The physics of induction stoves

Transcription

1 The physics of uction stoves This is an aticle fom my home page: Contents 1. What is an uction stove...1. Including self-uctance The contibution fom the magnetic moments...6 Ole Witt-Hansen 9 (17)

2 The uction hub 1 1. What is an uction stove When I was teaching physics and mathematics in the Danish 9 1 gade high school in 9, a teache (with humanistic pofession) who had acquied a uction stove, appoached me and asked, why it did not wok with kitchen pots and pans which wee not magnetic. At that time, I actually knew nothing specific about uction hobs besides, moe geneally, that they wee based on Maxwell s second equation, when witten in integal fom, is bette known as Faaday s law. x E t A time dependent magnetic flux (magnetic field times the aea) though a closed cuve, will uce an emf (electomagnetic foce) aound the cuve. Integating both sides, using Stokes law, we have: emf : E ds aea x E da t aea da t t t ( Magnetic flux) So my fist answe was, that the efficiency of an uction hub, does not depend whethe the mateial is magnetic, as long as it is conducting, so that the uced emf fom an altenating magnetic flux can dive a cuent and heats the pan. Fom this point of view a mateial with high conductivity i.e. cobbe, should be most efficient. A cude calculation showed that the heat geneated in fact should be popotional to the specific conductivity of the mateial. This seems fom a physical point of view to be entiely plausible, but when I shotly aftewads (as a cheep offe) bought an uction hub, and tied it with cobbe and aluminium pots and pans it did not wok at all. (What I could also have ead in the use guide!). What I ealized was, that the Maxwell equations, that I knew of, ae actually only valid in vacuum, but also that thee ae lage diffeences, when fo example, thee ae applied in mateials with a magnetic dipole moment as ion o nickel. The magnetic flux in the metal, comes of couse fom the -field in the metal, which is popotional to H, the field outside the metal times the elative pemeability. H Then I went ahead with a calculation of the powe P, which is geneated fom an altenating magnetic field in the metal. Since the field lines fom the electic field that causes the eddy cuents in the metal can be consideed cicula, we slice the bottom of the metallic disc into ing with thickness d.

3 The uction hub The disc itself is assumed to have thickness w, and adius. See the figue below. l The esistance in a ing with adius is accoding to the fomula: R whee ρ is the A esistivity of the mateial, l is the length and A is the coss section aea, so we f fo dr. dr wd The flux though the ing is: H. Hee is the elative pemeability fo the disc, and H is the extenal magnetic field geneated by the coil. In each ing is uced an emf (electomagnetic foce). t Whee is the magnetic flux though the ing. The powe dp which is set aside in the ing is: dp whee dr 1 w d dr The powe can then be found by integation. The oute magnetic field is assumed to be given by: H it ( t) e. With H, and t Wethen get an expession fo the uced emf in a ing with adius. ie it

4 The uction hub 3 As it is the case with an altenation voltage, the effective value is obtained by taking the modulus multiplied by. 1 eff 4 when inseted in dp, with dr 1 w d gives: dr w 4 dp 3 d Integating this expession fom to, (the adius of the disc) we f: w P 16 4 You may wonde why we integate ove 1/dR instead of integating ove dr, but this is as it should be, since the ings can be consideed as esistances in paallel, accoding to the fomula: 1/R = 1/R 1 +1/R. In the popula desciption of uction cooking plates, you can ead that they ae diven by a high fequency magnetic field, but I have not been able to f a value fo the fequency. To asses whethe the calculation above beas any elation to eality, I have put (somewhat abitay). The thickness of the plate in the ion pot w = 5mm. The peak value of the extenal of the ucting magnetic field =.1 Wb/m. The cyclic fequency of the field ω = π 1kHz, and the adius of the pot = 1 cm. The elative pemeability fo ion is μ = 66, and the esistivity of ion is ρ = Ωm. When this is inseted in the expession fo the deliveed powe P, it gives 1887 W, which must be consideed as vey satisfactoy. ut the cucial point is of couse the magnitude and the fequency of the extenal magnetic field, which geneates the eddy cuents, since they ente as the squae in the fomula fo the geneated powe. So, fo example, if the fequency o the geneating field is 1 times less, then the geneated powe is 1 times less. The impotant point hee is that the powe geneated gows with the squae of the -field, the elative pemeability of the metal μ, and of the cyclic fequency ω. The elative pemeability fo ion μ = 66, while fo Aluminium it is. So the ation of the dissipation of powe in an Al plate and a Fe plate is: (/66) ( / 66) 1/ 1. Since coppe is diamagnetic and μ = -1, the elative pemeability of cobbe and Aluminium, seve as an definite explanation why an uction hub only wok on cookwae made of ion.

5 The uction hub 4 This is, howeve, not the whole stoy, since we have not yet consideed the possible influence of self-uctance, and the possible contibution fom the foced oscillations of the magnetic dipole moments of the ion atoms.. Including self-uctance In the calculations above, we chose in the fist place to ignoe the phenomenon of the self uctance coming fom the eddy cuents in the ion plate. Fo completeness, we shall include self-uction in the calculation, although it tuns out that it does not change the esult significantly. Howeve, when taking account the self-uctance the calculations pogess in a athe diffeent way. The setup is shown schematically above. The coil can be consideed as an ideal coil in seies with a esistance. The coil is imposed to a foced oscillation fom an extenal field accoding to the voltage equation. Whee we have put Then we have: Tying with a solution: di Ri L extenal ( t) dt dt e i t and di Ri L dt dt it i i e gives: Ri e i it ii Le i e R il it ie i it i R il R ( L) The effective value of i is thus: i eff i To apply this to the uction hub, we must eplace R with dr and the uctance L, with the uctance of a single tun. Unfotunately thee is no simple fomula fo the magnetic field fom a

6 The uction hub 5 single tun, but if we apply the expession fo the - field in the cente of a cicula conducto with adius, then it is pobably not entiely wong. (Cf. The Helmholtz coils, whee magnetic field in between the two coils is almost homogenous). The -field in the cente of a single coil having a i cuent i is given by:, as can be deived fom iot and Savat s law. The magnetic flux i 1 though the cicula coil with coss section A then becomes: A i. If we then compae the two expessions fo the uced emf. t i L t 1 i t i L t 1 Which shows that L. Replacing R with dr 1 togethe with L in the wd expession fo i eff : i eff we have: dr ( L) Which can be educed to: i eff i eff wd wd 1 w d In the denominato squae oot the last tem is seen to be vanishing compaed to 1, so if we discad that tem we simply get: And fo the powe we f: i eff w d dp= i eff dr = 1 w w 3 d which gives dp d wd 4 4 If the metal plate is a metal with the elative pemeability, then should be multiplied with that facto. We finally end up with exactly the same expession as we got befoe, though it was deived in a quite diffeent manne. This is of couse satisfactoy. At the same time we have shown that the self uctance plays a vey little ole in the calculation of the eleased powe in the uction hub. w 3 dp 4 d

7 The uction hub 6 3. The contibution fom the magnetic moments Wheeas self-uction gives a minimal contibution to the functionality of the uction hub, we can be almost cetain that thee comes a contibution to the heating fom the otational enegy of the magnetic moments of the ion atoms. The fomulas fo the potential enegy of and the toque on the magnetic moments μ fom a magnetic field ae: E pot H As we have demonstated above one can faily easy calculate the contibution to the heating of the plate fom the eddy cuents. It is, howeve, fa moe complicated to estimate the contibution fom the magnetic moments in an oscillating field. Fistly it is necessay to know the hysteesis cuve fo the ion as well as the magnetic dipole moments fo the atoms, and even when both ae well known, a calculation may be unealistic. elow ae sketched thee hysteesis cuves, whee only the fist two have anything to do with eality. ut fo the last two seem to be mathematically equivalent when calculating the loss in potential enegy. The potential enegy of a magnetic moment is E pot, and we shall evaluate the loss in potential enegy, when a magnetic moment is put though the cuve, stating and ending in the same point. Fo the fist cuve the loss will be zeo, and fo the last cuve, the contibution fom the hoizontal lines also be zeo, since the -field is constant. On the vetical lines, howeve, thee will be contibutions -μ and μ, so that Δ E pot = -4 μ. In the Feynmann Lectues II (fom 1964) thee is a athe compehensive and detailed desciption of Feo magnetism. Hee you can lean that feomagnetism is entiely due fom the magnetic moment of one electon, having the magnitude of one oh magneton. e m e Am Evaluating the numbe of ion atoms in an ion plate in the bottom of the peviously mentioned cooking pot, we f:

8 The uction hub 7 N n M N A m M Fe N A V M Fe Fe N A 1,331 5 We then f that E pot 4 N 49. 7J. Multiplying with a fequency of 1 khz, we f in this (pesumably unealistic calculation) a powe of 49.7 kw. So something is definitely wong, at least with the magnitudes of the magnetic field o the oscillating fequency of the extenal magnetic field. Also the atoms will otate, only when the fequency is nea esonance of the atoms. Futhemoe, the powe fom the eddy cuents gows with the squae of the magnetic field and the squae of the fequency, wheeas the powes fom the magnetic moments gow linealy with these quantities. It is of couse possible to chose an extenal magnetic field and a fequency that esults in a powe of about kw, but this is not eally inteesting, as long as we don t know the detailed dynamics of the magnetic moments. Although we have not eached a quantitative desciption of the functionality of the uction stove, we have accounted fo the physics of the two mechanisms that contibute to an uction hub.