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 ________ .

A uniform and constant magnetic field $$B=\widehat zB$$ exists in the $$\widehat z$$ direction in vacuum. A particle of mass m with a small charge q is introduced into this region with an initial velocity $$v=\widehat xv_x+\widehat zv_z$$. Given that B, m, q, v_x and v_z are all non-zero, which one of the following describes the eventual trajectory of the particle?

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?

Consider a region of silicon devoid of electrons and holes, with an ionized donor density of $${\mathrm N}_\mathrm d^+=10^{17}\;\mathrm{cm}^{-3}$$. The electric field at x = 0 is 0 V/cm and the electric field at x = L is 50 kV/cm in the positive x direction. Assume that the electric field is zero in the y and z directions at all points.

Given q = 1.6 × 10⁻¹⁹ coulomb, $$\varepsilon$$₀ = 8.85 × 10⁻¹⁴ F/cm, $$\varepsilon$$_r = 11.7 for silicon, the value of L in nm is ________.