GATE 2016 EC – Question 34
Concentric spherical shells of radii 2 m, 4 m, and 8 m carry uniform surface charge densities of 20 nC/m$^2$, $-4$ nC/m$^2$ and $\rho_s$, respectively. The value of $\rho_s$ (nC/m$^2$) required to ensure that the electric flux density $\vec{D} = \vec{0}$ at radius 10 m is ________
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Correct answer: -0.28 to -0.22
Explanation
By Gauss's law, $\vec{D} = 0$ at 10 m means the total enclosed charge is zero. Each shell holds $4\pi r^2 \rho$, so $4\pi[2^2(20) + 4^2(-4) + 8^2 \rho_s] = 0$. That gives $80 - 64 + 64\rho_s = 0$, so $\rho_s = -\frac{16}{64} = -0.25$ nC/m$^2$.