FARADAY S LAW. dw F dr qe dr. EMF E d. EMF v B d. dt dt

Transcription

1 FARADAY S LAW It is observed experimentally that if the magnetic flux through a circuit is changed a voltage is produced around the circuit in such a direction as to oppose the change. The magnetic flux is defined in the same way as electric flux: M BdA We put voltage in quotes since it is quite different than the voltage we have discussed before that voltage would be zero around a closed loop since E is a conservative field. This voltage has the same effect on a charge it gives the work/charge in moving between two points. In this case, it is moving around a closed loop. Since work is given by: dw F dr qe dr we define voltage (to which we give the name EMF) as: Thus EMF E d dm d EMF B da dt dt The first question is whether there is really anything new here. To see why this is a legitimate question, consider the Lorentz force law: F qv B This means that a charge moving in a magnetic field will experience a force equivalent to that produced by an electric field: E v B Thus it should experience an EMF around the loop of: EMF v B d v BdA (Stokes Theorem) M

2 MOTIONAL EMF As we will see this EMF will be non-zero in certain circumstances even without making any changes in the Maxwell equations we have so far considered. To see this we note first that the EMF will be zero unless at least one of B or v is not constant. Recalling that: AB B AB A A BA B we have EMF B v B v v B vb da We note that the first two terms are concerned only with variation in v with B constant, while the last two hold v constant while B varies. We will treat the two sets differently. is: Consider first the contribution to this integral due to changes in B. That part of the curl (recall that B0). But Thus: v BvB vb db B B dx B dy B dz dt t x dt y dt z dt B B B B B v x vy vz v B dt x dy z dt v B is that part of d B /dt arising from the loop s motion in an inhomogeneous B. Next consider the contribution due changes in v. These will result in a distortion in the shape or orientation of the loop. To find their effect we go back to the original form for the EMF: vb d Bv d B vd (we can interchange and ). But the change in area of the loop due to the shift of d is:

3 da vd sin dt ˆ where ˆ is in the direction of the change in area. We recall that the vector area is perpendicular to the surface and directed outward. The new chunk of area at d is the parallelogram defined by d and v. The outward direction is out of board (recall the right hand rule), and hence the direction of ˆ is that of v d andda vd dt. Hence da v d dt gives the change in A at d. Thus the change in the integral due to distortion of the loop is: da vb d B dt Thus putting the two results together we get: db B d d B EMF M da B da B da da dt t dt dt dt Comparing this with Faraday s Law we have: Thus the only thing that is new is the term: d B EMFF B da EMF M da dt t B da t To see the change this makes in Maxwell s equations we note that since it involves only change in B for a fixed loop we can write: Then since the loop is arbitrary we must have: B E d E da da t B E t

4 Thus a magnetic field which changes in time produces a non-conservative electric field. This has profound consequences, as we shall see. Note that there are now two sources of EMF a time varying magnetic field, and a loop which changes shape or orientation. The first is new, the second is not. Ultimately the EMF is always given by: d EMF dt but this can be calculated in several different ways. For example consider a coil of wire rotating in a constant magnetic field: M We can calculate the EMF around the loop either by finding M directly or by calculating: v B d If B were a function of time and we used the second method we would have to add a term: Let s see how this works. B da t

5 First find M directly. We have: M BdSByˆ 2 xsin ˆ tycos ˆ t 2 Bcost where Area B EMF B sin t cos t dt Next calculate: v B d around the loop. Here we must be careful. We go around the loop keeping it to our left. On sides (1) and (3) v is in or out of the paper resulting in v B in the + or z direction. Hence: v B d 0 On (2) v is into the paper and of magnitude (l/2). Since B is in the ŷ direction we get: and 0n (4). (Recall the diagram) v B Bsintzˆ 2 vb Bsintzˆ 2

6 But d zˆ on (2) and ẑ on (4). Thus 2 2 vbd Bsin t 2 Bsint 2 Hence method two is off by the term: 2 B cos t t If d B /dt = 0, the methods give the same result. MAXWELL S EQUATIONS We now have the equations: E, 0 E E t B0 B0J Maxwell recognized that these are inconsistent with charge conservation: J 0 t To see this take the divergence of: B 0J B 00J But J 0 t To fix this he added a term:

7 to the right hand side to get: 0 t B 0 J t But from E, 0 we have Thus E 0 E,, 0 t t t E B 0J, 0 t E D B0J0, 0 0J t t Hence the final form of the Maxwell equations is: E, 0 B E t B0 In a material medium these become: B J, 0 0 E t

8 To use these we must have D(E) and H(B). D B E t B0 D H J t