Two identical charged spheres are suspended by strings of equal lengths. The strings make an angle of $$37^{\circ}$$ with each other. When suspended in a liquid of density $$0.7 \mathrm{~g} / \mathrm{cm}^3$$, the angle remains same. If density of material of the sphere is $$1.4 \mathrm{~g} / \mathrm{cm}^3$$, the dielectric constant of the liquid is _______ $$\left(\tan 37^{\circ}=\frac{3}{4}\right)$$

An electron is moving under the influence of the electric field of a uniformly charged infinite plane sheet $$\mathrm{S}$$ having surface charge density $$+\sigma$$. The electron at $$t=0$$ is at a distance of $$1 \mathrm{~m}$$ from $$S$$ and has a speed of $$1 \mathrm{~m} / \mathrm{s}$$. The maximum value of $$\sigma$$ if the electron strikes $$S$$ at $$t=1 \mathrm{~s}$$ is $$\alpha\left[\frac{m \epsilon_0}{e}\right] \frac{C}{m^2}$$, the value of $$\alpha$$ is ___________.

Two charges of $$-4 \mu \mathrm{C}$$ and $$+4 \mu \mathrm{C}$$ are placed at the points $$\mathrm{A}(1,0,4) \mathrm{m}$$ and $$\mathrm{B}(2,-1,5) \mathrm{m}$$ located in an electric field $$\overrightarrow{\mathrm{E}}=0.20 \hat{i} \mathrm{~V} / \mathrm{cm}$$. The magnitude of the torque acting on the dipole is $$8 \sqrt{\alpha} \times 10^{-5} \mathrm{Nm}$$, where $$\alpha=$$ _________.

The electric potential at the surface of an atomic nucleus $$(z=50)$$ of radius $$9 \times 10^{-13} \mathrm{~cm}$$ is __________ $$\times 10^6 \mathrm{~V}$$.