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The line integral \( \int \mathbf{A} \cdot d\mathbf{l} \) of a field along a path measures

  • A.the divergence of the field integrated along the length of the path
  • B.the work the field would do along that path
  • C.the flux of the field through the surface spanning the path
  • D.the circulation of the field round the path, whether closed or not

Answer: B

For a force field it is literally work; for the electric field it is the potential difference with the sign reversed. Only when the path is closed does it become a circulation, and for a conservative field that closed value is zero.

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