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 ______.

A particle, of mass $${10^{ - 3}}$$ $$kg$$ and charge $$1.0$$ $$C,$$ is initially at rest. At time $$t=0,$$ the particle comes under the influence of an electric field $$\overrightarrow E \left( t \right) = {E_0}\sin \,\,$$ $$\omega t\widehat i,$$ where $${E_0} = 1.0\,N{C^{ - 1}}$$ and $$\omega = 10{}^3\,rad\,{s^{ - 1}}.$$ Consider the effect of only the electrical force on the particle. Then the maximum speed, in $$m{s^{ - 1}},$$ attained by the particle at subsequent times is _______________.

An infinitely long uniform line charge distribution of charge per unit length $$\lambda$$ lies parallel to the y-axis in the y-z plane at $$z = {{\sqrt 3 } \over 2}$$a (see figure). If the magnitude of the flux of the electric field through the rectangular surface ABCD lying in the x-y plane with its centre at the origin is $${{\lambda L} \over {n{\varepsilon _0}}}$$ ($${{\varepsilon _0}}$$ = permittivity of free space), then the value of n is

An infinitely long solid cylinder of radius R has a uniform volume charge density $$\rho$$. It has a spherical cavity of radius R/2 with its centre on the axis of the cylinder, as shown in the figure. The magnitude of the electric field at the point P, which is at a distance 2R from the axis of the cylinder, is given by the expression $${{23\rho R} \over {16k{\varepsilon _0}}}$$. The value of k is _____________.