MAGNETISM We now turn to magnetsm. Ths has actually been used for longer than electrcty. People were usng compasses to sal around the Medterranean Sea several hundred years BC. However t was not understood untl the frst half of the 19 th century. The trouble was that t appears to have two sources: magnetc materals and electrc currents. The frst was responsble for compasses etc, whle the second was not even observed untl the early part of the 19 th century. Ultmately, of course, they are the same, but that s hndsght. We wll start wth the effect of magnetc materals. MAGNETIC DIPOLES As seen n class the fundamental source of magnetsm appears to be not a pont charge but rather a pont dpole. We have already worked out the effect of dpoles n electrostatcs. Expermentally we fnd that magnetc dpoles work n exactly the same way except that the force constant, k, has a dfferent value: 7 km 110 n m /q where q s the unt of magnetc charge whch we wll defne later. It s mportant to realze that so far as we know THERE IS NO MAGNETIC SINGLE CHARGE. Such an entty would be called a magnetc monopole and has never been observed. A bt later we wll fnd the physcal source of the dpole. In order to determne the sze of the effects we should expect we have to know the sze of the dpoles. To fnd ths we have to consder the second source currents. FIELD PRODUCED BY CURRENT In the 180 s Ampere studed the forces between current carryng wres and found the followng result. If you thnk about t ths s qute an achevement. Just magne how hard t would have been to deduce these equatons based on the equpment he had to work wth. It s really true that we stand on the shoulders of gants! Imagne two lttle current elements as shown: He found that the force on element two due to element one was gven by: df Id k Id 1 m r r
Followng what we dd for electrc forces we break ths up nto two parts: the dsturbed envronment produced by current one, and the force the dsturbance produces on current two: k Id r df Id Id B r r 1 m wth k mid1 r db r r We then get the total feld by summng over the entre current I 1 B r k m1 I d r r r r where Ths s the basc result for the feld produced by a current. We can compare ths wth the correspondng result for electrostatcs: de r kq r r r r
E r kq r r r r The smlartes are obvous. We now connect these two dfferent approaches to magnetc feld by consderng the feld produced by a small current loop. Suppose the loop s a crcle and les n the xy plane wth center at the orgn. Let s fnd the feld on the axs of the loop at the pont (0,0,z). Assume the loop has radus R and carres a current I. Then: Consder a tny pece of the loop Idl IN s perpendcular to r r. Thus Id IN r r Id r r c ˆ As we go around the crcle, all components wll cancel except those n ẑ. But the z component s where r r' dbz kmid sn r r' sn R r r'
Hence db k Id R m z r r' Addng these up around the crcle we get Now suppose we make z >> R. Then and we get B B0,0,z kmir z r r' r r' z k m z IR But ths s exactly the feld we found for a pont dpole wth p IR IA The drecton s gven by the rght-hand rule. Curl the fngers of rght hand n drecton of current and the thumb wll pont n the drecton of dpole. In our case p IAzˆ Ths provdes the couplng between the two sources of magnetc feld. Note that any current loop can be thought of as a collecton of dpoles: Note that all the nteror currents cancel so that all that s left s the current around the loop. But we have now replaced t wth an nfnte set of dpoles. To fnd the feld we can use ether the current or the dpole approach.
DIPOLE MOMENT OF BAR MAGNET We can now estmate the dpole moment of the bar magnets shown n class. We frst realze that the magnet s not really a dpole unless we are far away from t. If we are, we can consder t to be made up of a collecton of ndvdual atomc dpoles, each due to a current loop caused by the electron movng around the nucleus. Then we need the number of atoms where Then We have qv qv qvr I p R zˆ R R volume A # N (A#) r Aw Aw (A#) = Avogadros number = 6 10 6 /kg mole (Aw) = atomc weght = 57 for ron ρ = densty = 8000 kg/m for ron l = 7 =.18 m r = 5 10 - m Q = 1.6 10-19 coul R 10-10 m v 10 6 m/sec qvr A # p r zˆ Aw 19 6 1.610 10 10 10 6 10 6.18 p 810 510 57 57 8.6 amp m zˆ
p 8.6 6.010 amp / m vol.005.18 We can now fnd the force between two such magnets placed a dstance d apart. From our prevous result n electrostatcs we expect: 7 4 6kmm1m 6 10 8.6 4.9 10 4 F 4.9 10 n 4 4 4 d d d d1m Ths s clearly a small force. Estmate what t would do No wonder we can t see t!! mg mg T cost cos FM Tsnmgtan 4 F 4.910 4 4.410 rad.05 mg 810.005.189.8 MAGNET AGAINST PLATE We now consder the famlar case of a magnet aganst a steel plate. To do ths we recall the effect of an electrc feld on a delectrc. The result was to polarze the delectrc resultng n a net charge at the surface of delectrc:
If the dpole moment/volume s P then the charge/area on the surface s P ˆn where ˆn was a unt vector pontng outward from the surface. To see ths let l be the length of the dpole. Then Then the dpole contaned n the volume V A where A s the area of the surface, s PA Thus the charge s PA q PA Hence the charge/area s P Ths means that we can pcture the bar magnet as Now consder the ron plate. The atoms can t move, but ther magnets can lne up, just as for the delectrc. Hence we expect
We expect ths to occur untl the charge on the surface of the ron s equal and opposte to that on the magnet. Note that these are fcttous magnetc charges. However, ths s ok snce we have seen that the math s the same as f we had looked at current loops. Hence Fnally, we recall that the force on a conductor was F k k M 10 10 A 7 6 m 7 6 F 10 10.005.18 16n attractve. To do ths we have assumed equal and opposte charges, and neglected the effect of the charges at the other end of the magnet. The frst approxmaton s very good for ferromagnetc materals such as ron. The second depends on the magnet beng long. In our case, snce the force falls off as 1/R 4 t s a pretty good approxmaton. AMPERE S LAW In general to fnd the feld produced by currents we have to perform the sum found above. Ths equaton s known as the Bot Savart Law. Unfortunately ths n general nvolves calculus just as for electrostatcs. However there s an analogue to Gauss s Law whch can be used n specal symmetres to get the result easly. Ths result s known as Ampere s Law. To derve t from the Bot Savart Law requres calculus so I wll smply state the result: Bd 4kmout I where I out s the current comng out through the cap coverng the loop. The sgns matter. You must go around the loop keepng t to your left. Then the current s postve and out of the page. Just as wth Gauss s Law, ths s always true, but normally only useful f we can get B out of the sum. It s nterestng to compare the two laws: Gauss Law Qn E da 4kQn 0
Ampere s Law B d 4 I I out 0 out We can now use ths to fnd the magnetc feld n two very mportant specal cases.