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GATE 2020 EE – Question 50

Electromagnetic Fields · Coulomb's Law, Electric Field Intensity, Electric Flux Density, Gauss's Law, Divergence · 2 marks · Multiple choice

Let $\mathbf a_r$, $\mathbf a_\phi$ and $\mathbf a_z$ be unit vectors along r, $\phi$ and z directions, respectively in the cylindrical coordinate system. For the electric flux density given by $\mathbf D=(\mathbf a_r\,15+\mathbf a_\phi\,2r-\mathbf a_z\,3rz)$ Coulomb/m$^2$, the total electric flux, in Coulomb, emanating from the volume enclosed by a solid cylinder of radius 3 m and height 5 m oriented along the z-axis with its base at the origin is:

  1. $54\pi$
  2. $90\pi$
  3. $108\pi$
  4. $180\pi$

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Correct answer: (D) $180\pi$

Explanation

By Gauss's law the flux equals $\iiint\nabla\cdot\mathbf D\,dV$. $\nabla\cdot\mathbf D=\frac1r\frac{\partial(15r)}{\partial r}+0-3r=\frac{15}{r}-3r$. So the flux is $2\pi\times5\int_0^3\left(\frac{15}{r}-3r\right)r\,dr=10\pi\int_0^3(15-3r^2)dr=10\pi(45-27)=180\pi$.