Laws of Electromagnetism

Transcription

1 There are four laws of electromagnetsm: Laws of Electromagnetsm The law of Bot-Savart Ampere's law Force law Faraday's law magnetc feld generated by currents n wres the effect of a current on a loop of flux whch t threads the force on an electron movng through a magnetc feld the voltage nduced n a crcut by magnetc flux cuttng t Our unfed theory deduces each of these from ts basc assumpton that magnetc felds exst to gve elementary charged partcles the property of nertal mass. Classcal Physcs calls these "emprcal laws" meanng that they have been formed from observaton. That s to say that they have no theoretcal justfcaton; they just work. Our dervaton of these laws rases ther status, but also rases questons about ther physcal meanng. What nature does Drectons are mportant n electromagnetsm and nature usually does thngs at rght angles. Any proper descrpton must nvolve vectors and the vector cross product descrbed n the secton on felds. Magnetc felds contan energy. That s ther functon n nature. The factor whch determnes ther nteracton wth movng charges s ther energy densty. There s no unversally agreed symbol for energy densty so we have chosen to use for the energy densty of magnetc felds. Q m Q m = 1 B H or Q m = 1 B H where H = v D If we use the propertes of permeablty µ and our defnton of magnetc ntensty as the sum of the actons of the movng electrc felds, we can wrte: whch expands to Q m = 1 µ H Q m = 1 µ ( ) v D ( ) v D Ths absolutely ghastly bt of unversty maths s the key to how everythng works. Snce there s no way that we can ever work t out because t nvolves more sums than there are atoms n the earth, t should be possble to talk about t wthout gettng too worred about how horrble t looks. The thng to understand s that the movng electrc flux of every electron s nteractng wth the movng flux of tself and every other electron and quark. Each one of these adds or subtracts a lttler bt to the energy densty at each pont n space. Energy Movement Cube of the magnetc feld Force Velocty Force Cone of electrc flux Energy Movement Cone of electrc flux Let us consder just two electrons. Velocty Page 1 of 5

2 ˆ We can draw the two cones of electrc flux whch come from the two electrons and meet n the tny volume of magnetc flux. We have made the drawng so that they meet at rght angles n a cube, but they could meet at any angle and the cube could be any shape. The contrbuton to the energy densty of each electron depends on ts velocty and all the angles between ts velocty, the cone of electrc flux and the magnetc flux. Any change n one of these results n a change to ts contrbuton to the energy n that tny cube. Ths requres energy to move back and forth along the tube of electrc flux. Now a consderable length of the tube of electrc flux passes through smlar tny mss-shaped cubes and the change n energy n each these adds to the energy whch needs to flow up or down the cone. The cone ends n a lttle square of the surface of the electron and that s where net amount of ths energy whch moves up and down the cone has to be ether generated or adsorbed. Ths results n a force. The force could be dong work or adsorbng energy; t all depends on the relatve drecton of the force and the velocty. One thng we can be certan of: each of these forces s tangental to the surface. What happens s that the whole of the surface les at the end of ts own cone of electrc flux and each s resonsble for a tny force. All these tny contrbutons add up to gve a net force on the electron. But the unverse s not qute that smple. If the magnetc feld s tryng to dump energy, the electron has to be free to move allowng the force to do work. If we stop the electron from movng, the magnetc feld has to fnd another electron whch s free to move. Ths acton can be seen n transformers and swtch mode power supples. If the unverse dd not work ths way, none of our televsons, computers or moble phones would work. If we could look closely at a movng electron and could see the magnetc feld generated by ts moton, we would see energy denstes mllons and bllons of tmes greater than any found n the magnetc feld of a magnet. The nteracton we have descrbed s completely domnated by the electrons own contrbuton. The magnetc feld contans ts knetc energy and the force generated s the nertal force whch ressts ts acceleraton. The maths even gves us Newton's law F = m a. As we buld machnes whch explot ths nteracton, the geometry and acton of each results n a specal law to ft the crcumstances. Tradtonal TV sets contan a tube n whch a beam of electrons scans the screen generatng the pcture. The beam s deflected by a powerful magnetc feld generated by currents n the feld cols. The law of Bot-Savart enables us to calculated the magnetc feld and the force law allows us to calculate how t bends the beam. Ampere's law allows us to calculate the energy contaned n a magnetc feld and together wth Faraday's nducton law s used n the desgn of the transformer or swtch mode power supply whch powers the TV from the mans. The law of Bot-Savart Ths gves the magnetc feld generated by a current I flowng through a small length of wre δl. The law s heavly dependent on drectons and can only be descrbed by a vector equaton. The flux densty B s: B = µ I δl r 4π r The basc assumpton of our unfed theory s that movng electrc flux generates a magnetc ntensty H = v D. We consder a short secton of wre and all of the electrons and quarks n t. The magnetc ntensty generated by ther moton s gven by the sum of all of ther ndvdual contrbutons H = v D. At frst the veloctes v are measured relatve to the background formed by the electrc felds of all the Page of 5

3 ˆ electrons and quarks n the unverse. But f we group together all the electrons and quarks of an atom, the sum of ther magnetc ntenstes s equal to H = u D where u s now the velocty of a conducton band electron relatve to ts parent atom. Thus the magnetc ntensty due to all the conducton band electrons n the length δl of wre s: δh = u D Then wth a bt more maths, we can show that the sum depends on the current I and the length δl of the bt of wre. We need to nclude the drecton of the current along the wre for the vector cross product and the end result s the law of Bot-Savart: B = µ I δl r 4π r Force law Classcal Physcs gves the emprcal law that the force on a charge q movng at velocty v through a magnetc feld of flux densty B s gven by the school and unversty maths formulae: F = B e v F = q v B If the electron's velocty s perpendcular to the flux then the magntude of the force s gven by the school maths F = B e v n whch e s now the charge of the electron. Classcal physcs makes no attempt to explan how the force s generated. Modern Physcs attempts to explan t wth relatvty usng a "Lorentz transform" to turn the magnetc feld nto an electrc feld, but n the author's opnon, ths s a maths fudge. In our unfed theory, there s an nteracton between the whole of the magnetc feld and the whole of the electron's electrc feld. Ths nteracton results n an energy transfer between the magnetc feld and the electron generatng a net force. When we look closely at the stuaton, we fnd that equal amounts of energy are flowng nto and out of the electron wth resultng forces spread over ts surface. Because they are not all n the same drecton, when we add them, we fnd that there s a net force. Ths s perpendcular to the velocty, so does no work and the speed and knetc energy reman unchanged. In our unfed theory, the electron s surrounded by ts own magnetc flux generated by ts velocty, so the flux of the magnetc feld through whch t s passng must be pushed asde. Ths means that t does not "feel" any local presence of the magnetc flux. Ths does not matter because n our unfed theory, the force s not generated by a local acton. The maths s dffcult unversty stuff, but amazngly t all smplfes to gve: F = µ q v H We should not talk of the "bev" force, but of the "mu-hev" force because t s ths mathematcal artefact whch comes out of the maths. Because H s a mathematcal artefact, t s not subject to the lmtaton mposed on the flux densty B by the fact that the flux s real and sngular. H That havng been sad, t s obvous that the force on the electron s perpendcular to ts velocty. Ths beng so, the acceleraton t mparts always causes the electron to follow a crcular or a spral path. Page 3 of 5

4 Faraday's Law Ths s the most mportant law. Wthout t, we would only have batteres. There would be no mans electrcty, no electrc motors. Ths s the law whch once understood allowed us to generate electrcty. The dagram shows a sketch of Faraday's orgnal apparatus whch conssted of a rng of soft ron wth two cols of wre wrapped around t. It s n fact the world's frst transformer. In our unfed theory, we try to descrbe how nature works at the most fundamental level and dentfy the actual physcal processes through whch she acts. Classcal Physcs as taught n the unversty Electrcal and Electroncs Engneerng Departments has the very smple dea that the ron rng contans magnetc flux and that any change n the flux content envolves flux cuttng through each turn of the wre. Modern Physcs should f t s honest to tself follow Ensten's doctrne that magnetc flux does not really exst. The truth les somewhere between these. Magnetc flux s real stuff, but t does not cut the wres. Faraday's law follows on from the force law. We saw how the movng electron does not actually pass through the magnetc flux, because t s surrounded by ts own magnetc feld and ths pushes the flux of the background feld asde. Once a current starts to flow, the turns of wre are each surrounded by ther own magnetc feld. A loop of flux can only move past the wre by jonng wth one of the flux loops around the wre, then breakng from t on the other sde as shown n the sequence below Lght blue represents the flux strands encrclng the wre and dark blue the strands of the background flux. As the wre moves two flux strands each break and rejon wth each other so that the strand now passes on the other sde. As the wre passes further away from ths strand, an new strand of flux shown n magenta emerges from the wre to replace the lost encrclng lght blue strand. The reverse process occures when the wre moves the rght so from 6 to 1, a pont s reached where the magenta loop of flux s adsorbed nto the wre 4 3 and then splts n two places rejonng wth tself 3 to allow the adsorbed encrclng loop to be replaced. So there s no drect acton between the loop of flux and the wre or the conducton band electrons wthn t. It s an ndrect acton nvolvng the nteracton of the electrc feld of each conducton band electron wth the whole the background magnetc feld. Ths nteracton nvolves energy transfers between the magnetc feld and each conducton band electron resultng n net force F = µ q v H on each one. It s only when we try to add up these forces that Faraday's law emerges. Page 4 of 5

5 w The "trck" s to express the velocty v of an electron relatve to the magnetc flux as the sum of the the velocty of the electron relatve to the wre and the velocty of the wre relatve to the flux. F = µ q v H becomes 1 = µ q w H F F = µ q (w + u) H Now, we can splt the force nto two parts F and. When we add up the all the F 1 forces for a short length of wre, we get a force actng on the wre at rght angles to t. When we add up all the F forces, we get a force along the length of the short bt of wre. If we now add up the forces on the short lengths, the F forces all add to zero and the F forces add up to gve the nduced voltage. 1 = µ q u H = µ q u H We need to understand how the equaton F relates to natures' actons. The actual magnetc flux does not pass through the wre, but the mathematcal artefact H whch s the magnetc ntensty of the background magnetc feld can be descrbed n exactly the same way as f t were a flux and ts flux passes through the wre. We have to use a suffx to dstngush t from the magnetc ntensty H w generated by any current n the wre. The actual magnetc flux threadng the crcut formed by the wre results form the sum of the two magnetc ntenstes. We need some college maths to descrbe how we calculate the amount of flux threadng the crcut by ntegratng µ (H + H w) over the area enclosed by the crcut. Nature s busy breakng and jonng, then breakng flux strand that they move form one sde of the wre to the other, so along the length of the crcut flux s beng lost or ganed resultng a net rate of change of the amount Φ of flux threadng the crcut. When all the maths s worked out, we fnd that the voltage nduced n the crcut s: Where the term dφ dt V = dφ dt descrbes the rate at whch the flux threadng the crcut s changng wth tme. Ampere's law Ampere's law s mportant when we consder magnetc felds n transformers and nductances. It nvolves the concept of magnetomotve force ( mmf) whch s gven by an ntegral around a closed path. Ampere's law states that the mmf s equal to the current threadng the path: mmf = H The ntal assumpton of our unfed theory s that movng electrc flux generates a magnetc feld. Ths leads to the descrpton of magnetc ntensty generated by the moton of an electron H = v D. We saw how ths could be summed for a current I n a short length δl of wre to gve the law of Bot-Savart. For a partcular crcut, t s possble to calculate the magnetc feld produced by the current. It s then possble to ntegrate H dl around a partcular path and show t s equal to the current. However, t only smple enough to do n the case of some very specal geometres. It does not consttute a general proof. We can derve the law when we consder the nteracton between a sngle electron and a loop of flux. It then becomes qute easy to sum the actons of all the conducton band electrons n a crcut and deduce the law. But there s a lot of geometry n t, so we have refer the reader to the dervaton at physcst level. dl = I Page 5 of 5