Magnetic Vector Potential

The Vector Potential A

Because the magnetic field is solenoidal ($\nabla \cdot \mathbf{B} = 0$), Helmholtz's theorem assures us that $\mathbf{B}$ can be expressed as the curl of a vector function. We define this function as the magnetic vector potential, denoted by $\mathbf{A}$:

\mathbf{B} = \nabla \times \mathbf{A}

While $\mathbf{B}$ is physically measurable, $\mathbf{A}$ is a theoretical construct that simplifies calculations. By imposing the Coulomb Gauge condition ($\nabla \cdot \mathbf{A} = 0$), Ampère's law transforms into a vector Poisson equation. The general solution for a localized current distribution $\mathbf{J}$ is given by:

\mathbf{A}(\mathbf{r}) = \frac{\mu_0}{4\pi} \int \frac{\mathbf{J}(\mathbf{r}')}{|\mathbf{r} - \mathbf{r}'|} d\tau'

This integral suggests that the vector potential is parallel to the current flow, unlike the magnetic field which curls around it.

Deep Dive: Gauge Freedom

Multipole Expansion of the Vector Potential

When calculating the magnetic effects of a localized current loop at large distances ($r \gg r'$), it is often easier to approximate the potential using a multipole expansion in powers of $1/r$.

The first term (monopole) is always zero because magnetic monopoles do not exist ($\oint \mathbf{J} \, d\tau = 0$ for steady currents). The dominant term is usually the magnetic dipole term:

\mathbf{A}_{\text{dip}}(\mathbf{r}) = \frac{\mu_0}{4\pi} \frac{\mathbf{m} \times \hat{\mathbf{r}}}{r^2}

Here, $\mathbf{m}$ is the magnetic dipole moment, defined as $I \int d\mathbf{a}$. This approximation reveals that far from a current loop, the field structure is determined solely by the loop's orientation and total magnetic moment, regardless of the loop's specific shape.

Deep Dive: Multipole Expansion of A

Boundary Conditions

Just as the scalar potential $V$ is continuous in electrostatics, the vector potential $\mathbf{A}$ is continuous across a surface boundary carrying a surface current density $\mathbf{K}$. However, the derivative of the potential undergoes a discontinuity.

At an interface, the conditions are:

\mathbf{A}_{\text{above}} = \mathbf{A}_{\text{below}} $$ $$ \frac{\partial \mathbf{A}_{\text{above}}}{\partial n} - \frac{\partial \mathbf{A}_{\text{below}}}{\partial n} = -\mu_0 \mathbf{K}

This implies that while the "height" of the potential matches on both sides of the sheet, the "slope" (normal derivative) kinks sharply by an amount proportional to the surface current. This ensures that $\mathbf{B} = \nabla \times \mathbf{A}$ properly reflects the discontinuity in the tangential magnetic field.

Deep Dive: Boundary Conditions