# A Dynamical Theory of the Electromagnetic Field

A Dynamical Theory of the Electromagnetic Field was the third of James Clerk Maxwell's papers concerned with electromagnetism. The theory was the first paper in which Maxwell's equations appeared. The concept of displacement current was introduced, so that it became possible to derive equations of electromagnetic wave.

 Contents

## The original equations

In the original paper, because compact notations based on vectors had not yet been introduced, Maxwell formulated the equations in terms of 20 equations in 20 unknowns, as described in the Maxwell's equations article. Maxwell also included several equations now considered auxiliary to the core of Maxwell's equations.

In modernized notation, the equations originally listed by Maxwell were:

1. [itex]\nabla \times \mathbf{H} = \mu_0 \mathbf{J}_{tot}[itex]
2. [itex]\mathbf{J}_{tot} = \mathbf{J} + \frac{\partial\mathbf{D}}{\partial t}[itex]
3. [itex]\nabla \cdot \mathbf{D} = \rho[itex]
4. [itex]\mu \mathbf{H} = \nabla \times \mathbf{A}[itex]
5. [itex]\mathbf{E} = -\nabla \phi - \frac{\partial\mathbf{A}}{\partial t}[itex]
6. [itex]\mathbf{E} = \frac{1}{\epsilon} \mathbf{D}[itex]
7. [itex]\mathbf{E} = \frac{1}{\sigma} \mathbf{J}[itex]
8. [itex]\nabla \cdot \mathbf{J} = -\frac{\partial\rho}{\partial t}[itex]

Note that Maxwell did not consider completely general materials; his initial formulation used linear, isotropic, nondispersive permittivity ε and permeability μ, although he also discussed the possibility of anisotropic materials.

Here, [itex]\mathbf{H}[itex] is the magnetic field, which Maxwell called the "magnetic intensity". [itex]\mathbf{J}[itex] is the electric current density (with [itex]\mathbf{J}_{tot}[itex] being the total current including displacement current). [itex]\mathbf{D}[itex] is the displacement field (called the "electric displacement" by Maxwell). [itex]\rho[itex] is the free charge density (called the "quantity of free electricity" by Maxwell). [itex]\mathbf{A}[itex] is the magnetic vector potential (called the "electromagnetic momentum" by Maxwell). [itex]\mathbf{E}[itex] is the electric field (called the "electromotive force" by Maxwell, not to be confused with the scalar quantity that is now called electromotive force). [itex]\phi[itex] is the electric potential (which Maxwell also called "electric potential"). [itex]\sigma[itex] is the electrical conductivity (Maxwell called the inverse of conductivity the "specific resistance", what is now called the resistivity).

Note also that Maxwell further includes a [itex]\mu \mathbf{v} \times \mathbf{H}[itex] term in his expression for the "electromotive force", which corresponds to the magnetic force per unit charge on a moving conductor with velocity [itex]\mathbf{v}[itex]. Due to this term, his expression was not strictly equal to the modern electric field except in the rest frame of the conductor.

## Quotes

This velocity is so nearly that of light, that it seems we have strong reason to conclude that light itself (including radiant heat, and other radiations if any) is an electromagnetic disturbance in the form of waves propagated through the electromagnetic field according to electromagnetic laws. Maxwell, Dynamical Theory of the Electromagnetic Field. 1865.
... we have strong reason to conclude that light itself -- including radiant heat, and other radiations if any -- is an electromagnetic disturbance in the form of waves propagated through the electromagnetic field according to electromagnetic laws. Maxwell, Dynamical Theory of the Electromagnetic Field. 1864.

• Maxwell, James Clerk, "A Dynamical Theory of the Electromagnetic Field", Philosophical Transactions of the Royal Society of London 155, 459-512 (1865). (This article accompanied a December 8, 1864 presentation by Maxwell to the Royal Society.)
• Maxwell, James Clerk, "A Dynamical Theory of the Electromagnetic Field", Vol. CLV, 1865.
• James C. Maxwell, Thomas F. Torrance, "A Dynamical Theory of the Electromagnetic Field (http://www.engineering-shop.com/A_Dynamical_Theory_of_the_Electromagnetic_Field_1579100155.html)". March, 1996. ISBN 1579100155
• Niven, W. D., "The Scientific Papers of James Clerk Maxwell", 2 vols. Dover, New York, 1952, Vol. 1.

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