GATE 2021 EC – Question 47
For a vector field $\mathbf D=\rho\cos^2\varphi\,\mathbf a_\rho+z^2\sin^2\varphi\,\mathbf a_\varphi$ in a cylindrical coordinate system $(\rho,\varphi,z)$ with unit vectors $\mathbf a_\rho,\mathbf a_\varphi$ and $\mathbf a_z$, the net flux of $\mathbf D$ leaving the closed surface of the cylinder ($\rho=3$, $0\le z\le2$) (rounded off to two decimal places) is ________.
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Correct answer: 56.27 to 56.83
Explanation
By the divergence theorem the flux is $\iiint\nabla\cdot\mathbf D\,dV$ with $\nabla\cdot\mathbf D=\frac1\rho\frac{\partial(\rho D_\rho)}{\partial\rho}+\frac1\rho\frac{\partial D_\varphi}{\partial\varphi}=2\cos^2\varphi+\frac{2z^2}{\rho}\sin\varphi\cos\varphi$. The second term integrates to zero over $\varphi$, and $\int_0^{2\pi}2\cos^2\varphi\,d\varphi=2\pi$. So the flux is $2\pi\int_0^3\rho\,d\rho\int_0^2dz=2\pi\times4.5\times2=18\pi\approx56.55$.