Magnetostatics and the vector potential

Equipment

• Field model around the wire (two view modes A/B)
• Sliders for amplitude/source, wavelength/scale, frequency/velocity, viewing angle
• "Model A/B" toggle buttons and "Pause"

Procedure

1. Observe the B field around the wire

   In figure 1 "Magnetic field around the wire", watch field lines encircle the current axis. Raise the "Amplitude / source" slider (larger current I) and note the field-line density grow: $B \propto I$. Change "Wavelength / scale" to zoom and see $B$ fall off like $1/r$.

2. Compare the two display models

   Press "Model A" and "Model B" to switch the field visualization (e.g., field lines versus vector field). In figure 2 "Vector potential A and gauge", watch $\mathbf{A}$ run along the wire and curl: $\mathbf{B} = \nabla \times \mathbf{A}$. Note that adding $\nabla \chi$ to $\mathbf{A}$ leaves $\mathbf{B}$ unchanged — gauge freedom.

3. Check the flux theorem

   Vary "Frequency / velocity" and "Viewing angle / coefficient" in figure 2, then press "Pause" for a static view. Picture a closed loop around the wire: $\oint \mathbf{A}\cdot d\boldsymbol{\ell}$ equals the total flux $\Phi_B = \int \mathbf{B}\cdot d\mathbf{S}$ threading it — a visual form of Stokes' theorem and the basis of the Aharonov–Bohm effect.

Simulation

Experiment history

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