Magnetic Vector-Potential Gauge

Magnetic Vector-Potential Gauge

In classical electromagnetism, the magnetic field B is defined as the curl of the magnetic vector potential A, i.e., B = ∇ × A. Because the curl of a gradient is always zero, we can add the gradient of any scalar field χ to the vector potential without changing the observable magnetic field: A' = A + ∇ χ. This flexibility is known as gauge freedom.

For a uniform magnetic field B_z in the z-direction, a common choice is the symmetric gauge, where A = (-(B_z y)/2, (B_z x)/2, 0). In this module, we explore how adding different scalar functions χ affects the vector potential while preserving the curl and circulation around a loop.

Experiment

In the interactive module, you can select different gauges and observe how the vector field arrows change. You can configure:

1. Gauge: Choose between Symmetric, Cross gradient, or Radial gradient gauges.
2. Uniform B: Set the strength of the uniform magnetic field B_z.
3. Point x, y: Set the target coordinate to see the exact vector potential calculated there.
4. Loop radius: Change the size of the integration loop to verify that the circulation ∮ A · dl remains invariant across different gauges for the same magnetic field.

Worked Example

Let's calculate the vector potential and other properties for a specific configuration.

Inputs:
Gauge: Radial gradient
Uniform B: 1
Point x: 1
Point y: 0
Loop radius: 1

1. Calculate the symmetric gauge vector potential:

   A_sym = (-B * y / 2, B * x / 2) = (-1 * 0 / 2, 1 * 1 / 2) = (0, 0.5)

2. Calculate the scalar function χ for the Radial gradient:

   chi = (x² + y²) / 2 = (1² + 0²) / 2 = 0.5

3. Calculate the gradient of χ:

   grad(chi) = (x, y) = (1, 0)

4. Add the gradient to the symmetric vector potential to find the new vector potential:

   A_new = A_sym + grad(chi) = (0 + 1, 0.5 + 0) = (1.0, 0.5)

5. The divergence of the new vector potential:

   div(A_new) = div(A_sym) + div(grad(chi)) = 0 + 2 = 2

6. The loop circulation for a circle of radius 1:

   Circulation = B * π * r² = 1 * 3.141592653589793 * 1² = 3.141592653589793

The independently calculated visible results match our expectations: Vector potential x is 1.0, Vector potential y is 0.5, and the circulation is 3.141592653589793.

Limitations

This module considers only magnetostatics with a uniform magnetic field in the z-direction. It does not cover time-varying fields or more complex, non-uniform spatial magnetic fields. Additionally, this module focuses strictly on classical gauge freedom and does not address quantum mechanical implications, such as the Aharonov-Bohm effect or gauge-fixing dynamics in quantum field theory. The visualization acts in a 2D plane and ignores the z-component of the vector potential.

References

The concepts presented in this module concerning magnetostatics, vector potentials, and gauge transformations are foundational to classical electromagnetism.

• Feynman, R. P., Leighton, R. B., & Sands, M. (1964). The Feynman Lectures on Physics, Vol. II. California Institute of Technology. Chapter 14: The Magnetic Field in Various Situations.