Respuesta :
Curl is defined as:
[tex]\bigtriangledown \times F=\frac{ \int F.dr}{Area}[/tex]
Which can be rewritten as:
[tex]\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}[/tex]
This is because C1 lies on the xy plane and thus it's unit vector will be [tex]\widehat{k}[/tex].
By similar arguments the rest will follow too.
Now, area of each circle will be: [tex]\pi \times (0.1)^2=0.01 \pi[/tex]
Therefore, Curl, as per our definition will be:
[tex]\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}[/tex]
Thus, Curl=[tex]40\pi \widehat{i}+500\pi \widehat{j} +\widehat{k}[/tex]
Which is the required answer.
If a vector field F has circulation around C1 of 0.01π, around C2 of 0.4π, and around C3 of 5π then curl is [tex]40 \pi \hat{i}+500 \pi \hat{j}+\widehat{k}$[/tex]
What is curl ?
The curl is a vector operator that describes the infinitesimal rotation of a vector field in three-dimensional Euclidean space.
We can that formula of curl is
[tex]$\nabla \times F=\frac{\int F \cdot d r}{A r e a}$[/tex]
[tex]$\frac{\int F d r}{\text { Area }}=\frac{\int_{C_{1}} F d r}{\text { Area }} \widehat{k}+\frac{\int_{C_{2}} F d r}{\text { Area }} \widehat{i}+\frac{\int_{C_{3}} F d r}{\text { Area }^{j} \hat{j}}$[/tex]
And area of circle can be calculated as :
[tex]\pi \times(r)^{2} = \pi \times(0.1)^{2}=0.01 \pi[/tex]
Hence curl can be calculated as :
[tex]\frac{\int F d r}{\text { Arca }}=\frac{0.01 \pi}{0.01 \pi} \widehat{k}+\frac{0.4 \pi}{0.01 \pi} \widehat{i}+\frac{5 \pi}{0.01 \pi} \widehat{j}[/tex]
[tex]=40 \pi \hat{i}+500 \pi \hat{j}+\widehat{k}[/tex]
If a vector field F has circulation around C1 of 0.01π, around C2 of 0.4π, and around C3 of 5π then curl is [tex]40 \pi \hat{i}+500 \pi \hat{j}+\widehat{k}$[/tex]
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