USAP Science Practice Exam

Magnetic field lines don’t begin or end in empty space; they form closed loops because there are no isolated magnetic charges (no magnetic monopoles). This means there’s no local source or sink of magnetic field. The amount of field passing out of a tiny volume must balance the amount entering it, so the net outward flux from any point is zero. In math, that idea is written as ∇·B = 0, Gauss’s law for magnetism. The integral form, ∮ B·dA = 0, expresses the same thing as a flux through any closed surface being zero.

If magnetic monopoles did exist with some density, we'd have ∇·B = μ0ρm, and you'd see a nonzero divergence at regions with monopoles. Since that isn’t observed, the divergence of B is zero. The field can vary in space and still have zero divergence; being perpendicular to current isn’t required for zero divergence, and the divergence is a well-defined quantity.