Chapte F Magnetism Blinn College - Physics 46 - Tey Honan F. - Magnetic Dipoles and Magnetic Fields Electomagnetic Duality Thee ae two types of "magnetic chage" o poles, Noth poles N and South poles S. Playing with ba magnets demonstates that like poles epel and unlike attact. This is analogous to the situation we had with electic chage. This analogy is a deep one and is called Electomagnetic duality. Noth and South poles ae elated to the magnetic field B as positive and negative electic chages ae to the electic field. N and S ae to B as + and - ae to E The foce of an electic chage q in an electic field gives by analogy the foce of a magnetic pole of stength g in a magnetic field. F = q E and F = g B If g denotes the "magnetic chage" o pole stength then a Noth pole coesponds positive g and a South pole to negative. Gauss's law elates the total electic flux though a closed suface to the chage enclosed by the suface. Similaly, the magnetic flux though a closed suface coesponds to the total "magnetic chage" inside. E ÿ A = ε 0 q enclosed and B ÿ A = m 0 g enclosed whee analogous to ê ε 0, the electic constant of popotionality, is the magnetic constant m 0. E = q ` 4 p ε 0 and B = m 0 4 p g ` Magnetic Dipoles A pemanent magnet has both Noth and South poles sepaated by some distance. When it is placed in a field the N pole expeiences a foce in the diection of the field and the S pole has a foce opposite the field. If the field is unifom the net foce is zeo but thee is a net toque. This is analogous to an electic dipole and it will be called a magnetic dipole. The stength of a magnet can be descibed by its magnetic dipole moment m. Fo an electic dipole p the toque and potential enegy given by t = p äe and U = -p E. The coesponding expessions fo toque and potential enegy of a magnetic dipole in a magnetic field ae t = m äb and U = -m B. In addition to pemanent magnets being dipoles we will see that cuent loops also ae magnetic dipoles. In that case we will wite an expession fo the magnetic dipole moment. Gauss's Law fo Magnetism and the Absence of Isolated Poles Isolated magnetic poles could exist but so fa none have eve been obseved. Gauss's law in the electic case states that electic field lines begin at isolated positive chages and end at isolated negative chages. The absence of isolated magnetic poles implies that magnetic field lines neve begin o end; they eithe fom closed loops o go off to infinity The magnetic analog of Gauss's law is ò B ÿ A = m 0 g inside. The nonexistence of isolated magnetic poles implies that the ight hand side is zeo and Gauss's law fo magnetism becomes B ÿ A = 0 This is ou second of Maxwell's equations. If we apply Gauss's law to a magnet and put a Gaussian suface aound the Noth pole then thee is magnetic flux leaving the suface at the end of the magnet. Fo the flux to be zeo though this Gaussian suface the field lines inside the magnet
Chapte F - Magnetism magnetic flux leaving the suface at the end of the magnet. Fo the flux to be zeo though this Gaussian suface the field lines inside the magnet must close back on themselves and fom closed loops. Because of this if a ba magnet is cut in half then it doesn't split into a pai of isolated poles; it becomes two smalle dipoles. If someday isolated poles ae discoveed, then we may just modify Gauss's law by adding in a magnetic chage tem m 0 g inside to its ight hand side as was shown above. Othe modifications to Maxwell's equations associated with magnetic cuents will also be needed; these will be discussed late. F. - Foce on Moving (Electic) Chages and Cuents Electicity and magnetism ae not sepaate foces whee electic fields just exet foces on electic chages and magnetic fields exet foces on magnets. Instead electicity and magnetism ae aspects of the same foce called electomagnetism. Magnetic fields cause foces on moving (electic) chages and cuents. Magnetic Foce on Moving Chages If a chage Q is moving with a velocity v in a magnetic field B then the foce is given by F = Q v äb. Note that the coss poduct is a thee dimensional thing. The velocity and field vectos define a plane and the foce is in the diection pependicula to the plane. Note also that when a vecto is multiplied by a negative scala its diection changes, so negative chages expeience foces opposite that of positive ones. Foce on Cuents Conside the flow of chage caies of chage q with dift velocity v d though a staight segment of wie of length { with coss-section A. If the density of chage caies (numbe/volume) is n then the total numbe of chage caies is N = n µvolume = n A {. Summing ove all the chages gives the total foce on the wie in a unifom field F = N q v d äb = n A { q v d äb Using the expession fo cuent fom the pevious chapte, I = q n A v d, we get F = I { äb, whee the diection of the cuent is put into the diection of the vecto {. Note that the cuent diection is the same as the dift velocity when q is positive and it is opposite when q is negative. This is built into the above expession. To genealize this expession conside a cuved wie with an infinitesimal segment s. The foce on that segment is I s äb. Integating ove the length of the wie gives whee the field need not be unifom. F = I s äb, Hall Effect It is clea fom the pevious section that we cannot detemine the chage of the chage caies by measuing the magnetic foce on a wie; simultaneously changing the signs of q and v d gives the same cuent. We can, howeve, use the magnetic foce to find the chage of the chage caies by measuing the voltage acoss a conducting stip in a magnetic field. Conside a flat conducting stip with a cuent in a magnetic field. Take the width of the stip, the cuent and the field to be mutually pependicula as shown in the diagam.
Chapte F - Magnetism 3 B - + - + - + - + - + - + - + - + - + F I F v q < 0 q > 0 d v d + - + - + - + - + - + - + - + - + - L The magnetic foce will push eithe positive o negative chage caies towad the top of the wie. This will ceate a voltage acoss the stip. The polaity of the voltage depends on the chage of the chage of the chage caies. If they ae positive then the top is positive and othewise it is negative. The Hall voltage is the voltage acoss the stip. If the width of the stip is L then the wok on the chage caie q is F L. Since the foce is F = q v B we have The wok pe chage is the induced voltage, since DU = qdv. W = q v d B L V Hall = W êq ï V Hall = v d B L Motion of Chaged Paticles Any foce that acts pependiculaly to the velocity of a paticle doesn't affect the speed of the paticle; it only altes its diection. This is the case with the magnetic foce F = Q v äb. Suppose a paticle with speed v is shot into a egion of unifom magnetic field with the velocity pependicula to the field then the magnitude of the foce is just F = Q v B. Since the speed and the magnitude of the foce ae constant and the foce and velocity ae pependicula, the motion will be unifom cicula motion. Using the acceleation fo unifom cicula motion a c = v ê and Newton's second law we get: F = m a ï Q v B = m v ï = m v Q B. The angula fequency w is elated to the speed and adius by w = vê which gives an expession known as the cycloton fequency w = Q B m. If a chaged paticle moves in a unifom magnetic field with a velocity that is not pependicula to the field, then the pependicula component changes as befoe and the paallel component is unchanged. The esulting motion is a combination of linea and cicula motion, giving a helix. The geneal shape of the path of a chaged paticle in a unifom magnetic field is helical. An electomagnetic field is a combination of both electic and magnetic fields. The foce of a chaged paticle in an electomagnetic field is the sum of both electic and magnetic foces and is called the Loentz foce law F = Q IE + v äbm. F.3 - Souces of Magnetic Fields In ou discussion of electic fields we have discussed the foce on chages due to fields. Analogously, we have found the magnetic foce on moving chages and cuents. Ou discussion of electic fields is moe complete, howeve, since we have ways to calculate electic fields due to souces, electic chages. We now need to addess the souces of magnetic fields. The Biot-Savat Law will be intoduced as the analog of the Coulomb's Law integals ove continuous distibutions to get electic fields. In cases of symmety we could use Gauss's Law to find electic fields; as the magnetic analog of this we will intoduce Ampee's law. Foce on Q Ho IL Electic Fields F = Q E Magnetic Fields F = Q v äb F = I Ÿ s äb
4 Chapte F - Magnetism Field due to Q Ho IL ` E = k e q ò E ÿ A = ε 0 Q enclosed Biot Savat Law Ampee' s Law Coulomb's Law and Gauss's Law ae mathematically equivalent fo electostatics. Similaly, we will see that the Biot-Savat Law is equivalent to Ampee's Law fo magnetostatics. Electostatics allows no movement and thus no cuents. Magnetostatics allows cuents but equies all cuents to be steady. Gauss's Law is fully coect even beyond electostatics and is one of Maxwell's equations. Ampee's Law, as discussed in this chapte, is only coect in the context of magnetostatics. Next chapte we will intoduce Maxwell's addition to Ampee's Law; this will make it geneally coect and it will become one of Maxwell's equations. F.4 - Biot-Savat Law E = The Biot-Savat law elates the magnetic field at some point P to the cuent in a wie. The analogous expession fo electic fields is 4 p ε 0 ` Savat law is q. The souce is a cuent I though an infinitesimal segment of wie s. Take the vecto to be fom the souce to P. The Biot- B = m 0 4 p I s ä `. I P s Deivation Using a Test Pole F PW g 0 I B B WP s F WP Exploiting duality symmety we can deive the Biot-Savat Law. To do this intoduce a test magnetic pole of stength g 0 at the position P. The fact that these poles have neve been obseved need not distub us. The field at the wie due to the pole is B WP = m 0 -` 4 p g 0. The negative sign is thee because the vecto in the diagam is pointing towad the pole whee we usually take as pointing away fom the chage o pole. The foce on the wie due to the pole is then F WP = I s äb WP = - m 0 4 p g 0 I s ä `. Using Newton's thid law we can elate this to the foce on the pole due to the wie
Chapte F - Magnetism 5 F PW = - F WP = m 0 4 p g 0 I s ä `. The foce on a magnetic pole g 0 in a field B is F = g 0 B so using the pole as a test pole we can wite the field at P due to the wie as B = F PW = m 0 g 0 4 p I s ä `, which is just the Biot-Savat Law. Note that the esult is independent of ou test pole. Examples Field at the cente of a cicula ac I s q P Conside a cicula ac of adius R and of angle q in the xy-plane with a counteclockwise cuent. The vecto is fom s to the oigin, which is the point P. B = m 0 4 p I s ä` Using A äb = A B sin q ù we get Fo evey point on the ac we have = R = const. giving: The integal is just the total ac length Ÿ s = R q giving s ä` = s ÿ ÿ z`. B = z` m0 I 4 p B = z` R s. m 0 I 4 p R q. Field at a pependicula distance z 0 fom the cente of a cicle
6 Chapte F - Magnetism z z 0 P 0 I x R q Now conside a full cicle of adius R in the xy-plane with the cente at the oigin and a counteclockwise cuent. Choose the point P to be at z 0 along the positive z-axis. Integate aound the cicle by vaying q fom 0 to p. The position as a function of q is given by the vecto. s = XR cos q, R sin q, 0\ The s is the infinitesimal change in this vecto unde an infinitesimal change in angle, q. s = = X-R sin q q, R cos q q, 0\ The vecto is the vecto fom s, which is at, to P which is at 0 = X0, 0, z 0 \. This gives By the Pythagoean theoem, the magnitude of is just Using ` = we get an expession fo the field. 3 = 0 - = X-R cos q, -R sin q, z 0 \. = R + z 0. B = m 0 I s ä 4 p 3 The coss poduct can now be explicitly evaluated using the deteminant method s ä = Since is a constant we can bing the tem out of the integal giving 3 x` ỳ z` -R sin q q R cos q q 0 -R cos q -R sin q z 0 = x` Hz 0 R cos q q - 0L -ỳ H-z 0 R sin q q - 0L +z` IR sin q q -- R cos q qm = Yz 0 R cos q, z 0 R sin q, R ] q y This gives thee simple integals. B = m 0 I 4 p HR + z 0 L 3ê 0 p Yz 0 R cos q, z 0 R sin q, R ] q. 0 p z 0 R cos q q = 0
Chapte F - Magnetism 7 0 pz 0 R sin q q = 0 0 p R q = p R The final esult can, finally, be witten B = z` m0 I R HR + z 0 L 3ê. Field due to a line segment y a P f x s x x+ x I x x Fo some staight-line segment choose the x-diection to be the diection of the cuent and take the segment to be between x and x. The point P is on the y-axis at y = a, whee x = 0 is the point on the line closest to P. The vecto s is the vecto fom x to x + x And the vecto is fom the s to P. s = x` x = -x x` + a ỳ The magnetic field at P is Evaluating the coss poduct B = m 0 I s ä 4 p 3 and using = x + a gives s ä = x` x ä H-x x` + a ỳl = z` a x B = z` a m 0 I 4 p x This can be evaluated using a tig substitution. Define the angle f as shown. The substitution is x x = a tan f. x Hx + a L 3ê. The diffeential becomes and becomes a x = cos f f Define f and f as the f values coesponding to x and x = x + a = a cos f ï = Hx + a L 3ê cos 3 f a 3
8 Chapte F - Magnetism This gives the final esult B = z` a m 0 I f cos 3 f 4 p f a 3 a cos f f = z` m0 I f cos 4 p a f f. f This can also be witten in tems of the oiginal x vaiables B = z` m0 I 4 p a Hsin f - sin f L. B = z` m0 I 4 p a x x + a - x x + a. Note that with the choice of f given hee that a negative x value coesponds to a negative f value. Field of a long staight wie An impotant special case of this is that of a long staight wie. The field magnitude a distance fom a long staight wie becomes x Ø - and x Ø ï f Ø - p = -90 and f Ø p = 90 Hsin f - sin f L Ø - H-L =. B = m 0 I p. To get the diection of the field fo a long staight wie, o fo that matte fo a segment, put the thumb of you ight hand in the diection of the cuent. The field ciculates aound the wie in the diection given by you finges. F.5 - Magnetic Foces, m 0 and the Definition of the Coulomb We have not yet assigned a value to the constant m 0. We also haven't given a definition of the Coulomb. The Coulomb will be defined in tems of the Ampee, C = A ÿs, and then the Ampee's definition will be established when we assign a value to the constant m 0. To gain a physical undestanding of these definitions we will conside the magnetic foce between paallel wies and deive a magnetic analog to Coulomb's Law. Foces between Paallel Wies { B B I F I a Conside a long wie with cuent I and a paallel segment of length { a distance a fom the long wie. The paallel segment will have a cuent I ; its cuent will be supplied by pependicula wies coming fom infinity. The magnetic foces on these pependicula segments will cancel and the net foce will just be the foce on the segment. To find this foce takes two steps: Define B to be the field due to I at I and then define F as the foce on I due to B. B = m 0 I p a The diection of B is out of the page. We can then find the magnetic foce. The diection of this foce is towad the othe wie and its magnitude is F = I { äb
Chapte F - Magnetism 9 F = I { B = m 0 p I I { a. We can make a geneal statement about magnetic foces between cuents. Paallel cuents attact and anti-paallel cuents epel. m 0 and the Ampee In the expession fo the foce both { and a ae lengths; it follows that the SI units of m 0 ae NêA. It should now be clea that edefining an Ampee will change the numeical value of m 0 ; assigning a value to it will then povide a definition of the Ampee. The constant m 0 could be emoved completely by defining its value to be but fo histoical easons we choose diffeently. The value of m 0 is m 0 = 4 p µ0-7 N A. If an expeiment wee set up with the aangement above, then it could be used to explicitly calibate an ammete. If the two cuents ae foced to be equal I = I = I, then measuing the foce and using the expeiment's values fo { and a would give the cuent in Ampees. F.6 - Ampee's Law Ampee's law is mathematically equivalent to the Biot-Savat law fo magneto-statics, whee all cuents ae steady giving constant fields. This equivalence cannot be demonstated at this level. We will use Ampee's law similaly to Gauss's law. In cases of symmety we will use it to find magnetic fields fom cuents. Ampee's law is B = m 0 I enclosed. The integal is aound a closed contou and I enclosed is the total cuent enclosed by that contou. If the integal is ove some closed contou then thee ae many diffeent sufaces (an infinite numbe) that have that contou as its bounday. An example is the Eath's equato that has the nothen hemisphee, the southen hemisphee and a disk though the Eath's cente as diffeent sufaces that shae it as thei boundaies. The cuent I is the cuent piecing any suface that has the contou as its bounday. We can elate the oientation of the bounday to the oientation of the contou (the diection of integation aound the contou.) Cylindical Symmety In any case of cylindical symmety choose a cicula contou. By symmety and using Gauss's law fo magnetism we get that the field otates aound the axis. This gives B = p B. Inseting this into Ampee's law gives the geneal expession fo cylindical symmety Note that the field fo a long wie is a tivial special case of this. B = m 0 I enclosed. p Long Solenoid
0 Chapte F - Magnetism { 4 3 I B Fo an (infinitely) long solenoid take the cuent to be I and the density of tuns to be n. # of tuns n = length The field inside the solenoid is unifom and the field outside is zeo. (The field outside appoaches zeo as the length become infinite.) Choose the contou to be fou segments as shown Thee ae n { tuns though the contou giving B = B + B + 3 = B { + 0 + 0 + 0 I enclosed = n { I It follows fom Ampee's law that the field anywhee inside a long solenoid is B = m 0 n I. B + 4 B F.7 - Cuent Loops as Magnetic Dipoles The net foce on a cuent loop in a unifom magnetic field is zeo. The field does affect the loop, though. Thee is a toque on it. F net = I s äb = I K s O äb = 0 We saw ealie using electomagnetic duality (the analogy between electic and magnetic fields and chages) that the toque on a magnetic dipole is t = m äb. In showing thee is a toque on a cuent loop we will demonstate that a cuent loop is a magnetic dipole. This is a second souce of magnetic dipoles; in addition to pemanent magnets being dipoles, we will now see that cuent loops ae dipoles as well. To calculate the dipole moment of a loop o N-tun coil, we will fist find the moment of a single tiangula loop by calculating the toque on a tiangula loop in a unifom field. A Single Tiangula Loop The definition of toque about an oigin due to some foce is t = äf, whee is the vecto fom the oigin to whee the foce F acts. Conside fist a line segment of length { with cuent I. In finding the foce on this segment we took { to be in the diection of the cuent and the foce is F = I { äb. The toque on the segment becomes that of the foce acting at the midpoint of the segment mid t = I mid äi{ äbm. Conside a single tiangula loop caying a cuent I in a unifom magnetic field B. Toque depends on one's choice of oigin, but wheneve the net foce vanishes the net toque is independent of the choice of oigin. We will choose the oigin to be at the cente of one side; this
Chapte F - Magnetism the net foce vanishes the net toque is independent of the choice of oigin. We will choose the oigin to be at the cente of one side; this emoves the contibution of that segment to the toque since mid = 0. The two sides that do contibute ae labeled { and {, and take thei diections to be the diection of the cuent. { { - { { t = I - { We can simplify this using an identity satisfied by coss poducts. äi{ äbm + I { äi{ äbm + 0 A äib äcm + B äic ä AM + C äia äbm = 0 This identity can be ewitten as A bit of algebaic manipulation A Ø {, B Ø { and C Ø B gives A äib äcm - B äia äcm = IA äbm äc. t = I { ä { äb. Recall that the magnitude of the coss poduct is the aea of a paallelogam. A tiangle is half of that. The diection of the coss poduct is pependicula to the two vectos and thus to the tiangle. It follows that the aea vecto of the tiangle is A = { ä {. Note that the ight-hand ule gives the outwad nomal as the diection of A and the ight-hand ule also associates that diection with a counteclockwise cuent. We can now wite toque as t = I A äb. Since the toque on a magnetic dipole is t = m äb, we can wite the magnetic moment of a single tiangula cuent loop as Plana Cuent Loops o Coils with N Tuns m = I A. We can now genealize this esult to a geneal plana loop. Any loop may be boken up into infinitesimal pieces each of aea A. Summing ove all these infinitesimal aeas gives A = Ÿ A. The aea vecto has a magnitude A and its diection is nomal to the loop. A = A ǹ To get the pope diection use a ight-hand ule. Wap you finges aound loop in the diection of the cuent. The thumb of you ight hand points in the diection of the unit nomal ǹ that gives the diection of A. The magnetic moment is then m = I A. HSingle tun loopl If thee ae N tuns then each tun contibutes I A to the magnetic moment. The total magnetic dipole moment is m = N I A. HN tun coill N is the numbe of tuns, I is the cuent and A = A ǹ is the aea vecto defined as above.