Six point charges are kept $60^{\circ}$ apart from each other on the circumference of a circle of radius $R$ as shown in figure. The net electric field at the center of the circle is $\_\_\_\_$ .

( $\epsilon_0$ is permittivity of free space)

Consider two identical metallic spheres of radius $R$ each having charge $Q$ and mass $m$. Their centers have an initial separation of $4R$. Both the spheres are given an initial speed of $u$ towards each other. The minimum value of $u$, so that they can just touch each other is:

(Take $k = \frac{1}{4 \pi \epsilon_0}$ and assume $kQ^2 > Gm^2$ where $G$ is the Gravitational constant)

A point charge of $10^{-8} \mathrm{C}$ is placed at origin. The work done in moving a point charge $2 \mu \mathrm{C}$ from point $A(4,4,2) \mathrm{m}$ to $B(2,2,1) \mathrm{m}$ is $\_\_\_\_$ J. $\left(\frac{1}{4 \pi \epsilon_{\mathrm{o}}}=9 \times 10^9\right.$ in SI units)

Two metal spheres of radius R and 3R have same surface charge density σ. If they are brought in contact and then separated, the surface charge density on smaller and bigger sphere becomes σ₁ and σ₂, respectively. The ratio $ \frac{\sigma_1}{\sigma_2} $ is