Question

Three small circles C1, C2, and C3, each with radius 0.1 and centered at the origin are in the xy-, yz-, and xz-planes, respectively. The circles are oriented counterclockwise when viewed from the positive z-, x-, and y-axes, respectively. A vector field F⃗ has circulation around C1 of 0.01π, around C2 of 0.4π, and around C3 of 5π. Estimate curl(
f. at the origin.

Answer 1

Curl is defined as:

$\bigtriangledown \times F=\frac{ \int F.dr}{Area}$

Which can be rewritten as:

$\frac{ \int F.dr}{Area}=\frac{ \int_{C_1} Fdr}{Area}\widehat{k}+\frac{ \int_{C_2} Fdr}{Area}\widehat{i}+\frac{ \int_{C_3} Fdr}{Area}\widehat{j}$

This is because C1 lies on the xy plane and thus it's unit vector will be $\widehat{k}$.

By similar arguments the rest will follow too.

Now, area of each circle will be: $\pi \times (0.1)^2=0.01 \pi$

Therefore, Curl, as per our definition will be:

$\frac{\int F.dr}{Area}=\frac{0.01\pi}{0.01\pi}\widehat{k}+\frac{0.4\pi}{0.01\pi}\widehat{i}+\frac{5 \pi}{0.01\pi}\widehat{j}$

Thus, Curl=$40\pi \widehat{i}+500\pi \widehat{j} +\widehat{k}$

Which is the required answer.