Charge density on a conducting needle

American Journal of Physics

Volumn 64, Issue 6, 1996, Pages 706-714

  Griffiths, David J ^a   Li, Ye ^a  

^a REED COLLEGE   (United States)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 0030556146     PISSN: 00029505     EISSN: None     Source Type: Journal    
DOI: 10.1119/1.18236     Document Type: Article

References (22)

1. This is already a somewhat surprising conclusion. The charge density on a thin conducting disk, for example, can be obtained by a variety of limiting procedures, all of which yield the same result [for a particularly ingenious method, see R. Friedberg, "The electrostatics and magnetostatics of a conducting disk," Am. J. Phys. 61, 1084-1096, (1993)]. Evidently, there is some deep pathology in the reduction to a one-dimensional object that does not infect the reduction from three to two.
2. William R. Smythe, Static and Dynamic Electricity, 3rd ed., revised printing (Hemisphere, New York, 1989), Sec. 5.02.
3. This was noted by Max Abraham and Richard Becker, Classical Theory of Electricity and Magnetism, 2nd ed. (Hafner, New York, 1949); our derivation is based on Jeffrey Gima's unpublished Reed College senior thesis (1993).
4. W. R. Smythe, "Charged Right Circular Cylinder," J. Appl. Phys. 27, 917-920 (1956). See also Ref. 2, Sec. 5.39.
5. T. T. Taylor, "Electric Polarizability of a Short Right Circular Conducting Cylinder," J. Res. NBS B 64, 135-143 (1960). Taylor is concerned with the problem of an uncharged cylinder in an external electric field, but it is straightforward to adapt his method to the case of a charged cylinder with no external field. See also T. T. Taylor, "Magnetic Polarizability of a Short Right Circular Conducting Cylinder," J. Res. NBS B 64, 199-210 (1960).
6. T. T. Taylor, "Electric Polarizability of a Short Right Circular Conducting Cylinder," J. Res. NBS B 64, 135-143 (1960). Taylor is concerned with the problem of an uncharged cylinder in an external electric field, but it is straightforward to adapt his method to the case of a charged cylinder with no external field. See also T. T. Taylor, "Magnetic Polarizability of a Short Right Circular Conducting Cylinder," J. Res. NBS B 64, 199-210 (1960).
7. W. R. Smythe, "Charged Right Circular Cylinder," J. Appl. Phys. 33, 2966-2967 (1962); P. C. Waterman, "Matrix Methods in Potential Theory and Electromagnetic Scattering," J. Appl. Phys. 50, 4550-4566 (1979); B. D. Popović, M. B. Dragović, and A. R. Djordjević, Analysis and Synthesis of Wire Antennas (Research Studies Press, Chichester, 1982); P. K. Wang, "Calculation of Electrostatic Fields Surrounding Finite Circular Cylindrical Conductors," IEEE Trans. Antennas Propag. 32, 956-962 (1984); A. R. Djordjević, "Comments on 'Calculation of Electrostatic Fields Surrounding Finite Circular Cylindrical Conductors,'" IEEE Trans. Antennas Propag. 33, 683-684 (1985); P. K. Wang, C. H. Chuang, and N. L. Miller, "Electrostatic, Thermal and Vapor Density Fields Surrounding Stationary Columnar Ice Crystals," J. Atmos. Sci. 42, 2371-2379 (1985).
8. W. R. Smythe, "Charged Right Circular Cylinder," J. Appl. Phys. 33, 2966-2967 (1962); P. C. Waterman, "Matrix Methods in Potential Theory and Electromagnetic Scattering," J. Appl. Phys. 50, 4550-4566 (1979); B. D. Popović, M. B. Dragović, and A. R. Djordjević, Analysis and Synthesis of Wire Antennas (Research Studies Press, Chichester, 1982); P. K. Wang, "Calculation of Electrostatic Fields Surrounding Finite Circular Cylindrical Conductors," IEEE Trans. Antennas Propag. 32, 956-962 (1984); A. R. Djordjević, "Comments on 'Calculation of Electrostatic Fields Surrounding Finite Circular Cylindrical Conductors,'" IEEE Trans. Antennas Propag. 33, 683-684 (1985); P. K. Wang, C. H. Chuang, and N. L. Miller, "Electrostatic, Thermal and Vapor Density Fields Surrounding Stationary Columnar Ice Crystals," J. Atmos. Sci. 42, 2371-2379 (1985).
9. W. R. Smythe, "Charged Right Circular Cylinder," J. Appl. Phys. 33, 2966-2967 (1962); P. C. Waterman, "Matrix Methods in Potential Theory and Electromagnetic Scattering," J. Appl. Phys. 50, 4550-4566 (1979); B. D. Popović, M. B. Dragović, and A. R. Djordjević, Analysis and Synthesis of Wire Antennas (Research Studies Press, Chichester, 1982); P. K. Wang, "Calculation of Electrostatic Fields Surrounding Finite Circular Cylindrical Conductors," IEEE Trans. Antennas Propag. 32, 956-962 (1984); A. R. Djordjević, "Comments on 'Calculation of Electrostatic Fields Surrounding Finite Circular Cylindrical Conductors,'" IEEE Trans. Antennas Propag. 33, 683-684 (1985); P. K. Wang, C. H. Chuang, and N. L. Miller, "Electrostatic, Thermal and Vapor Density Fields Surrounding Stationary Columnar Ice Crystals," J. Atmos. Sci. 42, 2371-2379 (1985).
10. W. R. Smythe, "Charged Right Circular Cylinder," J. Appl. Phys. 33, 2966-2967 (1962); P. C. Waterman, "Matrix Methods in Potential Theory and Electromagnetic Scattering," J. Appl. Phys. 50, 4550-4566 (1979); B. D. Popović, M. B. Dragović, and A. R. Djordjević, Analysis and Synthesis of Wire Antennas (Research Studies Press, Chichester, 1982); P. K. Wang, "Calculation of Electrostatic Fields Surrounding Finite Circular Cylindrical Conductors," IEEE Trans. Antennas Propag. 32, 956-962 (1984); A. R. Djordjević, "Comments on 'Calculation of Electrostatic Fields Surrounding Finite Circular Cylindrical Conductors,'" IEEE Trans. Antennas Propag. 33, 683-684 (1985); P. K. Wang, C. H. Chuang, and N. L. Miller, "Electrostatic, Thermal and Vapor Density Fields Surrounding Stationary Columnar Ice Crystals," J. Atmos. Sci. 42, 2371-2379 (1985).
11. W. R. Smythe, "Charged Right Circular Cylinder," J. Appl. Phys. 33, 2966-2967 (1962); P. C. Waterman, "Matrix Methods in Potential Theory and Electromagnetic Scattering," J. Appl. Phys. 50, 4550-4566 (1979); B. D. Popović, M. B. Dragović, and A. R. Djordjević, Analysis and Synthesis of Wire Antennas (Research Studies Press, Chichester, 1982); P. K. Wang, "Calculation of Electrostatic Fields Surrounding Finite Circular Cylindrical Conductors," IEEE Trans. Antennas Propag. 32, 956-962 (1984); A. R. Djordjević, "Comments on 'Calculation of Electrostatic Fields Surrounding Finite Circular Cylindrical Conductors,'" IEEE Trans. Antennas Propag. 33, 683-684 (1985); P. K. Wang, C. H. Chuang, and N. L. Miller, "Electrostatic, Thermal and Vapor Density Fields Surrounding Stationary Columnar Ice Crystals," J. Atmos. Sci. 42, 2371-2379 (1985).
12. W. R. Smythe, "Charged Right Circular Cylinder," J. Appl. Phys. 33, 2966-2967 (1962); P. C. Waterman, "Matrix Methods in Potential Theory and Electromagnetic Scattering," J. Appl. Phys. 50, 4550-4566 (1979); B. D. Popović, M. B. Dragović, and A. R. Djordjević, Analysis and Synthesis of Wire Antennas (Research Studies Press, Chichester, 1982); P. K. Wang, "Calculation of Electrostatic Fields Surrounding Finite Circular Cylindrical Conductors," IEEE Trans. Antennas Propag. 32, 956-962 (1984); A. R. Djordjević, "Comments on 'Calculation of Electrostatic Fields Surrounding Finite Circular Cylindrical Conductors,'" IEEE Trans. Antennas Propag. 33, 683-684 (1985); P. K. Wang, C. H. Chuang, and N. L. Miller, "Electrostatic, Thermal and Vapor Density Fields Surrounding Stationary Columnar Ice Crystals," J. Atmos. Sci. 42, 2371-2379 (1985).
13. W. R. Smythe, Ref. 2, p. 209.
14. A. R. Djordjević, personal communication.
15. Material in this section is based on Ye Li's unpublished Reed College senior thesis (1994).
16. The electrical force on the outermost charges (at ±a) is not zero, of course, because they are subject to the extra constraining force holding the charges on the wire. By the way, one can (equivalently) calculate the total potential energy of the configuration, and minimize it to determine the positions of the charges.
17. J. B. Ross, "Plotting the charge distribution of a closed-loop conducting wire using a microcomputer," Am. J. Phys. 55, 948-950 (1987). Ross used the potential at each charge (due to all the others), but for open segments it is better to use the midpoints, since the end charges are subject to nonelectrostatic forces.
18. Smythe, Ref. 2, Sec. 4.22.
19. 2) for all b - but (unlike the finite ellipsoid) it is not independent of x.]
20. It is easy to check that the total linear charge density [∫σ(x)dx] is Λ.
21. We have pushed the fixed-position bead model up to n = 300 with barely detectable changes in the curve; the best fit of the form (3.7) occurs for A = 0.384049, B = 0.088295.
22. This method cannot be applied to the Coulomb problem, of course, because of the nasty singularity in the integrand.