Consider the time-varying vector $$I = \,\hat x\,\,15\,\cos \,(\omega \,t) + \,\hat y\,5\,sin(\omega \,t)$$ in Cartesian coordinates, where $$\omega $$ > 0 is a constant. When the vector magnitude $$\left| I \right|$$ is at its minimum value, the angle $$\theta $$ that I makes with the x axis (in degree, such that $$0\, \le \,0 \le \,180$$) is _________________

A microwave circuit consisting of lossless transmission lines $$T_1$$ and $$T_2$$ is shown in the figure. The plot shows the magnitude of the input reflection coefficient $$\Gamma $$ as a function of frequency f. The phase velocity of the signal in the transmission lines is $$2\, \times \,{10^8}$$ m/s

Light from free space is incident at an angle $${\theta _1}$$ to the normal of the facet of a step-index large core optical fibre. The core and cladding refractive indices are $${n_1}\,\, = \,\,1.5$$ and $${n_2}\,\, = \,\,1.4$$ respectively.

The maximum value of $${\theta _1}$$ (in degrees) for which the incident light will be guided in the core of the fibre is ________ .

The parallel-plate capacitor shown in the figure has movable plates. The capacitor is charged so that the energy stored in it is E when the plate separation is d. The capacitor is then isolated electrically and the plates are moved such that the plate separation becomes 2d.

At this new plate separation, what is the energy stored in the capacitor, neglecting fringing effects?