Q 12-01-117JEE MainJEE Main 2021 (22 Jul, Shift 1)Medium
The total charge enclosed in an incremental volume of $2\times10^{-9}\ \text{m}^3$ located at the origin is ______ nC, if electric flux density of its field is found as $\vec D = e^{-x}\sin y\,\hat i - e^{-x}\cos y\,\hat j + 2z\,\hat k\ \text{C m}^{-2}$.
Numerical value type. Enter your answer.
Answer: 4
By Gauss's law in differential form, the charge density is $\rho = \nabla\cdot\vec D$:
$$\nabla\cdot\vec D = -e^{-x}\sin y + e^{-x}\sin y + 2 = 2\ \text{C m}^{-3}$$
$Q = \rho\,\Delta V = 2\times2\times10^{-9} = 4\times10^{-9}$ C $= 4$ nC.
Solution by Sreeraj P, M.Sc Physics