The total charge enclosed in an incremental volume of $$2 \times 10^{-9}$$ m$$^3$$ located at the origin is ___ nC, if electric flux density of its field is found as $${D} = e^{-x}\sin y \hat{i} - e^{-x}\cos y \hat{j} + 2z\hat{k}$$ C m$$^{-2}$$

The total charge enclosed in a volume is related to the electric flux density $$\vec{D}$$ through Gauss's law in differential form: $$\rho_v = \nabla \cdot \vec{D}$$.

Given $$\vec{D} = e^{-x}\sin y\,\hat{i} - e^{-x}\cos y\,\hat{j} + 2z\,\hat{k}$$, the divergence is:

$$\nabla \cdot \vec{D} = \frac{\partial}{\partial x}(e^{-x}\sin y) + \frac{\partial}{\partial y}(-e^{-x}\cos y) + \frac{\partial}{\partial z}(2z)$$

$$= -e^{-x}\sin y + e^{-x}\sin y + 2 = 2\,\text{C\,m}^{-3}$$

The divergence is constant (equals 2 everywhere, independent of position), so the volume charge density at the origin is $$\rho_v = 2\,\text{C\,m}^{-3}$$.

The total charge enclosed in the incremental volume $$\Delta v = 2 \times 10^{-9}\,\text{m}^3$$ is:

$$Q = \rho_v \cdot \Delta v = 2 \times 2 \times 10^{-9} = 4 \times 10^{-9}\,\text{C} = 4\,\text{nC}$$