Question

In the x-y plane, a vector is given by
\(\overrightarrow𝐹 (𝑥, 𝑦) = \frac{−𝑦𝑥̂ + 𝑥𝑦̂}{𝑥^ 2 + 𝑦^2} . \)
The magnitude of the flux of \(\overrightarrow∇ ×\overrightarrow 𝐹\) , through a circular loop of radius 2, centered at the origin, is:

• A. π
• B. 2π
• C. 4π
• D. 0

Answer 1

The Correct Option is B

The flux of the curl of a vector field through a surface is given by:

\[ \Phi = \iint_S (\nabla \times \vec{F}) \cdot d\vec{A} \]

Using Stokes’ Theorem, this flux can be related to the circulation around the boundary of the surface, which in this case is a circular loop of radius 2 centered at the origin.

Since \( \vec{F}(x, y) \) is a vector field in the x-y plane, the curl of \( \vec{F} \) in the z-direction can be evaluated using the given components.

Upon solving for the flux, we find that the magnitude of the flux through the circular loop is:

\[ 2\pi \]

Conclusion:

Thus, the correct answer is (B).