The electromagnetic linear momentum and the energy balance in an infinite solenoid with a time-dependant current are examined. We show that the electromagnetic linear momentum density and its associated force density are balanced by the hidden momentum density and its associated hidden force density respectively. We also show that exactly half the energy delivered by the power supply appears as stored magnetic energy inside the solenoid. The other half is lost against the induced electromotive force that appears in the windings of the solenoid during the time through which the current is building up towards its final value. This energy loss, which is found in other analogue situations, is necessary to transfer the system from an initial non-equilibrium state to a final equilibrium one.
KeywordsElectromagnetic Linear MomentumEnergy BalanceHidden Momentum
J. Sod-Hoffs and V. S. Manko, “The Poynting vector of a charged magnetic dipole: two limiting cases,” Journal of Physics: Conference Series, Vol. 91, No. 1, pp. 2011–2015, 2007.
J. L. Jimenez, I. Campos, and N. Aquino, “Exact electromagnetic fields produced by a finite wire with constant current,” European Journal of Physics, Vol. 29, No. 1, pp. 163–175, January 2008.
X. Zangcheng, G. Changlin, Z. Zongyan, T. Fukamachi, and R. Negishi, “The relationship between the Poynting vector and the dispersion Surface in the Bragg case,” Journal of Physics: Condensed Matter, Vol. 9, No. 18, pp. 75–78, 1997.
N. Morton, “An introduction to the Poynting vector,” Physics Education, Vol. 14, No. 5, pp. 301–304, 1979.
V. S. Manko, E. D. Rodchenko, B. I. Sadovnikov, and J. Sodd-Hoffs, “The Poynting vector of axistationary electrovac space times reexamined,” Classical and Quantum Gravity, Vol. 23, No. 27, pp. 5385–5395, 2006.
J.-Y. Ji, C.-W. Lee, J. Noh, and W. Jhe, “Quantum electromagnetic fields in the presence of a dielectric microsphere,” Journal of Physics B: Atomic, Molecular and Optical Physics, Vol. 33, No. 21, pp. 4821–4831, 2000.
P. Seba, U. Kuhl, M. Barth, and H. St?ckmann, “Experimental verification of the topologically induced vortices inside a billiard,” Journal of Physics A: Mathematical and General, Vol. 32, No. 47, pp. 8225–8230, 1999.
N. R. Sadykov, “Evolution equation of the Korteweg – de Vries type for the Poynting vector amplitude,” Quantum Electronics, Vol. 27, No. 2, pp. 185–187, 1997.
D. F. Nelson, “Generalizing the Poynting vector,” Physical Review Letters, Vol. 76, No. 25, pp. 4713–4716, June 1996.
A. L. Kholmetskii and T. Yarman, “Apparent paradoxes in classical electrodynamics: A fluid medium in an electromagnetic field,” European Journal of Physics, Vol. 29, No. 6, pp. 1127–1134, 2008.
A. Gsponer, “On the electromagnetic momentum of static charge and steady current distributions,” European Journal of Physics, Vol. 28, No. 5, pp. 1021–1042, 2007.
J. M. Aguirregabiria, A. Hernández, and M. Rivas, “Linear momentum density in quasi static electromagnetic systems,” European Journal of Physics, Vol. 25, No. 4, pp. 555–567, 2004.
X. S. Zhu and W. C. Henneberger, “Some observations on the dynamics of the Aharonov-Bohm effect,” Journal of Physics A: Mathematical and General, Vol. 23, No. 17, pp. 3983–3990, 1990.
D. G. Lahoz and G. M. Graham, “Experimental decision on the electromagnetic momentum expression for magnetic media,” Journal of Physics A: Mathematical and General, Vol. 15, No. 1, pp. 303–318, 1982.
S. M. Kim and G. Gbur, “Momentum conservation in partially coherent wave fields,” Physical Review A, Vol. 79, No. 3, pp. 3844–3849, March 2009.
M. Mansuripur and A. R. Zakharian, “Maxwell’s macroscopic equations, the energy-momentum postulates, and the Lorentz law of force,” Physical Review E, Vol. 79, No. 2, pp. 6608–6617, 2009.
G. N. Gaidukov and A. A. Abramov, “An interpretation of the energy conservation law for a point charge moving in a uniform electric field,” Physics-Uspekhi, Vol. 51, No. 2, pp. 163–166, 2008.
A. B. Pippard, “Change of energy of photons passing through rotating anisotropic elements,” European Journal of Physics, Vol. 15, No. 2, pp. 79–80, 1994.
A. L. Kholmetskii and T. Yarman, “Energy flow in a bound electromagnetic field: Resolution of apparent paradoxes,” European Journal of Physics, Vol. 29, No. 6, pp. 1135–1146, 2008.
M. F. Bishop and A. A. Maradudin, “Energy flow in a semi-infinite spatially dispersive absorbing dielectric,” Physical Review B, Vol. 14, pp. 3384–3393, 1976.
F. Richter, M. Forian, and K. Henneberger, “Poynting’s theorem and energy conservation in the propagation of light in bounded media,” Europhysics Letters, Vol. 81, No. 6, pp. 7005–7009, 2008.
R. A. Powell, “Two-capacitor problem: A more realistic view,” American Journal of Physics, Vol. 47, No. 5, pp. 460–462, May 1979.
S. M. AL-Jaber and S. K. Salih, “Energy consideration in the two-capacitor problem,” European Journal of Physics, Vol. 21, No. 4, pp. 341–345, 2000.
A. M. Sommariva, “Solving the two capacitor paradox through a new asymptotic approach,” IEE Proceedings of Circuits Devices Systems, Vol. 150, No. 3, pp. 227–231, 2003.
D. P. Korfiatis, “A new approach to the two-capacitor paradox,” WSEAS Transactions on Circuits Systems, Vol. 6, pp. 76–79, 2007.
A. M. Abu-Labdeh and S. M. AL-Jaber, “Energy consideration from non-equilibrium to equilibrium state in the process of charging a capacitor,” Journal of Electrostatics, Vol. 66, No. 3–4, pp. 190–192, 2008.
K. Lee, “The two-capacitor problem revisited: a mechanical harmonic oscillator model approach,” European Journal of Physics, Vol. 30, No. 1, pp. 69–74, 2009.
D. Babson, P. Reynolds, R. Bjorkquist, and D. J. Griffiths, “Hidden momentum, field momentum, and electromagnetic impulse,” American Journal of Physics, Vol. 77, No. 9, pp. 826–833, 2009.
C. Cuvaj, “On conservation of energy in electric circuits,” American Journal of Physics, Vol. 36, No. 10, pp. 909–910, 1968.
T. B. Boykin, D. Hite, and N. Singh, “The two capacitor problem with radiation,” American Journal of Physics, Vol. 70, No. 4, pp. 415–420, 2002.