Question

The total charge enclosed in an incremental volume of \(2\times10^{-9}\ \mathrm{m^3}\) located at the origin is ________ nC, if electric flux density of its field is found as
\(\vec{D} = e^{-x}\sin y\,\hat{i} - e^{-x}\cos y\,\hat{j} + 2z\,\hat{k}\ \ \mathrm{C/m^2}\)

Hint:

By Gauss's law in differential form, \(\rho_v = \nabla\cdot\vec{D}\) and \(Q = \rho_v\,\Delta v\).

Answer 1

Correct Answer: 4

Differential form of Gauss's law: \(\nabla\cdot\vec{D} = \rho_v\).
\(\dfrac{\partial D_x}{\partial x} = \dfrac{\partial}{\partial x}(e^{-x}\sin y) = -e^{-x}\sin y\).
\(\dfrac{\partial D_y}{\partial y} = \dfrac{\partial}{\partial y}(-e^{-x}\cos y) = e^{-x}\sin y\).
\(\dfrac{\partial D_z}{\partial z} = 2\).
\(\rho_v = -e^{-x}\sin y + e^{-x}\sin y + 2 = 2\ \mathrm{C/m^3}\) (the same everywhere, including the origin).
\(Q = \rho_v\,\Delta v = 2\times 2\times10^{-9} = 4\times10^{-9}\ \mathrm{C} = 4\ \mathrm{nC}\).
Answer: 4