Question

2. Demonstrate that the following is a valid vector potential, satisfying the Coulomb gauge, for a homogeneous magnetic field: $\mathbf{A} = -\frac{1}{2}\mathbf{r} \times \mathbf{B}$. Hint: you may find the following identity useful: $\nabla \times (\mathbf{v} \times \mathbf{w}) = \mathbf{v}(\nabla \cdot \mathbf{w}) - \mathbf{w}(\nabla \cdot \mathbf{v}) + (\mathbf{w} \cdot \nabla)\mathbf{v} - (\mathbf{v} \cdot \nabla)\mathbf{w}$. N.B. \"homogeneous\" means that $\mathbf{B}$ has the same magnitude and direction at all points in space.

          2. Demonstrate that the following is a valid vector potential, satisfying the Coulomb gauge,
for a homogeneous magnetic field:

$\mathbf{A} = -\frac{1}{2}\mathbf{r} \times \mathbf{B}$.

Hint: you may find the following identity useful:
$\nabla \times (\mathbf{v} \times \mathbf{w}) = \mathbf{v}(\nabla \cdot \mathbf{w}) - \mathbf{w}(\nabla \cdot \mathbf{v}) + (\mathbf{w} \cdot \nabla)\mathbf{v} - (\mathbf{v} \cdot \nabla)\mathbf{w}$.

N.B. \"homogeneous\" means that $\mathbf{B}$ has the same magnitude and direction at all points
in space.

2. Demonstrate that the following is a valid vector potential, satisfying the Coulomb gauge,
for a homogeneous magnetic field:

𝐀 = -(1)/(2)𝐫×𝐁.

Hint: you may find the following identity useful:
∇× (𝐯×𝐰) = 𝐯(∇·𝐰) - 𝐰(∇·𝐯) + (𝐰·∇)𝐯 - (𝐯·∇)𝐰.

N.B. ḧomogeneousm̈eans that 𝐁 has the same magnitude and direction at all points
in space.

Added by Nathan B.

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