# 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 |

### 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.

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) |

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

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

## 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)