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Suppose that I have a situation where I can use the magnetic scalar potential.

Given the magnetic scalar potential, is there a simple way to calculate the magnetic vector potential from the magnetic scalar potential?

(The integral equations for the scalar and vector potentials are similar, but I don't quite see how to get from one to the other.)

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For a given elementary dipole source of strength $\mathfrak{m}$ the answer is very simple: $$\textbf{A} = \frac{1}{4\pi} \frac{\mathfrak{m}\times \textbf{r}^0}{r^2} $$ and $$\phi = \frac{1}{4\pi} \frac{\mathfrak{m}\cdot \textbf{r}^0}{r^2}$$ The induction field is then $$\textbf{B} = \text{curl}\textbf{A} = -\text{grad}\phi = \frac{1}{4\pi} \frac{-\mathfrak{m}+3\mathfrak{m}\cdot \textbf{r}^0\textbf{r}^0}{r^3}.$$ Of course, for a continuous dipole distribution you can sum this with an integral. Anyhow, first you have to find the dipole field from the scalar potential and then you can get the vector potential.

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