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The flux of \( \mathbf{F} \) through a surface \( S \) is

  • A.\( \displaystyle\iint_{S} \mathbf{F} \times \mathbf{n}\,dS \)
  • B.\( \displaystyle\iint_{S} \mathbf{F}\,dS \)
  • C.\( \displaystyle\iint_{S} \mathbf{F} \cdot \mathbf{n}\,dS \)
  • D.\( \displaystyle\iint_{S} |\mathbf{F}|\,dS \)

Answer: C

Only the component crossing the surface counts, so each element contributes \( \mathbf{F} \cdot \mathbf{n}\,dS \) and

\[ \Phi \;=\; \iint_{S} \mathbf{F} \cdot \mathbf{n}\,dS. \]

A field lying wholly in the surface carries no flux through it, which the dot product records and an integral of the magnitude would not

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