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For a superconductor we have the nice equation [itex]\nabla^2 \mathbf B=\frac{1}{\lambda^2}\mathbf B[/itex]. Using this equation we can find the B-field inside the superconductor if we have the boundary values. But what about an ordinary conductor(or dielectric) ? If I know what the field is outside the object, what is the equation I can solve to find the field inside?