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The current density in a medium is given by $\vec{J}=\frac{400\sin\theta}{2\pi(r^2+4)}\,\hat{a}_r\ \mathrm{A\,m^{-2}}$. The total current and the average current density flowing through the portion of a spherical surface $r=0.8$ m, $\pi/2\le \theta\le \pi/4$ and $0\le \phi\le 2\pi$ are, respectively,

1. 15.09 A, 12.86 A m−2
2. 18.73 A, 13.65 A m−2
3. 12.86 A, 9.23 A m−2
4. 10.28 A, 7.56 A m−2

Correct answer: 12.86 A, 9.23 A m−2

Solution

The current through the spherical patch is obtained by integrating $\vec{J}\cdot d\vec{S}$ over the given angular limits. Using $dS=r^2\sin\theta\,d\theta\,d\phi$ and the specified region gives the total current $I=12.86$ A. Dividing by the area of the patch gives the average current density $9.23$ A m$^{-2}$.

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