Exploring Magnetic and Kinetic Energy in Current-Carrying Inductors, Assignments of Physics

Download Exploring Magnetic and Kinetic Energy in Current-Carrying Inductors and more Assignments Physics in PDF only on Docsity! How Much of Magnetic Energy Is Kinetic Energy? Kirk T. McDonald Joseph Henry Laboratories, Princeton University, Princeton, NJ 08544 (September 12, 2009; updated June 27, 2019)) 1 Problem How much of the “magnetic” energy stored in a current-carrying inductor is due to the kinetic energy of the moving charges? Consider the example of a toroidal coil of major radius a and minor radius b  a that carries current I in its N turns, which current is due to electrons of charge −e and (rest) mass m. 2 Solution The “magnetic” energy U stored in an inductor can be written as, U = 1 2 LI2 = ∫ B2 8π dVol, (1) where L is the self inductance, and the latter form assumes all media have unit (relative) permeability and is expressed in Gaussian units. For the example of a toroidal inductor, the magnetic field B0 along its circular axis follows from Ampère’s law as, B0 = 2NI ac , (2) where c is the speed of light in vacuum. Thus, the stored “magnetic” energy is, U ≈ B2 0 8π 2π2ab2 = N2I2b2 ac2 ( and hence L ≈ 2N2b2 ac2 ) . (3) Supposing that all conduction electrons have the same speed v, the current I is related to the number density n of conduction electrons per unit length along the (spiral) conductor according to, I = nev. (4) The total length of the conductor is 2πNb, so the kinetic energy T of the conduction electrons is, T = 2πNnb mv2 2 = πNb mI2 ne2 = πNbI2 nc2r0 , (5) where r0 = e2/mc2 ≈ 3 × 10−13 cm is the classical electron radius. The ratio of the kinetic energy to the “magnetic” energy is, T U ≈ πa Nn b r0 . (6) 1 Since only the product nv is determined by eq. (4), the result (6) is ambiguous. To go further, we suppose that there is one conduction electron per atom in the copper conductor, such that the volume density of conduction electrons ne ≈ 8 × 1022/cm3. Then, the linear number density is n = πned 2/4, where d  b is the diameter of the copper conductor. Equation (6) now becomes, T U ≈ 4a Nne b d2 r0 ≈ a 6Nbd2 10−9, (7) for a, b and d measured in cm. We could also suppose that the N turns are tightly wound on the toroid, such that Nd = 2πa. Then, T U ≈ 1 12πb d 10−9, (8) As an example, suppose b = 1 cm and d = 1 mm = 0.1 cm, for which T/U ≈ 3 × 10−10. 3 Comments This problem was of interest to Maxwell, who did not have a vision of currents as due to the motion of electrons. In Art. 551 of his Treatise [1], he wrote: It appears, therefore, that a system containing an electric current is a seat of energy of some kind; and since we can form no conception of an electric current except as a kinetic phenomenon, its energy must be kinetic energy, that is to say, the energy which a moving body has in virtue of its motion. We have already shewn that the electricity in the wire cannot be consider as the moving body in which we are to find the energy, for the energy of a moving body does not depend on anything external to itself, whereas the presence of other bodies near the current alters its energy. We are therefore led to enquire where there may not be some motion going on in the space outside the wire, which is not occupied by the electric current, but in which the electromagnetic effects of the current are manifested. .... What I now propose to do is to examine the consequences of the assumption that the phenomena of the electric current are those of a moving system, the motion being commu- nicated from one part of the system to another by forces, the nature and laws of which we do not yet even attempt to define, because we can eliminate those forces from the equations of motion by the method given by Lagrange for any connected system. .... I have chosen this method because I wish to shew that there are other ways of viewing the phenomena which appear to be more satisfactory, and at the same time more consistent ... than those which proceed on the hypothesis of direct action at a distance. It appears that although Maxwell did not make a calculation like that of eq. (8) he correctly understood that the (ordinary) kinetic energy of “electricity” in the wire of an inductor can account for only a small part of the “magnetic” energy of that inductor. This 2