GATE 2021 EC – Question 11
The vector function $\mathbf F(\mathbf r)=-x\,\hat\imath+y\,\hat\jmath$ is defined over a circular arc $C$ shown in the figure.
The line integral of $\int_C\mathbf F(\mathbf r)\cdot d\mathbf r$ is
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Correct answer: (A) $\frac12$
Explanation
$\mathbf F=\nabla\phi$ with $\phi=\frac{y^2-x^2}{2}$, so the integral depends only on the end points. The arc runs from $(1,0)$ to $\left(\frac1{\sqrt2},\frac1{\sqrt2}\right)$: $\phi(1,0)=-\frac12$ and $\phi\left(\frac1{\sqrt2},\frac1{\sqrt2}\right)=0$. So the integral is $0-\left(-\frac12\right)=\frac12$.