Q 12-01-072JEE MainJEE Main 2023 (30 Jan, Shift 2)Medium
As shown in figure, a cuboid lies in a region with electric field $\vec E=2x^2\hat i-4y\hat j+6\hat k\ \text{N C}^{-1}$. The magnitude of charge within the cuboid is $n\varepsilon_0$ C. The value of $n$ is ______ (if dimension of cuboid is $1\times2\times3\ \text{m}^3$).
Numerical value type. Enter your answer.
Answer: 12
By Gauss's law $q=\varepsilon_0\displaystyle\int(\nabla\cdot\vec E)\,dV$, with $\nabla\cdot\vec E=4x-4$.
$$\int_0^1(4x-4)\,dx=2-4=-2,\qquad q=\varepsilon_0(-2)(2\times3)=-12\varepsilon_0$$
Magnitude $12\varepsilon_0$, so $n=12$.
Solution by Sreeraj P, M.Sc Physics