# 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.

 Eqn:1.1

### Gauss’s Law (magnetic)

Flux of B through a closed surface equals zero.

 Eqn:1.2

Line integral of E around a loop plus the time rate of change of the flux of B through the loop equals zero.

 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

 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)
 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

 Eqn:1.5
 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):

 Eqn:1.7
 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

## References

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