A tiny ball of mass $$\mathrm{m}$$ and charge $$\mathrm{q}$$ is suspended from the fixed support using an insulating string of length $$1 \mathrm{~m}$$. The horizontal uniform electric field $$\mathrm{E}$$ is switched on. The angle made by the string with vertical when the ball is in equilibrium is $$45^{\circ}$$. The magnitude of uniform electric field is

Figure shows three arrangements of electric field lines. In each arrangement, a proton is released from rest at point $$\mathrm{P}$$ and then accelerated through point $$\mathrm{Q}$$ by the electric field. Points $$\mathrm{P}$$ and $$\mathrm{Q}$$ have equal separations in the three arrangements. If $$p_1 p_2$$ and $$p_3$$ are linear momentum of the proton at point $$\mathrm{Q}$$ in the three arrangement respectively, then

3 point charges each of $$-\mathrm{q}$$ are placed on the circumference of a circle of diameter $$2 \mathrm{a}$$ at $$\mathrm{A}, \mathrm{B}$$ and $$\mathrm{C}$$ respectively as shown in figure. The electric field at $$\mathrm{O}$$ is

A uniformly charged solid sphere of radius $$\mathrm{R}$$ has potential $$\mathrm{V}_0$$ (measured with respect to infinity) on its surface. For this sphere the equipotential surfaces with potentials $$\frac{3 \mathrm{~V}_0}{2}, \frac{\mathrm{V}_0}{1}, \frac{3 \mathrm{~V}_0}{4}$$ and $$\frac{\mathrm{V}_0}{4}$$ have radius $$\mathrm{R}_1, \mathrm{R}_2, \mathrm{R}_3$$ and $$\mathrm{R}_4$$ and respectively, then