1967, No. 9 271 The skin effect H. B. G. Casimir and J. Ubbink 1. Introduction; the current distribution for various configurations IT. The skin effect at high frequencies Ill. The skin effect in superconductors "It was discovered by mathematical reasoning that when an electric current is started in a wire, it begins entirely upon its skin, infact upon the outside ofits skin; and that, in conse- quence, sufficiently rapidly impressed fluctuations of the current keep to the skin of the wire, and do not sensibly penetrate its interior. Now very few (if any) unmathematical electricians can understand this fact; many of them neither understand it nor believe it. Even many who do believe it do so, I believe, simply because they are told so, and not because they can in the leastfeel positive about its truth of their ownknowledge. As an eminent practician remarked, after prolonged seep- ticism, 'When Sir W. Thompson says so, who can doubtit? " These were the wordsof Heaviside in apleafor the use of mathematical methods in 1891. Now, seventy-five years later, the skin effect is such common knowledge that one sometimes thinks one understands it even without "mathematical reasoning". The expression for this effect put forward in 1886 by Rayleigh and now often used as a matter of course, does however have its limitations. For example, its application to pure metals at very highfrequencies leads to incorrect results, as noted by H. London in 1940. Superconductors are another special case, where theformula predicts an infinitely thin skin layer. These are afew of the problems which will be dealt with in this article, showing as far as possible their inter-relationship. The article is divided into three parts, the first of whichfollows here. I. Introduction; the current distribution for various conûgurations If a direct current flows in a conducting wire, it will that the charges in the wall of the cage are mobile. be distributed uniformly over the cross-section. With The tendency of the current to flow at the surface is alternating current, however, the current distribution is closely connected with the stable character of the not homogeneous and, if the frequency, conductivity electromagnetic phenomena. We see this in the follow- and dimensions of the conductor satisfy certain con- ing way. Let us assume that inside the metal there is a ditions, to be dealt with later, the current flows mainly filamentary current I which is increasing in strength in a thin layer at the surface of the conductor. This (fig. 1). This current is associated with a rotational phenomenon is called the skin effect. It is an electro- magnetic field H around it which is also increasing. A dynamic effect, that is to say, it is a result of the way in changing magnetic field induces a rotational electric which time-varying electric and magnetic fields and field E which, in turn, induces a current in the metal. electriccurrents areinterrelated. The skin effect phenom- According to Lenz's law, the direction of E is such that enori is quite different from the action of a Faraday it opposes the increase of I, thus keeping the situation cage, for instance, which acts as a barrier to a static stable. The figure shows that simultaneously at some electric field purely and simply as a result of the fact distance from I a current is generated parallel to I. The net result therefore is that the current is forced out- Prof. Dr. H. B. G. Casimir is a member of the Board of Manage- ment of N. V. Philips' Gloeilampenfabrieken; Dr. J. Ubbink is with wards. Furthermore, we see that this effect increases Philips Research Laboratories, Eindhoven. with frequency (E is larger the more rapid the change L.._ ~ ~~__ ~_ PHILIPS TECHNICAL REVIEW VOLUME28 subject to many cycles of the alternating field between two collisions and within the mean time that it spends t in the skin layer. Broadly speaking, the field then "sees" in effect a layer of free electrons. Finally, at still higher frequencies, the "plasma fre- quency" of the metal will be reached above which the metal becomes transparent to the radiation. These skin effect complications at high frequencies form the subject of part IL Fig. 1. A filamentary current J is accompanied by a magnetic Metals in the superconducting state form a special field H. As J and H increase, an electric field E is produced class of conductors. These will be discussed in part "rIL whose direction near J is such' astto oppose the increase in J, whereas further away from J the induced field E produces a We consider the two-fluid model, in which the electrons current in the conducting medium parallel to J. are divided into two types, normal and superconduct- ing. The superconducting electrons, although they dissi- pate no energy, do have screening properties (even at ofH) and with conductivity (the larger the conductivi- cu = 0). They-therefore cause a skin effect: fields can ty, the larger the current caused by E). penetrate only to the London penetration depth, In this :first article, we shall consider in some detail which is independent ofthe frequency, and the "super- the configuration of the current for one or two special current" in this layer does not cause energy losses. The situations. normal electrons within this layer do, however, absorb The skin effect can be a disadvantage in transporting electromagnetic enemy for cu=!= 0, givmg some alternating current energy along a wire or cable. To (small) high frequency losses. There is, however, a keep the resistance low, the cross-section of the con- frequency limit above which the superconducting ductor should be as large as is practicable but the effect electrons also absorb energy. This absorption is due of increasing the diameter is far less than with direct to the transfer of electrons from the superconducting current. For alternating currents it is advantageous to to the normal state by the radiation, via a quantum use hollow cables (in power engineering), or braided process. A superconductor differs very little from an cable (in radio engineering). At microwave frequencies ordinary metal at frequencies above this frequency the skin effect can be put to good use: the effect makes limit (which lies in the microwave range). it possible to transport and store electromagnetic Since the skin effect is based entirely upon the dy- energy without radiation losses by using closed wave- namic properties of electromagnetic fields and currents guides and resonant cavities. as given by Maxwell's equations, it will be useful to At high frequencies the skin layer may be regarded as set down the four equations here: a layer screening electromagnetic radiation incident upon the metal: as a result ofthe conducting properties curl H = oD/of + J, (1) of the metal the radiation penetrates into the metal no curl E = -oB/Ot, (2) further than the depth of the skin layer. Even a single electron, whether bound or free, possesses screening div B = 0, (3) properties to a certain extent: incident radiation is div D = (J. (4) scattered by the electron so that the power travelling straight on is less than the incident power. The following points should be noted: As cu(the angular frequency) or (J (the conductivity) a) In what follows we shall in general regard the mat- increases, the penetration depth ~ decreases. In simple erial as a medium with a given relative dielectric con- electron theory, (J is proportional to the mean free stant and permeability er and pr (so that in the material path I of the conduction electrons. As (J or cuincreases, D = eE, B = pH, with e = ereO,p = prpo), in which there will therefore be an instant at which ~ becomes the free electrons carry the current. er and pr are gener- smaller than I. The current density at a given point ally of the order of unity for non-ferromagnetic will then no longer be dètermined simply by the materials. Interesting complications which may arise local field intensity and the static conductivity, and the when pr becomes much greater than unity (ferromag- simple theory of the skin effect will no longer apply. netism) will be discussed in the last section of part 1. This situation is referred to as the "anomalous skin b) Over a very wide-frequency range, the term oD/Ot effect". (the ·"displacement. current") in the metal is negligible . As the frequency increases, other effects may become with respect to the current density J and may therefore significant, namely, relaxation effects: the electron is be ignored. When J can be represented simply as (JE, 1967, No. 9 SKIN EFFECT, I 273 this amounts to taking w as negligible with respect tb the wires are thin and long compared with their sep- a/e. For copper at room temperature for instance, aration: rw« a e; L (rw is the radius of the wires, 8 a F::::i 10 (Qm)-l and a/e F::::i 1019 S-l, which is very much L their length and a is the spacing). It follows from (1), higher than the frequencies with which we shall be using Stokes's theorem, that if H is thè.field. at-a dis-, dealing in this article. 'öD/'öt can become comparable tance x from a wire carrying a current I: to J only in the relaxation range, where J becomes smaller than (jE; even then 'öD/'öt begins to become 2nxH = I, hence H = I/2nx, really significant only at frequencies near the plasma and the total flux through a surface bounded by two frequency. values of x, x = p and x = q, is: Parallel wires The case of a number of parallel wires lying in a plane and connected in parallel may be used as a simple illustration of the essential features of the skin effect.