One end of a spring of negligible unstretched length and spring constant k is fixed at the origin (0, 0). A point particle of mass m carrying a positive charge q is attached at its other end. The entire system is kept on a smooth horizontal surface. When a point dipole $$\overrightarrow p $$ pointing towards the charge q is fixed at the origin, the spring gets stretched to a length l and attains a new equilibrium position (see figure below). If the point mass is now displaced slightly by $$\Delta $$l << l from its equilibrium position and released, it is found to oscillate at frequency $${1 \over \delta }\sqrt {{k \over m}} $$. The value of $$\delta $$ is ______.

Consider one mole of helium gas enclosed in a container at initial pressure P₁ and volume V₁. It expands isothermally to volume 4V₁. After this, the gas expands adiabatically and its volume becomes 32V₁. The work done by the gas during isothermal and adiabatic expansion processes are W_iso and W_adia, respectively. If the ratio $${{{W_{iso}}} \over {{W_{adia}}}}$$ = f ln 2, then f is ______.

Shown in the figure is a semicircular metallic strip that has thickness t and resistivity $$\rho $$. Its inner radius is R₁ and outer radius is R₂. If a voltage V₀ is applied between its two ends, a current I flows in it. In addition, it is observed that a transverse voltage $$\Delta $$V develops between its inner and outer surfaces due to purely kinetic effects of moving electrons (ignore any role of the magnetic field due to the current). Then (figure is schematic and not drawn to scale)

A particle of mass m moves in circular orbits with potential energy V(r) = Fr, where F is a positive constant and r is its distance from the origin. Its energies are calculated using the Bohr model. If the radius of the particle’s orbit is denoted by R and its speed and energy are denoted by v and E, respectively, then for the n^th orbit (here h is the Planck’s constant)