# Conceptual origin of the magnetic vector potential....?

• Michael Lazich
In summary, the introduction of the magnetic vector potential in classical electrodynamics was justified based on the mathematical relationship between the divergence of a curl and the definition of a magnetic field. This allowed for the addition of curl-less components and the concept of different gauges without affecting the consistency of Maxwell's equations. The physics and mathematics were interlinked, with the potentials serving as auxiliary quantities to simplify the solution of the equations. The physical meaning of the solution is given by the electromagnetic field, rather than the potentials themselves.
Michael Lazich
In Griffiths, it seems that the conceptual introduction of the magnetic vector potential to electrodynamics was justified based on the fact that the divergence of a curl is zero; so we can define a magnetic field as the curl of another vector A and still maintain consistency with Maxwell's equations.

Further, curl-less components could be added to A (introducing the concept of different gauges) and still obtain the same results as well.

My question is, basically: was it a purely mathematical justification for introducing the physical concept of the magnetic vector potential? I.e., was it just a question of noticing "Hey, I can make B the curl of another vector!"?

So essentially I guess I'm asking: did the physics drive the mathematics or vice versa?

My assumption is that the mathematical relationship was noticed first, followed by the introduction of physical concepts, gauges, etc.; but wondering if others may know differently?

Thanks.

I was curious about your question myself, because the textbooks I've used don't go into much detail on the history of classical electrodynamics. So I did a Google search on "magnetic vector potential history" and this turned up on the first page:

http://wwwphy.princeton.edu/~kirkmcd/examples/EP/wu_ijmpa_21_3235_06.pdf (A. C. T. Wu, U of Michigan; C. N. Yang, Chinese U of Hong Kong and Tsinghua U of Beijing)

This struck my eye because I remember Dr. Wu from when I was a grad student at U of M, and Dr. Yang is a Nobel Prize winner. So it might be worth your reading...

jtbell said:
I was curious about your question myself, because the textbooks I've used don't go into much detail on the history of classical electrodynamics. So I did a Google search on "magnetic vector potential history" and this turned up on the first page:

http://wwwphy.princeton.edu/~kirkmcd/examples/EP/wu_ijmpa_21_3235_06.pdf (A. C. T. Wu, U of Michigan; C. N. Yang, Chinese U of Hong Kong and Tsinghua U of Beijing)

This struck my eye because I remember Dr. Wu from when I was a grad student at U of M, and Dr. Yang is a Nobel Prize winner. So it might be worth your reading...
Thanks, pretty much exactly what I was looking for...

Wu and Yang have marvelous papers. One of my favorites is

T. T. Wu and C. N. Yang. Concept of nonintegrable phase factors and global formulation of gauge fields. Phys. Rev. D, 12:3845, 1975.

For classical electrodynamics the potentials (or relativistically spoken the four-vector potential) are auxilliary quantities to simplify the solution of the Maxwell equations. For given charge-current distributions they reduce a first-order set of differential equations for the 6 components of the electromagnetic field to a second-order set plus a gauge-fixing constraint. They are not physical, because they are only defined up to a gauge transformation, i.e., a physical situation is represented by an entire class of four-vector potentials, all connected by an appropriate gauge transformation. The choice of the appropriate gauge constraint for a given problem can be the key idea of its solution. The physical meaning of the solution is, however, given by the electromagnetic field, not immediately by the potentials.

prosteve037

## 1. What is the concept of the magnetic vector potential?

The magnetic vector potential is a mathematical concept used in electromagnetism to describe the magnetic field. It is a vector quantity that is defined as the curl of the magnetic field. In simpler terms, it represents the direction and strength of the magnetic field at any given point in space.

## 2. How is the magnetic vector potential related to the magnetic field?

The magnetic vector potential is mathematically related to the magnetic field through the equation A = μ₀J/4πr, where A is the magnetic vector potential, μ₀ is the permeability of free space, J is the current density, and r is the distance from the current source. This relationship shows that the magnetic field is directly proportional to the magnetic vector potential.

## 3. What is the significance of the magnetic vector potential in electromagnetism?

The magnetic vector potential plays a crucial role in understanding and describing electromagnetic phenomena. It is used in Maxwell's equations, which are fundamental equations in electromagnetism, to calculate the behavior of electric and magnetic fields. The concept of the magnetic vector potential also helps in understanding the principles of electromagnetic induction, which is the basis for technologies such as electric generators and motors.

## 4. How was the concept of the magnetic vector potential developed?

The concept of the magnetic vector potential was first introduced by the French physicist André-Marie Ampère in the early 19th century. However, it was not widely accepted until the 1860s when James Clerk Maxwell incorporated it into his equations to describe the behavior of electromagnetic fields. Since then, the concept has been further developed and refined by many scientists, including Oliver Heaviside, Hendrik Lorentz, and Albert Einstein.

## 5. How is the magnetic vector potential used in practical applications?

The magnetic vector potential has various practical applications in fields such as electrical engineering, physics, and geophysics. It is used to design and analyze magnetic circuits, calculate the forces on magnetic materials, and study the behavior of magnetic materials under different conditions. It is also used in medical imaging techniques, such as magnetic resonance imaging (MRI), to create detailed images of internal body structures.

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