For a vector field $\overrightarrow{\mathbf{D}}=\rho \cos ^2 \phi \hat{\mathbf{a}}_{\boldsymbol{\rho}}+z^2 \sin ^2 \phi \hat{\mathbf{a}}_\phi$ in a cylindrical coordinate system $(\rho, \phi, z)$ with unit vector $\hat{\boldsymbol{a}}_\rho, \hat{\boldsymbol{a}}_\phi$ and $\hat{\boldsymbol{a}}_z$, the net flux of $\overrightarrow{\mathbf{D}}$ leaving the closed surface of the cylinder $(\rho=3,0 \leq z \leq 2)$ (Round off to 2 decimal places) is $\_\_\_\_$ .

The vector function $F(r) = -x \hat{i} + y \hat{j}$ is defined over a circular arc C shown in the figure,

The line integral of $\int\limits_{C} \mathbf{F(r)} \cdot d\mathbf{r}$ is

Consider the vector field $\overline{\mathbf{F}}=\hat{\mathbf{a}}_{\mathbf{x}}\left(4 y-c_1 z\right)+\hat{\mathbf{a}}_{\mathbf{y}}(4 x+2 z)+\hat{\mathbf{a}}_{\mathbf{z}}(2 y+z)$ in a rectangular coordinate system $(x, y, z)$ with unit vectors $\hat{\mathbf{a}}_{\mathbf{x}}, \hat{\mathbf{a}}_{\mathbf{y}}, \hat{\mathbf{a}}_{\mathbf{z}}$. If the field $\mathbf{F}$ is irrotational (conservative), then the constant $c_1$ (in integer) is $\_\_\_\_$ .

The magnetic field of a uniform plane wave in vacuum is given by

$$ \vec{H}(x, y, z, t)=\left(\hat{a}_x+2 \hat{a}_y+b \hat{a}_z\right) \cos (\omega t+3 x-y-z) . $$

The value of $b$ is $\_\_\_\_$ .