Maxwell’s Equations
The Maxwell’s equations define the divergence and curl of the electric and magnetic fields.
The Helmholtz theorem states that given appropriate boundary conditions a field is uniquely determined by its divergence and curl. (from Griffiths pp.52-53.) Therefore, the classical theory of electromagnetic fields is described by the Maxwell’s equations.
Maxwell’s equations
From Balinis pp. 2-3, 6, 104; (In units: MKS, SI) and Feynman vol. II Table 18.1 “Classical Physics” Also, Taflove2005 pp.51-54 (emphasis on Maxwell’s equations for FDTD) and Griffiths pp. 175, 269, 326-327, 330; And see Griffiths page 330 for Maxwell’s equations for electromagnetic fields in matter
Gauss’s Law (electric)
Flux of D through a closed surface equals the charge inside.
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Eqn:1.1 |
Gauss’s Law (magnetic)
Flux of B through a closed surface equals zero.
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Eqn:1.2 |
Faraday’s Law
Line integral of E around a loop plus the time rate of change of the flux of B through the loop equals zero.
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Eqn:1.3 |
Ampere’s Law
Line integral of H around a loop minus the time rate of change of the flux of D through the loop equals the current through the loop
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Eqn:1.4 |
Terms Defined
electric flux density (coulombs / square meter) |
electric charge density (coulombs / cubic meter) |
magnetic flux density (webers / square meter) |
magnetic charge density (webers / cubic meter), (usually 0, no magnetic monopoles, Griffiths p.327) |
electric field intensity (volts / meter) |
impressed (source) magnetic current density (volts / square meter), (usually 0) |
magnetic field intensity (amperes / meter) |
electric current density (amperes / square meter) |
conduction electric current density (amperes / square meter) |
impressed (source) electric current density (amperes / square meter) |
electric permittivity (farads/meter) |
electric permittivity (free space) (farads/meter) |
magnetic permeability (Henrys/meter) |
magnetic permeability (free space) (Henrys/meter) |
Electromagnetic fields in matter (The Constitutive Relations)
The “Constitutive relations” are the equations that define the relationships between B, H, D, E. The Constitutive Parameters are: electricPermittivity, magneticPermeability and electricConductivity. (Balanis page 7)
In free space
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Eqn:1.5 |
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Eqn:1.6 |
For linear materials
If Material is linear, isotropic, non-dispersive (i.e. materials having field-independent, direction-independent, and frequency-independent electric and magnetic properties).
From Griffiths pp.179-180, 274, 275, 330; Balinis p.8 (For non-linear dispersive materials see Griffiths page 401; Balanis pp. 76,77):
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Eqn:1.7 |
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Eqn:1.8 |
Constitutive relations in Dissipative Materials
From Taflove pp. 52-53, GriffithsED pp. 285,393, Balanis page 104.; Taflove2000 pp.68-70 If Material can dissipate electromagnetic fields (due to conversion to heat energy) JmResistive, H, JeConduction, E, are related by:
TBD
See Also
References
- Balanis – Advanced Engineering Electromagnetics, 1989
- GriffithsED – Introduction to ElectroDynamics, (3rd edition) 1999
- Taflove2005 – Computational Electrodynamics, 2005
External Links