Abstract

We derive an alternative formulation of the field equations for macroscopic electromagnetic fields in a linear magneto-dielectric medium as an identity of the Maxwell–Minkowski equations, complementing a variety of other representations including the Ampère, Chu, Lorentz, and Minkowski formulations of continuum electrodynamics. In the new formulation of the macroscopic field equations, the material properties are carried as a renormalization of the temporal and spatial coordinates instead of as independent material constants. The new representation of the field equations raises some interesting physical issues with relativity and boundary conditions.

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References

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  1. T. K. Simpson, Maxwell on the Electromagnetic Field: A Guided Study (Rutgers Univ. Press, 1997).
  2. N. Wheeler, “Theories of maxwellian design,” http://www.reed.edu/physics/faculty/wheeler/documents/Electrodynamics/Miscellaneous-Essays/Maxwellian-Theories.pdf .
  3. J. D. Jackson, Classical Electrodynamics (John Wiley, 1975).
  4. J. B. Marion, Classical Dynamics of Particles and Systems (Academic, 1970).
  5. D. J. Griffiths, Introduction to Electrodynamics (Prentice-Hall, 1981).
  6. A. Zangwill, Modern Electrodynamics (Cambridge Univ. Press, 2011).
  7. J. P Penfield and H. A. Haus, Electrodynamics of Moving Media (MIT Press, 1967).
  8. B. A. Kemp, “Resolution of the abraham–minkowski debate: Implications for the electromagnetic wave theory of light in matter,” J. Appl. Phys. 109(11), 111101 (2011).
    [Crossref]
  9. B. A. Kemp, “Macroscopic theory of optical momentum,” Prog. Opt. 60, 437–488 (2015).
    [Crossref]
  10. M. E. Crenshaw, “Application of axiomatic formal theory to the abraham–minkowski controversy,” http://arxiv.org/pdf/1502.05997.pdf (2015).
  11. M. E. Crenshaw, “Representation independent boundary conditions for a piecewise-homogenous linear magneto-dielectric medium,” AIP Adv. 9(7), 075102 (2019).
    [Crossref]
  12. M. E. Crenshaw, “Reconciliation of the rosen and laue theories of special relativity in a linear dielectric medium,” Am. J. Phys. 87(4), 296–300 (2019).
    [Crossref]
  13. G. Weinstein, “Albert einstein and the fizeau 1851 water tube experiment,” http://arxiv.org/pdf/1204.3390.pdf (2012).
  14. M. Laue, “Die mitfuhrung des lichtes durch bewegte kopernach dem relativitatsprinzip,” Ann. Phys. 328(10), 989–990 (1907).
    [Crossref]
  15. N. Rosen, “Special theories of relativity,” Am. J. Phys. 20(3), 161–164 (1952).
    [Crossref]
  16. H. Fizeau, “On the hypotheses relating to the luminous aether, and an experiment which appears to demonstrate that the motion of bodies alters the velocity with which light propagates itself in their interior,” Philos. Mag. 2(14), 568–573 (1851).
    [Crossref]

2019 (2)

M. E. Crenshaw, “Representation independent boundary conditions for a piecewise-homogenous linear magneto-dielectric medium,” AIP Adv. 9(7), 075102 (2019).
[Crossref]

M. E. Crenshaw, “Reconciliation of the rosen and laue theories of special relativity in a linear dielectric medium,” Am. J. Phys. 87(4), 296–300 (2019).
[Crossref]

2015 (1)

B. A. Kemp, “Macroscopic theory of optical momentum,” Prog. Opt. 60, 437–488 (2015).
[Crossref]

2011 (1)

B. A. Kemp, “Resolution of the abraham–minkowski debate: Implications for the electromagnetic wave theory of light in matter,” J. Appl. Phys. 109(11), 111101 (2011).
[Crossref]

1952 (1)

N. Rosen, “Special theories of relativity,” Am. J. Phys. 20(3), 161–164 (1952).
[Crossref]

1907 (1)

M. Laue, “Die mitfuhrung des lichtes durch bewegte kopernach dem relativitatsprinzip,” Ann. Phys. 328(10), 989–990 (1907).
[Crossref]

1851 (1)

H. Fizeau, “On the hypotheses relating to the luminous aether, and an experiment which appears to demonstrate that the motion of bodies alters the velocity with which light propagates itself in their interior,” Philos. Mag. 2(14), 568–573 (1851).
[Crossref]

Crenshaw, M. E.

M. E. Crenshaw, “Reconciliation of the rosen and laue theories of special relativity in a linear dielectric medium,” Am. J. Phys. 87(4), 296–300 (2019).
[Crossref]

M. E. Crenshaw, “Representation independent boundary conditions for a piecewise-homogenous linear magneto-dielectric medium,” AIP Adv. 9(7), 075102 (2019).
[Crossref]

M. E. Crenshaw, “Application of axiomatic formal theory to the abraham–minkowski controversy,” http://arxiv.org/pdf/1502.05997.pdf (2015).

Fizeau, H.

H. Fizeau, “On the hypotheses relating to the luminous aether, and an experiment which appears to demonstrate that the motion of bodies alters the velocity with which light propagates itself in their interior,” Philos. Mag. 2(14), 568–573 (1851).
[Crossref]

Griffiths, D. J.

D. J. Griffiths, Introduction to Electrodynamics (Prentice-Hall, 1981).

Haus, H. A.

J. P Penfield and H. A. Haus, Electrodynamics of Moving Media (MIT Press, 1967).

Jackson, J. D.

J. D. Jackson, Classical Electrodynamics (John Wiley, 1975).

Kemp, B. A.

B. A. Kemp, “Macroscopic theory of optical momentum,” Prog. Opt. 60, 437–488 (2015).
[Crossref]

B. A. Kemp, “Resolution of the abraham–minkowski debate: Implications for the electromagnetic wave theory of light in matter,” J. Appl. Phys. 109(11), 111101 (2011).
[Crossref]

Laue, M.

M. Laue, “Die mitfuhrung des lichtes durch bewegte kopernach dem relativitatsprinzip,” Ann. Phys. 328(10), 989–990 (1907).
[Crossref]

Marion, J. B.

J. B. Marion, Classical Dynamics of Particles and Systems (Academic, 1970).

Penfield, J. P

J. P Penfield and H. A. Haus, Electrodynamics of Moving Media (MIT Press, 1967).

Rosen, N.

N. Rosen, “Special theories of relativity,” Am. J. Phys. 20(3), 161–164 (1952).
[Crossref]

Simpson, T. K.

T. K. Simpson, Maxwell on the Electromagnetic Field: A Guided Study (Rutgers Univ. Press, 1997).

Weinstein, G.

G. Weinstein, “Albert einstein and the fizeau 1851 water tube experiment,” http://arxiv.org/pdf/1204.3390.pdf (2012).

Wheeler, N.

N. Wheeler, “Theories of maxwellian design,” http://www.reed.edu/physics/faculty/wheeler/documents/Electrodynamics/Miscellaneous-Essays/Maxwellian-Theories.pdf .

Zangwill, A.

A. Zangwill, Modern Electrodynamics (Cambridge Univ. Press, 2011).

AIP Adv. (1)

M. E. Crenshaw, “Representation independent boundary conditions for a piecewise-homogenous linear magneto-dielectric medium,” AIP Adv. 9(7), 075102 (2019).
[Crossref]

Am. J. Phys. (2)

M. E. Crenshaw, “Reconciliation of the rosen and laue theories of special relativity in a linear dielectric medium,” Am. J. Phys. 87(4), 296–300 (2019).
[Crossref]

N. Rosen, “Special theories of relativity,” Am. J. Phys. 20(3), 161–164 (1952).
[Crossref]

Ann. Phys. (1)

M. Laue, “Die mitfuhrung des lichtes durch bewegte kopernach dem relativitatsprinzip,” Ann. Phys. 328(10), 989–990 (1907).
[Crossref]

J. Appl. Phys. (1)

B. A. Kemp, “Resolution of the abraham–minkowski debate: Implications for the electromagnetic wave theory of light in matter,” J. Appl. Phys. 109(11), 111101 (2011).
[Crossref]

Philos. Mag. (1)

H. Fizeau, “On the hypotheses relating to the luminous aether, and an experiment which appears to demonstrate that the motion of bodies alters the velocity with which light propagates itself in their interior,” Philos. Mag. 2(14), 568–573 (1851).
[Crossref]

Prog. Opt. (1)

B. A. Kemp, “Macroscopic theory of optical momentum,” Prog. Opt. 60, 437–488 (2015).
[Crossref]

Other (9)

M. E. Crenshaw, “Application of axiomatic formal theory to the abraham–minkowski controversy,” http://arxiv.org/pdf/1502.05997.pdf (2015).

G. Weinstein, “Albert einstein and the fizeau 1851 water tube experiment,” http://arxiv.org/pdf/1204.3390.pdf (2012).

T. K. Simpson, Maxwell on the Electromagnetic Field: A Guided Study (Rutgers Univ. Press, 1997).

N. Wheeler, “Theories of maxwellian design,” http://www.reed.edu/physics/faculty/wheeler/documents/Electrodynamics/Miscellaneous-Essays/Maxwellian-Theories.pdf .

J. D. Jackson, Classical Electrodynamics (John Wiley, 1975).

J. B. Marion, Classical Dynamics of Particles and Systems (Academic, 1970).

D. J. Griffiths, Introduction to Electrodynamics (Prentice-Hall, 1981).

A. Zangwill, Modern Electrodynamics (Cambridge Univ. Press, 2011).

J. P Penfield and H. A. Haus, Electrodynamics of Moving Media (MIT Press, 1967).

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Equations (53)

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×H1cDt=0
B=0
×E+1cBt=0
D=0,
×Hc1cEct=1cPct
Hc=Mc
×Ec+1cHct=1cMct
Ec=Pc,
D=ε(r)E
B=μ(r)H
×H1c(εE)t=0
(μH)=0
×E+1c(μH)t=0
(εE)=0
ne(r)=ε
nm(r)=μ
nm×Hnmc(neneE)t=0
(nmnmH)=0
ne×E+nec(nmnmH)t=0
(neneE)=0.
nm×(nmH)+nec(neE)t=nmnmnm×(nmH)
nm(nmH)=nmnmnm(nmH)
nm×(neE)nec(nmH)t=nenmne×(neE)
nm(neE)=nenmne(neE).
Π=ne(r)E
β=nm(r)H
nm×β+necΠt=nmnmnm×β
nmβ=nmnmnmβ
nm×Πnecβt=nenmne×Π
nmΠ=nenmneΠ.
x¯0=x0ne=ctne
nm×β+Πx¯0=1nmnmnm×β
nmβ=1nmnmnmβ
nm×Πβx¯0=1nmnene×Π
nmΠ=1nmneneΠ.
¯=(1nmx,1nmy,1nmz)
¯×β+Πx¯0=¯nmnm×β
¯β=¯nmnmβ
¯×Πβx¯0=¯nene×Π
¯Π=¯neneΠ.
x¯=nmx
y¯=nmy
z¯=nmz
¯×β+Πx¯0=0
¯β=0
¯×Πβx¯0=0
¯Π=0,
¯=(x¯,y¯,z¯)
¯×(¯×A)+x¯0(Ax¯0)=0
β=¯×A
Π=Ax¯0
×(×A)+n2c22At2=0.
γd=11n2v2/c2