Determine the strength $E$ of the electric field at the centre of a hemisphere produced by charges uniformly distributed with a density, $\sigma$ over the surface of this hemisphere.

The strength of the electric field produced by charges uniformly distributed over the surface of a hemisphere at its centre $O$ is $E_0$. A part of the surface is isolated from

this hemisphere by two planes passing through the same diameter and forming an angle $\alpha$ with each other determine the electric field strength $E$ produced at the same point $O$ by the charges located on the isolated surface (on the "mericarp").

Two parallel-plate capacitors are arranged perpendicular to the common axis. The separation, $d$ between the capacitors, is much larger than the separation, $l$ between their plates and their size. The capacitors are charged to $q_1$ and $q_2$ respectively. Find the force, $F$ of interaction between the capacitors. 

Determine the force $F$ of interaction between two hemispheres of radius $R$ touching each other along the equator if one hemisphere is uniformly charged with a surface density ${\sigma }_1$ and the other with a surface density ${\sigma }_2$. 

The minimum strength of a uniform electric field that can tear a conducting uncharged thin-walled sphere into two parts is known to be $E_0$. Determine the minimum electric field strength $E_1$ required to tear the sphere of twice as large radius if the thickness of its walls is the same as in the former case. 

Three small identical neutral metal balls are at the vertices of an equilateral triangle. The balls are in turn connected to an isolated large conducting sphere whose centre is on the perpendicular erected from the plane of the triangle and passing through its centre. As a result, the first and second balls have acquired charges, $q_1$ and $q_2$ respectively. Determine the charge $q_3$ of the third ball.

A metal sphere of radius $r_1$ charged to potential $V_1$ is then placed in a thin-walled uncharged conducting spherical shell of radius $r_2$. Determine the potential acquired by the spherical shell after it has been connected for a short time to the sphere by a conductor.

A very small earthed conducting sphere is at a distance $a$ from a point charge ${\dot{q}}_1$and at a distance $b$ from a point charge $q_2\left(a<b\right)$. At a certain instant, the sphere starts expanding so that its radius grows according to the law $R=\nu t$. Determine the time dependence $I\left(t\right)$ of the current in the earthing conductor, assuming that the point charges and the centre of the sphere are at rest, and in due time the initial point charges get into the expanding sphere without touching it (through small holes).

Three uncharged capacitors of capacitance $C_1,C_2$, and $C_3$ are connected as shown in Fig. to one another and to points $A,B$, and $D$ at potentials ${\varphi }_A, {\varphi }_B$, and ${\varphi }_D$. Determine the potential ${\varphi }_0$ at point $O$.