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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.

Contents 1 Maxwell’s equations 1.1 Gauss’s Law (electric) 1.2 Gauss’s Law (magnetic) 1.3 Faraday’s Law 1.4 Ampere’s Law 2 Terms Defined 3 Electromagnetic fields in matter (The Constitutive Relations) 3.1 In free space 3.2 For linear materials 3.3 Constitutive relations in Dissipative Materials 4 See Also 5 References 6 External Links

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.

Gauss’s Law (magnetic)

Flux of B through a closed surface equals zero.

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.

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

Terms Defined

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

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):

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

• Fdtd Main Index Page (with example code)
• Maxwell’s Equations
• The Yee Algorithm
• 2D UPML Algorithm and Theory
• The 2D FDTD Wave Algorithm and Theory
• The 2D FDTD Wave Hz Algorithm
• Analytical Solution: Infinite Magnetic Line-source
• Numerical Methods
• Spherical and Cartesian coordinates
• Vector Algebra
• Fundamental and Derived Units
• Glossary
• FDTD References
• FDTD Resources

References

• Balanis – Advanced Engineering Electromagnetics, 1989
• GriffithsED – Introduction to ElectroDynamics, (3rd edition) 1999
• Taflove2005 – Computational Electrodynamics, 2005

External Links

• Maxwell’s equations (wikipedia)