As shown in figure, a cuboid lies in a region with electric field $$E = 2x^2\hat{i} - 4y\hat{j} + 6\hat{k}$$ 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 $$1 \times 2 \times 3$$ m$$^3$$)

Use Gauss’s law in differential form:

$$\nabla\cdot\vec{E}=\frac{\rho}{\varepsilon_0}$$

and total charge enclosed

$$Q=ε_0∭(∇⋅E⃗)dV$$

Given

$$E = 2x^2\hat{i} - 4y\hat{j} + 6\hat{k}$$ N C$$^{-1}$$.

Divergence is

$$\nabla\cdot\vec{E}=\frac{\partial(2x^2)}{\partial x}+\frac{\partial(-4y)}{\partial y}+\frac{\partial(6)}{\partial z}$$

$$=4x−4+0$$

$$=4x−4$$

Now cuboid dimensions from figure:

$$0\le x\le1$$

$$0\le y\le2$$

$$0\le z\le3$$

So

$$Q=ε_0\int\int\int(4x−4)dzdydx$$

First integrate over z:

$$=ε_0\int\int3(4x−4)dydx$$

Integrate over y:

$$=ε_0\int6(4x−4)dx$$

$$=6\varepsilon_0\int_0^1(4x-4)dx$$

$$=6\varepsilon_0\left(2x^2-4x\right)_0^1$$

$$=6\varepsilon_0(2-4)$$

$$=-12\varepsilon_0$$

Magnitude of charge:

$$|Q|=12\varepsilon_0$$