Technical Article

# Maxwell’s Equations in Present Form

March 18, 2016 by Marie Christiano

## Maxwell's equations mathematically link electricity, magnetism and light. In 1864, there were 20 equations solving for 20 variables, now we have 4. A look at the equations through the years.

In 1864, there were 20 equations solving for 20 variables, now we have 4. A look at the equations through the years.

### Introduction

Michael Faraday noted in the 1830s that a compass needle moved when electrical current flowed through wires near it. With that observation, the sciences of Electricity and Magnetism started to be merged. Something was affecting objects 'at a distance' and researchers were looking for answers.

Eventually, the 'something' affecting the objects was considered to be a 'field', with lines of force that could affect objects through the air. James Clerk Maxwell was among the first to translate the forces Faraday first noticed into a mathematical form. The equations rooted in classical physics explained electromagnetic fields and provided the basis for radio, television, and our cellular phones.

### The Equations Today

Today we usually think of Maxwell's Equations as a set of four partial differential equations using vector notation. You may see them in differential or integral form as shown in Figure 1.

##### Figure 1. Maxwells Equations

Also referred to as Gauss's Law, Gauss's Law of Magnetism, Faraday's Law of Induction and Ampere's Law (with this latter having a 'Maxwellian Correction'.)

These use the divergence (∇●) and curl (∇x) operators of vector notation; given in terms of d/dt. The equations describe Faraday's concepts mathematically, giving how electrical and magnetic fields are generated, sustained, and affected by electric charges, moving current and magnets.

However, in 1864, when James Clerk Maxwell, a professor at King's College in London presented his groundbreaking theory to the Royal Society of London, there was no mention of Gauss, there were no vectors, and there were 20 equations clearly called out.

### A New Field

The paper 'A Dynamical Theory of the Electromagnetic Field' was published in the Philosophical Transaction of the Royal Society of London in1865. In it, Maxwell introduced the theory of an electromagnetic field, explained why he considered it dynamic, considering it matter in motion. He put forth what he refered to as the General Equations of the Electromagnetic Field. He also concluded that the properties seen in light make it a form of electromagnetic field, and included a theory for the propagation of light.

Relying on research from 1846 through 1864, Maxwell cited 21 researchers, his own early attempts to explain what was happening, as well as the experiments of the Committee of British Association on Standards of Electric Resistance in his paper. He even mentions research he became aware of between the time he submitted the paper in October of 1864 and the time he presented it to the society in December of that year. What Maxwell put forth was a unifying theory that brought all the research and experiments together and could explain what was being seen in the various research labs at the time.

Researchers and labs as they appear in Maxwell's 1864 paper:

• M. W. Weber
• C. Neumann
• Geissler
• Professor W. Thomson
• M. Verdet
• Mr. F. Jenkin
• Helmholtz
• Kohlrausch
• Felici
• Ampere
• Pouillet
• Daniell
• Knoblauch
• M. Faucault
• M. Gaugain
• M.Plateau
• Mr. C. Hockin
• Leyden
• Green
• Fizeau
•

Although two of the four Maxwell's Equations are commonly referred to as the work of Carl Gauss, note that Maxwell's 1864 paper does not mention Gauss. Gauss's Law (Gauss's flux theorem) deals with the distribution of electric charge and electric fields. Although formulated in 1835, Gauss did not publish his work until 1867, after Maxwell's paper was published.

In addition to the equations, Maxwell addressed the possibility of Light being a form of electromagnetic energy. A result of working through the equations, Maxwell found the velocity of electromagnetic waves, given in Figure 2,  appeared to be near the velocity of light. Maxwell recognized the number and proposed that light was a form of electromagnetic wave.

$$v = \frac{1}{\sqrt{\mu_{0} \epsilon_{0}}}$$

##### Figure 2.  The velocity of electromagnetic waves given in terms of electric charge and magnetic permeability.

In Part III, "General Equations of the Electromagnetic Field" Maxwell details the equations which require solving for 20 unknowns using 20 equations:

3 equations for electrical currents, 3 for electromotive force, 3 for electric elasticity, 3 for magnetic force, 3 for total currents, 3 for electrical resistance,
1 for electromagnetic momentum, and 1 for magnetic intensity.

For the 20 unknowns, the following symbols are used:

Electromagnetic momentum: $$F$$ $$G$$ $$H$$

Magnetic intensity: $$\alpha$$ $$\beta$$ $$\gamma$$

Electromotive force: $$P$$ $$Q$$ $$R$$

Current due to true conduction: $$p$$ $$q$$ $$r$$

Electric displacement: $$f$$ $$g$$ $$h$$

Total current: $$q'$$ $$q'$$ $$r'$$

Quantity of free electricity: $$e$$

Electric potential: $$\Psi$$

Maxwell set up 20 partial differential equations to solve for these 20 unknowns.

Remember the last time you had to solve 3 equations for 3 unknowns? Without the help of MATLab or other computer aided tools, it's an effort. In 1864, the computers we rely on were decades away. Twenty equations of any type were daunting. Also, during this time, there were many theories being proposed. Even in his paper, Maxwell mentioned a competing theory and explained why he couldn't agree with it. To really understand Maxwell's theory with its intense math was an effort. Consequently, for years, Maxwell's insights and equations made no impact.

Even Maxwell recognized the math was a hurdle for most people and the first attempt to simplify the 1864 equations was made by Maxwell himself. He left his position at Kings College and set to work on an in-depth treatise on his proposed Electromagnetic Field. In 1873, he published a two-volume work, "A Treatise on Electricity and Magnetism". In the Preface, Maxwell states it constitutes the "Science of Electromagnetism". He also mentioned Gauss prominently. The work reads like a collection of "what we know now at this point in time of the science of Electricity and Magnetism". Maxwell included all known facets of research. In Volume 2, Chapter IX, he included another "General Equations of the Electromagnetic Field", and in Art. 618 and 619 he mentioned quaternions.

Quaternions are an extension of complex numbers of the form:

$$H = a + bi + cj + dk$$;

where $$a$$, $$b$$, $$c$$, $$d$$ are real and $$i2 = j2 = k2= ijk = −1$$

Quaternions were invented by William Hamilton in 1843 and can be used to describe 3D rotations. Maxwell reformulated his equations using quaternions: Quaternion Expressions for Electromagnetic Quantities and Quaternion Equations of the Electromagnetic Field. Maxwell used what he referred to as Hamiltonian vectors and ended up with 11 vectors ( 33 symbols), 4 scalars and well as C for conductance; K for the dialectic inductive capacitance and μ for magnetic inductance capacity. And that was how the math was left for a long time. Maxwell passed away in 1879.

### After Maxwell

In 1884, Oliver Heaviside studied Maxwell's Treatise. He realized that by using vector notation instead of quaternions, 12 of the equations could be replaced by 4, the four equations given at the beginning of this article. The remaining equations dealing with circuit analysis became a separate field of study. With the vector notation, the math and implications of Maxwell's theory finally came to be understood. In 1888, Heinrich Hertz validated Maxwell's theory. By creating electromagnetic waves outside the visible light range, radio and Television became feasible.

In 1905, Einstein mentioned Maxwell and Hertz in his theory of special relativity. As the scientific world was introduced to 'observers' and 'frames of reference', the speed of light is found to be constant. Looking at velocities now required Galilean transformations, referring to the classical Newtonian physics where velocities are much less than the speed of light and Lorentz transformations, equations that can be used at any speed. The velocity of electromagnetic waves is constant. The frequencies can differ, but propagation of the electromagnetic waves all occur at the same velocity.

With Quantum Physics developed in the 1920s, photons entered the discussion and Quantum Electrodynamics caused the equations take on a microscopic aspect (dealing with currents and charges at the atomic level of materials) and a macroscopic aspect (using alternatives to the complex atomic-level calculations). The research continues toward the goal of a unified theory.

### Summary

Although we refer to Maxwell's equations as the 4 partial differential equations using vector notation, since Maxwell introduced his equations to the world in 1864 their form, content, and mathematical expression has changed. The equations in present form serve as placeholders for all of the insights Maxwell provided. Each generation has 'owned' the equations keeping them as relevant to the emerging sciences as Electricity, Magnetism, and Light itself. • Barbo April 04, 2016