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.

7f0c08587f38596672fc365bc7b81a98.png Eqn:1.1

Gauss’s Law (magnetic)

Flux of B through a closed surface equals zero.

031543d8b5efcb1f16a9beef6e353ce5.png 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.

b2d1dee3ba4e5467ad8bac47543d95d9.png 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

a92ac89e61a5e8251b0c5c6a5b41fc46.png Eqn:1.4

Terms Defined

37dfc68a38f6b3c39d9fe864b7c767b3.pngelectric flux density (coulombs / square meter)
259aebdd9c3efb4cb2aea976b3ade961.pngelectric charge density (coulombs / cubic meter)
3a823fab05bdbe0964b96286bd26c91e.pngmagnetic flux density (webers / square meter)
1f26e73142104b4f4a010c89d607b114.pngmagnetic charge density (webers / cubic meter), (usually 0, no magnetic monopoles, Griffiths p.327)
0898d6ef8f17e37365058e0f2f7335a3.pngelectric field intensity (volts / meter)
dfb872c55defcaed8dc729a7067f88c6.pngimpressed (source) magnetic current density (volts / square meter), (usually 0)
8f14501007f91398110881febb19de68.pngmagnetic field intensity (amperes / meter)
0490c4d4c29960fe2e2300b5c0430f8c.pngelectric current density (amperes / square meter)
b7996e787add51f52628cff44fb0474b.pngconduction electric current density (amperes / square meter)
9764aafce45a4d7bb33d393e067ec942.pngimpressed (source) electric current density (amperes / square meter)
a1f21f33e4cc224c1aa90a478a5339d5.pngelectric permittivity (farads/meter)
7da3363c48472901d1e800bfa833ec71.pngelectric permittivity (free space) (farads/meter)
2b3a144c31b05e0a773b0b60dc13e991.pngmagnetic permeability (Henrys/meter)
6615a9659f2dddee5d03c90e101594c7.pngmagnetic 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

ffa63bd10ff55db475eaa5fbafbd4b74.png Eqn:1.5
b0fbb7488be9baa7b0b95889febf6020.png 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):

15b6647e1650c18763544fd92424b3e5.png Eqn:1.7
04cfa811025cc9972b834fa57caa7e28.png 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:


See Also


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

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