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Electrically induced magnetic fields; a consistent approach
1.See R. A. Serway and R. J. Biechner, Physics for Scientists and Engineers, 5th ed. (Harcourt College Publishers, Orlando, FL, 2000), pp. 993–994 as an example of how many introductory texts perform this calculation.
2.See D. J. Griffiths, Introduction to Electrodynamics, 3rd ed. (Prentice Hall, Upper Saddle River, NJ, 1999), p. 308. Griffiths explains the quasistatic approximation.
2.Also see R. L. Reese, University Physics (Brooks/Cole Publishing Co., Pacific Grove, CA, 2000), pp. 895–952 for a consistent approach; Reese introduces Faraday’s law concurrently with Maxwell’s equations and electromagnetic radiation.
2.Also see R. Wolfson and J. M. Pasachoff, Physics, 2nd ed. (Harper Collins College Publishers, New York, 1995), p. 799 for an acknowledgement of the approximation.
3.See Ref. 1, pp. 979–1013. The fact that changing electric fields induce magnetic fields is only briefly mentioned on pp. 1000; the approximation is not recognized anywhere else in the chapter.
3.Also see P. A. Tipler, Physics, 4th ed. (W. H. Freeman and Co., New York, 1999), pp. 927–958;
3.V. D. Barger and M. G. Olsson, Classical Electricity and Magnetism (Allyn and Bacon, Inc., Newton, MA, 1987), pp. 240–280. The approximation is not mentioned in either book in the chapter covering Faraday’s law.
4.T. A. Abbott and D. J. Griffiths, “Acceleration without radiation,” Am. J. Phys. 53 (12), 1203–1204 (1985),
4.and C. Thevenet, “Electromagnetic radiation from sinusoidal currents,” Am. J. Phys. 67, 120–124 (1999).
5.See also integral derivation of differential equations in the Appendix. Introductory texts generally rely on integrals.
6.To be more rigorous, integrate Eq. (6) from to and take the limit as ε→0. Since the electric field is continuous, the last term is zero, and you are left with the boundary condition.
7.Equations (3) and (4) are also consistent with Maxwell’s other two laws. Namely, ∇⋅B=0 and where, for our problem, the charge density is 0.
8.Equations (8) and (9) are actually the Green’s function for Eq. (7) with an arbitrary source term.
9.I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, and Products (Academic, Orlando, FL, 1980), p. 967.
10.Reference 8, p. 969.
11.Reference 8, p. 961. Note the asymptotic dependence. This is consistent with the power radiated, which is proportional to To satisfy energy conservation, a point source, which has spherical symmetry, should have the power radiated per unit area as for an infinite solenoid with cylindrical symmetry, the power radiated per unit area should fall off as
12.This makes sense in two other ways: (a) As solutions to the differential equations, we could have chosen the Hankel functions instead of the Bessel and Neumann functions. Now, since and are equivalent to and respectively, they correspond to incoming and outgoing waves (for very large and . Thus we would need to get rid of if we only want outgoing waves. (b) In light of the vector potential, when the integral is performed in Ref. 5, the retarded Green’s function (for outgoing waves) gives Had the advanced Green’s function been chosen, which corresponds to incoming waves, would have been the solution.
13.The calculation of the self-capacitance of a finite solenoid is not the main focus of this paper. The interested reader can see R. G. Medhurst, Resistance and Self-Capacitance of Single-Layer Solenoids, Wireless Engineer, February 1947, pp. 35–43 and March 1947, pp. 80–92.
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