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True or false: a field that is both solenoidal and irrotational must be zero.

  • A.False, but only if the region under consideration is unbounded
  • B.False, a uniform field satisfying both conditions without vanishing
  • C.True, since a field with neither sources nor circulation cannot exist at all
  • D.True, the two conditions together making the Laplacian of the field zero

Answer: B

A uniform field has zero divergence and zero curl and is plainly not zero. What the two conditions together require is that the field derive from a potential satisfying Laplace's equation, which has many non-trivial solutions.

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