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Gauss's Law for Magnetism

Source lecture(s): SC134 Lec11

Intuition

There are no magnetic monopoles—every field line that enters a volume must leave it.

Formal Definition

\[\oint \vec{B}\cdot d\vec{A} = 0\]

Mathematical Formulation

Differential form: \(\nabla\cdot\vec{B} = 0\) Consequence: \(\vec{B} = \nabla\times\vec{A}\) (vector potential exists).

Derivation

No experimental evidence for isolated magnetic charge; field lines always close.

Worked Example

Bar magnet field lines exit north, loop through space, and enter south—closed loops.

Common Mistakes

  • Treating magnetic monopoles as real in introductory problems.
  • Confusing with Ampere's law.

Quiz

Q1. What would Gauss's law for magnetism look like if monopoles existed?

Answer

Right side would be \(\mu_0 q_m\).