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:

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:

cdquestions Admin · Accepted Answer

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).

Answer

Answer

Answer

Answer

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:

The Correct Option is B

Solution and Explanation

Conclusion:

Top Questions on Mathematical Methods

Questions Asked in IIT JAM exam