FI 2201 Electromagnetism

Transcription

1 FI 2201 Electomagnetism Alexande A. Iskanda, Ph.D. Physics of Magnetism and Photonics Reseach Goup Electodynamics ELETROMOTIVE FORE AND FARADAY S LAW 1

2 Ohm s Law To make a cuent flow, we have to push the chages. How fast they move depends on the natue of the mateial. Fo most substances, the cuent density J is popotional to the foce pe unit chage, f : J = σ f the popotionality constant σ is a second ank tenso, as ae the susceptibilities, but many common media ae linea in the sense that the conductivity σ can be consideed a scala. A pefect conducto has σ, and by contast, a esisto has small conductivity. The ecipocal of conductivity is called esistivity, ρ = 1 σ, which is a chaacteistics of the mateial. Alexande A. Iskanda Electomagnetism 3 Ohm s Law In pinciple, the foce could be anything (chemical etc), but we concentate on the electomagnetic foce F = q E + v f = E + v ( ) In eal substance, the velocity is vey small, thus the second tem above is neglected. Hence, we have J = σ E This elation is called Ohm s Law. Thee is no contadiction with the fact that inside a conducto the electic field is zeo, since in an electical l cicuit, the wies ae made of good conductos, thus J E = 0 σ Example 7.2, Poblem 7.3 Alexande A. Iskanda Electomagnetism 4 2

3 Ohm s Law Ohm s law should stike you as stange, at fist, because you also know that J = v Suppose J ρ is constant. Then v and E should be constant. ut if E is constant, chages should acceleate at a = qe m, endeing it impossible to have a constant v. So which elation is wong? What happens is that collisions between fee chages in a cuent (like electons) with fixed o slowe moving chages (like the ions the electons leave behind), and othe fee cuent caies, keep the acceleation fom going on fo vey long. Alexande A. Iskanda Electomagnetism 5 Ohm s Law In collisions with the ions, the kinetic enegy gained by the fee caies fom the field is lagely tansfeed to the ions, and the electon stats ove. We theefoe econcile Ohm s Law with the definition of cuent density by supposing that the collision pocess esults in a well-defined aveage velocity, v, also called the dift velocity which is a constant, and wite J = ρ v In fact, we don t need to suppose; we can show that that s the way it is, in a cude classical model of what is mostly a quantum phenomenon. Alexande A. Iskanda Electomagnetism 6 3

4 Electomotive Foce (emf) Note that in a typical electic cicuit (fo example, a light bulb) connected to a battey, the cuent is the same all the way aound the loop. Why is it constant aound the loop? Recall that the only diving foce on the chages, f, is s confined on the souce (battey). Suppose that thee is an accumulation of chages on some pat of the wie, such that the cuent is not constant. This accumulation of chages will ceate an electostatic field that will smooth out the flow of chages. Thus, we can wite f = f s + E electostatic Alexande A. Iskanda Electomagnetism 7 Electomotive Foce (emf) To calculate the wok done by this foce in taking a chage aound the loop, we line integate aound the closed loop to yield f dl = f + E d l = f d l ( ) E s This non-zeo esult is called the Electomotive foce (emf) of the cicuit. s Alexande A. Iskanda Electomagnetism 8 4

5 Motional emf Moving a conducting wie in a magnetic field can also esulted in motion of chages in the wie. This is called motional emf. onside the expeiment of moving a loop in a magnetic field. In this case, the foce that push the chages in motion is the magnetic foce v = v f s x h R Alexande A. Iskanda Electomagnetism 9 Motional emf The motional emf is calculated as befoe dx da dφ E = f s d l = vh = h = = the minus sign accounts fo the fact the ate of change of the aea a is negative and the magnetic flux is dφ = da. It tuns out that this last elation is valid much moe geneally independent of the shape of the loop, homogeneity of. Suppose inside id the wie the dift velocity is u, then the total magnetic foce on a chage can be deduced to be v f mag u w = w v u Alexande A. Iskanda Electomagnetism 10 f mag 5

6 Motional emf onside a loop of wie moving, and pehaps even changing shape, though a egion with a static, and follow the point A. In time it moves a distance v, and with the line element dl sweeps out an aea d a = v dl A dl θ v at time t at time (t + ) Alexande A. Iskanda Electomagnetism 11 da Motional emf The change in magnetic flux though the loop, that s admitted by the bode ibbon is d Φ = = = Φ ( t + ) Φ ( t) da ( v dll ) ibbon Now suppose a cuent uns in the loop. If the dift velocity of the caies (elative to the loop) is u, and thei total velocity w = v + u, then since u must be paallel to d l then v dl = w dl A dl θ v at time t at time (t + ) Alexande A. Iskanda Electomagnetism 12 da 6

7 Motional emf Hence, Thus, o dφ ( w dl) = dl ( w) = dl ( w ) = dl fmag = ( w dl) = ( w ) dl = dφ E = f mag dl Alexande A. Iskanda Electomagnetism 13 Electomagnetic Induction What if the field in the egion vaies, with the loop stationay? Relativity: as long as the elative motion is the same, the same emf must be obtained as befoe. (We see this expeimentally too.) In this case, though, it s no longe clea what exets the foce that moves the chages, since v = 0. Faaday gave an ingenious explanation to this. He postulate an induced, non-electostatic, electic field can be obtained fom changing of magnetic field : dφ E = E d l = With Stokes theoem d d E dl = ( E) da = da E = S S Alexande A. Iskanda Electomagnetism 14 7

8 Electomagnetic Induction d E = Faaday s Law of Induction This means that a non-static electic field can be induced by a nonstatic magnetic field. That is, a cuent can be induced in a loop of conducto by changing the flux of though it, no matte how the flux changes: motion of the loop, o change in. The minus sign in Faaday s law indicates that a changing magnetic flux will induce an electic field and cuent such that the magnetic field induces by the cuent leads to a flux change in the opposite diection. This is called Lenz s Law. Alexande A. Iskanda Electomagnetism 15 Induced Electic Field alculations of induced electic field E with Faaday s Law poceed just like calculations of fom steady cuents using Ampèe s Law. Note the following elations: d = µ 0 J E = o in integal fom dφ dl = µ 0I enc E dl = Also = 0 E = 0 ρ = 0, only cuents changes Appaently they e the same, with the intechange of d dφ E µ 0 J µ 0Ienc Alexande A. Iskanda Electomagnetism 16 8

9 Induced Electic Field We can use this similaity to calculate induced electic fields using the machiney of magnetostatics (namely Ampee s Law). Example 7.7, 7.8 Alexande A. Iskanda Electomagnetism 17 9