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 $\_\_\_\_$ .

The expression for an electric field in free space is $$E = {E_0}\left( {\widehat x + \widehat y + j2\widehat z} \right){e^{ - j\left( {\omega t - kx + ky} \right)}},$$ where $$x,{\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} y,{\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} z\,\,\,\,\,\,\,$$ represent the spatial coordinates, $$t$$ represents time, and $$\omega ,\,\,k$$ are contants. This electric field

The electric field of a uniform plane wave travelling along the negative $$z$$ direction is given by the following equation: $$$\overrightarrow E {}_w^i = \left( {{{\widehat a}_{_x}} + j{{\widehat a}_{_y}}} \right){E_0}{e^{jkz}}$$$

This wave is incident upon a receiving antenna placed at the origin and whose radiated electric field towards the incident wave is given by the following equation:

$$${\overrightarrow E _{_a}} = \left( {{{\widehat a}_{_x}} + 2{{\widehat a}_{_y}}} \right){E_1}{1 \over r}{e^{ - jkr}}$$$

The polarization of the incident wave, the polarization of the antenna and losses due to the polarization mismatch are, respectively,