Biot_Savart law for bounded current systems

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I have a PDE I don't know how to solve, which is a result of a physics problem related to the Biot-Savart equation.

The PDE in question is an equation for B:

$$\text{curl}(B)=\text{del}(\phi)$$

where $\phi$ is a known function in $3$D space.

  • Additional information:

div and curl of this equation are identically zero.

Any suggestions?

I can't apply a Biot-Savart approach to this equation as it is not an unbounded problem.

Thanks!