Consider a uniformly charged non-conducting cube of volume charge density $\rho .$ If the potential at one of the vertex is $4$ volt, find the potential at the centre of the cube (in volts).

Figure shows two conducting thin concentric shells of radii $r$ and $3r.$ The outer shell carries charge $q$ and the inner shell is neutral. The amount of charge that flows from the inner shell to the earth after the key $K$ is closed is equal to ${\left(\frac{1}{n}\right)}^{\mathrm{th}}$ of the charge on the outer shell. What is the value of $n?$

A frustum is cut from a right circular cone. The two circular faces have radii $R$and $2R$ and their centers are at $O_1$ and $O_2$ respectively. Height of the frustum is $h=3R$. When a point charge $Q$ is placed at $O_1$, the flux of electric field through the circular face of radius $2R$ is ${\varphi }_1$and when the charge $Q$ is placed at $O_2$ , the flux through the other circular face is ${\varphi }_2.$The ratio of $\left(\frac{{\varphi }_1}{{\varphi }_2}\right)$ is found to be $\left(\frac{\sqrt{13}-3}{\sqrt{10}-3}\right)\left(\sqrt{\frac{a}{b}}\right)$. The value of $(a+b)$ is

A particle having charge that of an electron and mass $1.6\times {10}^{-30} \mathrm{k}\mathrm{g}$ is projected with an initial speed $u$ at an angle ${45}^{\mathrm{o}}$ to the horizontal from the lower plate of a parallel plate capacitor as shown in the figure. The plates are sufficiently long and have separation $2 \mathrm{c}\mathrm{m}$. The maximum value of velocity of particle not to hit the upper plate is $\sqrt{W}\times {10}^6 \mathrm{m} {\mathrm{s}}^{-1}$. Take electric field between the plates as ${10}^3 \mathrm{V} {\mathrm{m}}^{-1}$ directed upward. Find the $W$.