cuent caying wies, we need to fist find the magnetic field ceated by the cuent in a wie. Then we apply equation (29.6) which we wite in ou new context as: (29.43) df2 = I2ds2 It gies us the foce on the cuent I 2 at location (2), ceated by the magnetic field which is due to a cuent in the paallel wie (). We will find an easy method to calculate the magnetic field suounding a wie with cuent I in a late chapte. It is equal to I = 2π and cicles aound the wie. The oientation of the magnetic field of cuent I follows the ight hand ule: The thumb indicates the diection of the cuent, and the finges cul aound it in the diection of the magnetic field. If we place a second wie paallel to the fist wie, the magnetic field is pependicula to the diection of the cuent. This means that the foce is pependicula to both the magnetic field and the diection of the cuent. This foce points fom one wie to the othe, and is pependicula to both paallel wies.

18 D. Fitz Wilhelm page 8 of 9 C:\physics\23 lectue\ch29 magnetic fields.docx; S: 5/3/29 2:: PM P: cuent I ceated by cuent I F 2 L II = 2 cuent I 2 df I ( ds ) fo paallel wies means that (29.44) = (29.44) the foce is pependicula to the wies. df = II dl F = II π L2 2π Whee we calculated the foce on the line segment of wie (2) pe unit length of the wie (2): Two cuents of A in paallel wies, and being one m apat attact each othe with the foce of 2-7 N pe mete. D) Calculating the magnetic field though the ecto potential: Just as a fun execise, let us calculate the magnetic field aound a wie by using ou insights into the magnetic potential A The situation is the same as with the electic potential aound a wie. Fo the electic field suounding a wie with linea chage λ we found that: λ (29.45) E () = 2πε Fom this we calculate the electic potential at distance fom the wie by: λ λ λ (29.46) V = E ds = d = ln = ln ; V ( ) = 2πε 2πε 2πε Now, we can again use the fact that the electic potential and the ecto potential follow the same equations, and theefoe, unde simila cicumstances must hae the same solutions : ρ -λ V = leads to V()= ln ε 2πε (29.47) dq dz A= j = ρ must lead to A z = - λ ln ; λ= = I 2π dz dt 2 2 We place the wie in the z-diection, and is gien by x + y

19 D. Fitz Wilhelm page 9 of 9 C:\physics\23 lectue\ch29 magnetic fields.docx; S: 5/3/29 2:: PM P: We hae eplaced ρ with λ dq dz dq = = =I and with dz dt dt ε. I I (29.48) A = z ln ln x y 2π = 2π Accoding to = A we need only to calculate the x and y components of the magnetic field, as the component in the z-diection (the diection of the cuent density) is. I 2 2 I 2y I y (29.49) x = Az = ln ( x + y ) = = y 2π y 2 2π 2 x + y 2π I I 2x I x = = + = = x 2π x 2 2π 2 x + y 2π Which means that cicles aound the wie and has the magnitude: I (29.5) = 2π 2 2 (29.5) y Az ln ( x y ) 2 2 2