A half wavelength dipole is kept in the x-y plane and oriented along $${45^ \circ }$$ from the x-axis. Determine the direction of null in the radiation pattern for $$0 \le \phi \le \pi $$. Here the angle $$\theta \left( {0 \le \theta \le \pi } \right)$$ is measured from the z-axis, and the angle $$\phi \left( {0 \le \phi \le 2\pi } \right)$$ is measured from the x-axis in the x-y plane.

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 voltage of an electromagnetic wave propagating in a coaxial cable with uniform characteristic impedance is $$V(l) = {e^{ - \gamma l\, + \,j\,\omega \,t}}$$ Volts, where $$l$$ is the distance along the length of the cable in metres, $$\gamma = (0.1\, + \,j40)\,\,{m^{ - 1}}$$ is the complex propagation constant, and $$\omega = \,2\,\pi \,\, \times \,\,{10^9}$$ rad/s is the angular frequency. The absolute value of the attenuation in the cable in dB/metre is ___________________.

An optical fiber is kept along the$$\widehat Z$$ direction. The refractive indices for the electric fields along $$\widehat X$$ and $$\widehat Y$$ directions in the fiber are $${n_x} = 1.5000$$ and $${n_y} = 1.5001$$, respectively ($${n_x} \ne {n_y}$$ due to the imperfection in the fiber cross-section). The free space wavelength of a light wave propagating in the fiber is $$1.5\,\,\,\mu m$$. If the light wave is circularly polarized at the input of the fiber, the minimum propagation distance after which it becomes linearly polarized, in centimeters, is _______ .