Gauss's law for magnetism implies what about magnetic monopoles?

Gauss's law for magnetism implies what about magnetic monopoles?

• A. ∮ B · dA = μ0 I_enc
• B. ∮ E · dA = 0
• C. ∇ · B = ρ_m
• D. ∮ B · dA = 0

Answer: (D)

Magnetic fields have no isolated magnetic charges; their field lines form closed loops and don’t begin or end at a point. Gauss's law for magnetism captures this by saying the net flux of B through any closed surface is zero. In other words, the integral of B over a closed surface vanishes: ∮ B · dA = 0. This also tells us ∇ · B = 0 everywhere, meaning there is no magnetic charge density to source or sink B. If magnetic monopoles did exist, you would have a nonzero ∇ · B equal to a magnetic charge density ρ_m, which would give a nonzero flux through some closed surfaces. The statement ∮ B · dA = 0 directly expresses the observed fact that there are no magnetic monopoles. The other options either describe different laws (Ampère’s law relating the line integral of B to current, or Gauss’s law for electricity relating E flux to charge) or posit magnetic charges (∇ · B = ρ_m). Those aren’t what Gauss's law for magnetism asserts.

Magnetic fields have no isolated magnetic charges; their field lines form closed loops and don’t begin or end at a point. Gauss's law for magnetism captures this by saying the net flux of B through any closed surface is zero. In other words, the integral of B over a closed surface vanishes: ∮ B · dA = 0. This also tells us ∇ · B = 0 everywhere, meaning there is no magnetic charge density to source or sink B.

If magnetic monopoles did exist, you would have a nonzero ∇ · B equal to a magnetic charge density ρ_m, which would give a nonzero flux through some closed surfaces. The statement ∮ B · dA = 0 directly expresses the observed fact that there are no magnetic monopoles.

The other options either describe different laws (Ampère’s law relating the line integral of B to current, or Gauss’s law for electricity relating E flux to charge) or posit magnetic charges (∇ · B = ρ_m). Those aren’t what Gauss's law for magnetism asserts.