Introducing Faraday's Law

Transcription

1 Introduing Faraday's Law L. L. Tankersley and Eugene P. Mosa Physis Department United tates Naval Aademy Annapolis, MD Faraday's Law Tankersley and Mosa 5/27/2013 page 1

2 This note is to propose that several points be more heavily emphasized in introdutory and intermediate presentations of Faraday's Law and that a ommonly used notation be avoided. The use of areful and omplete notation to ensure that the student is neither under-informed nor misled is most ruial. The Faraday flux rule relates the emf in a iruit to the rate of hange of the magneti flux through the iruit due both to the motion of the iruit and to the time variation of the magneti field. In this paper, ontributions to the net magneti emf assoiated with the motion of ondutors are referred to as motional emfs and those assoiated with the time dependene of the magneti field are referred to as indued emfs (indution). For iruits in whih a thin ondutor defines a losed boundary, the Faraday (filamentary iruit) flux rule an be expressed as: d d ˆ magneti da dt dt Bn E (1) but not as: d d d ˆ da E C dt dt B n (misleading form) (2) beause Equation 2 an be interpreted to mean that the eletri field irulates even for the ase of a purely motional emf in a stati magneti field (see Equation 9a). However, the eletri field irulates only in the ase of indution. Note that we are restriting our attention to emfs assoiated with magneti fields. Other soures, suh as hemial ells, generate emfs by distint means, and we refer you elsewhere for disussions of these subjets. i, ii, iii The important point is that when onsidering all emfs, a irulating eletri field exists only in the ase of indution. One of the first things that students learn in their introdution to eletriity and magnetism is that for stati situations the eletri field does not irulate. As a onsequene, an eletrostati potential an be defined suh that the eletri field is the negative of its gradient. The students must understand that this representation remains omplete as long as and only as long as the magneti field is stati. ome treatments introdue the Faraday flux rule and then analyze a motional emf devie suh as the slide wire generator in a stati magneti field to establish that the emf is equal to the (negative) time derivative of the magneti flux through a surfae bounded by the slidewire iruit. At this point, a simple statement that the irulation of the eletri field remains zero so long as the magneti field remains stati is not suffiient if an expression equivalent to Equation 2 for the irulation of the eletri field appears later. The magneti flux an be time-dependent even for ases in whih the Faraday's Law Tankersley and Mosa 5/27/2013 page 2

3 magneti field is stati, and so the student may onlude (inorretly) that Equation 2 predits that the eletri field an irulate even if the magneti field is stati everywhere. The relation B E (3) t frequently alled the differential form of Faraday's law, an be transformed into an integral form B E ˆ indution Ed da C n (4) t These integrals are to be omputed at a single instant in time on path C and on the surfae that C bounds with the fields E and B evaluated in an inertial frame iv at that same instant. using standard theorems. These integrals are to be omputed at the same instant on path C and surfae, and with fields E and B evaluated in an inertial frame v in whih C and are stati. We note that Equations 3 and 4 only desribe indution, and that expressions for motional emfs must be dedued separately using the Lorentz fore law F q E v B (5) This approah leads to an expression for motional emfs E motional v Bd (6) C where v is the veloity of the boundary element d. vi (At this point it should be emphasized that, in spite of the form of this expression, the magneti fore does no work. vii ) In ontrast to Equation 4, Equation 1 is a omplete statement of The Faraday flux rule whih inludes both motional emfs and indution, although Equation 5 is neessary to ompute motional emfs for problems in whih the iruit does not onsist of a thin wire. The orret physis is always given by Equations 3 and 5. viii Many students in intermediate eletromagnetism ourses need more instrution on the relation between the total time derivative of an integral and the partial time derivative of its integrand. As an example, we examine the ontent of Equation 1 using Leibnitz's rule for differentiating integrals. This rule expands the total time derivative of an integral as the sum of two ontributions, one assoiated with the motion of the boundaries of the integration interval and the other with the partial time derivative of the integrand. The Leibnitz rule is usually expressed as ix Faraday's Law Tankersley and Mosa 5/27/2013 page 3

4 d x2 x2 dx2 dx1 b( x, t) bxtdx (, ) bx ( 2, t) bx ( 1, t) dx dt x1 dt dt x 1 t (7) For a divergene-free field Br (,) t, this rule an be extended to a more general form x, xi where we identify d (,) (,) ˆ d t t da (,) t d ˆda dt r C dt Br Br n Br n (8) t dr dt as v, the veloity of the segment d of the path bounding the integration surfae. The ross produt represents the rate at whih area is swept out by d. By multiplying through Equation 8 by negative one and by hanging the order of the produts in the path integral we obtain d (,) t (,) t ˆda Br ˆ C d da dt Br n v B n (9a) t ubstituting in Equation 9a for the terms orresponding to the expressions in Equations 1, 4, and 6 we have E E E ( 9b) magneti motional indution In spite of the urious ompleteness of Equation 1 as revealed in Equation 9b, the generation of a motional emf and indution are "two different phenomena. xii The phenomena are learly related. This relation an be seen by omparing the measurements made by an inertial observer in the situation that a onduting loop moves with onstant veloity and fixed orientation in a stati, nonuniform magneti field with those of an inertial observer moving with the loop. The link between motional emf in one frame and the indued emf in the other is revealed not by the mathematis of the Leibnitz rule, but rather by the Priniple of Relativity. Introdutory presentations should give the distint natures of the two phenomena the same weight that is given to their unity. Motional emfs annot, in general, be transformed into examples of indution as there is no requirement that every point on a iruit be at rest in a single inertial frame. Construtive examples should be presented that identify the fields responsible for the work done on harges for motional and indutive phenomena. One might ontrast the operation of a slidewire generator and a betatron. For the slidewire generator, a quasi-stati Hall effet eletri field does work on the urrent arriers as they follow a non-losed path. xiii, xiv In the ase of a betatron, a irulating eletri field does work on the orbiting eletrons. Faraday's Law Tankersley and Mosa 5/27/2013 page 4

5 As with Equation 4, the desire for areful, omplete notation suggests that Maxwell's fourth equation be expressed as: E Bd ˆ C 0 J 0 nda (10) t This equation also relates the irulation integral of a field to the time variations in the flux of another field. Moving boundary ontributions are not to be inluded in this relation. Grouping the fields on the right hand side makes expliit that the surfae integrations for the two fields are over the same surfae bounded by the urve C. The Maxwell equations desribing the irulations of the fields are written, orretly and unambiguously, without total time derivatives in Equations 4 and 10. Following these onventions, the loal onservation of harge law is expressed as: ˆ da Jn dv V V t (11) In summary, we reommend avoiding inexat or ambiguous representations of Faraday's Law, suh as Equation 2, that blur the distintion between magneti emf (motional plus indued) and the irulation of the eletri field. The eletri field irulates only if there is a time-dependent magneti field. The fat that the magneti fore never does work should be reinfored whenever the magneti fore on harges is used as the basis for deriving expressions for motional emfs. The total time derivative of the flux of a divergene-free field through a surfae an differ from the flux of the partial time derivative of that field if the boundary of the surfae is not fixed in spae. In the evaluation of the total time derivative of the magneti flux through a iruit, the terms assoiated with the motion of the boundary do have an interpretation as the motional ontribution to the total magneti emf. However, they are not to be inluded in the expression for the irulation of the eletri field. Referene: L. L. Tankersley and Eugene P. Mosa, "Introduing Faraday's Law", AAPT pring Meeting (Washington, DC Apr 1994), presentation N11-8. Reommended: G. Giuliani, A general law for eletromagneti indution, EPL, 81 (2008) doi: / /81/60002 Faraday's Law Tankersley and Mosa 5/27/2013 page 5

6 Endnotes and Referenes: i ii iii iv v vi vii F. Reif, "Generalized Ohm's law, potential differene, and voltage measurements," Am. J. Phys. 50, (1982). D. J. Griffiths, Introdution to Eletrodynamis (Prentie-Hall, Englewood Cliffs, 1989), 2nd ed., pp. 277; 3rd ed., p E. M. Purel, Eletriity and Magnetism (MGraw-Hill, New York, 1965), pp ome authors [for example, see ref 10] use a form similar to Equation 2 in whih the eletri field is to be evaluated point by point in the loal omoving frame of the line element. This elegant approah reveals part of the beauty of eletromagneti field theory, but, for the introdutory student, it does not replae the inertial frame statement of Equation 4. ome authors [for example, see ref 10] use a form similar to Equation 2 in whih the eletri field is to be evaluated point by point in the loal omoving frame of the line element. This elegant approah reveals part of the beauty of eletromagneti field theory, but, for the introdutory student, it does not replae the inertial frame statement of Equation 4. ee Ref. 2, pp E. P. Mosa, "Magneti Fores Doing Work?," Am. J. Phys. 42, (1974). viii R. P. Feynman, R.B. Leighton, and M. ands, Addison-Wesley, Reading, 1964), p ix M. Boas, Mathematial Physis (John Wiley, New York, 1966), p. 162 x ee Ref.2, pp xi J. D. Jakson, Classial Eletrodynamis (John Wiley, New York, 1975) 2nd ed., p. 212; 3rd ed., p xii ee Ref. 7, p xiii ee Ref. 6, p. 297 xiv ee Ref. 2, p. 280 Faraday's Law Tankersley and Mosa 5/27/2013 page 6