If a vector field $\vec{F}$ is conservative (i.e., $\vec{F} = \nabla \phi$), then the circulation $\oint_C \vec{F} \cdot d\vec{r}$ along ANY closed path $C$ is:
Options:
(A) Always zero
(B) Equal to the enclosed area
(C) Equal to the divergence of $\vec{F}$
(D) Infinity
Answer Key:
A
Step-by-Step Solution & Explanation
For a conservative vector field, $\nabla \times \vec{F} = 0$. By Stokes' theorem, the line integral around any closed loop is identically zero: $\oint_C \vec{F} \cdot d\vec{r} = \iint_S (\nabla \times \vec{F}) \cdot \hat{n}\, dS = 0$.