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The Development of Maxwell's Equations
for Electric and Magnetic Fields

The Equations of Electricity and Magnetism Before Maxwell

The early work was in terms of charges and currents. During that time the laws concerning the forces between charges and between magnetic poles were developed. Then it was discovered by Ampére that currents in wires produce magnetic effects. This was around the beginning of the 19th century.

Later scientists began to thing of electrical and magnetic phenomena in terms of vector fields. Two vector fields were defined for an electric field, E and D, and two for a magnetic field, B and H. The relationships between them are

D = εE
B = μH

where ε is the dielectric constant of the material in which the electric field is located. Likewise μ is the permeability of the material in which the magnetic field is located.

The laws concerning the fields could be expressed in terms of the vector calculus operations of divergence, gradient and curl. Coulomb's Law is expressed as

Div(D) = ∇·D = 4πρ

where ρ is the charge density. (The 4π coefficient arises from the units used.)

There is a corresponding relationship for the magnetic field but since magnetic monopoles do not exist their density is always zero. Hence

div(B) = ∇·B = 0

In 1826 the French scientist André-Marie Ampére established that under steady state conditions the magnetic field intensity H is related to the current density J by the equation

curl(H) = ∇×H = (4π/c)J

where c is a parameter that turned out to be the speed of light.

It is to be emphasized that this is under steady state conditions.

It was Michael Faraday in 1831 who published the results of his experiments on field which changed over time. His law can be expressed as

curl(E) - ∇×E = −(1/c)(∂B∂t)

There was another relation that turned out to be critical to Maxwell's analysis. This is the equation for the continuity of charge

∇·J + (∂ρ∂t)= 0

From Coulomb's Law one finds that

(∂ρ∂t) = (1/4π)(∂∇·D∂t) = ∇·((1/4&pi:(∂D∂t))

Thus the equation of charge continuity takes the form of

∇·J + (∂ρ∂t) = ∇·(J + (1/4π)(∂D∂t)) = 0

This suggested to Maxwell that when the electric field is varying over time the charge current density J must be augmented by the term (1/4π)(∂D∂t) so Ampére's Law takes the form

∇×H = (4π/c)J + (1/c)(∂D∂t)

With the modified set of equations governing electric and magnetic fields Maxwell was able to deduce that the fluctuation of the electrial and magnetic fields satisfy a wave equation. Furthermore the speed of propagation of those fluctuation could be computed from the dielectric constant and the permeability. When that speed was computed using the dielectric constant and permeability of empty space the value was essentially identical to the speed of light. Maxwell then asserted that light was an electromagnetic wave.

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