The image you uploaded contains two problems related to electrostatics. Let's break down each one:

Part I:

This problem describes a system of two discs, $D_1$ and $D_2$, with the same radius $R$ and centers at $O_1$ and $O_2$, respectively. The discs are uniformly charged, with surface charge densities $\sigma$ and $-\sigma$, where $\sigma$ is positive and $-\sigma$ is negative.

• The discs are aligned along the $x$-axis, with their centers at $O_1$ and $O_2$, separated by a distance $O_1O_2 = 2D$.
• The goal is to express the electrostatic field at the center of the distribution in terms of the charge densities and distance.

The electric field at the center of the system (point $O$, midpoint of the line connecting $O_1$ and $O_2$) will depend on the charge distribution and geometry. The field contributions from both discs must be considered, and superposition will be used to compute the resulting field.

We’ll need to use the formula for the electric field due to a uniformly charged disc, which is given by:

$E = \frac{\sigma}{2\epsilon_0} \left( 1 - \frac{z}{\sqrt{z^2 + R^2}} \right)$

where $\sigma$ is the surface charge density, $R$ is the radius of the disc, $z$ is the distance from the disc along the axis perpendicular to the surface, and $\epsilon_0$ is the permittivity of free space.

However, for two discs with opposite charge densities, the fields will add up in the region near the center. The distance between the discs, and the symmetry of the problem, will play a role in the overall calculation.

Part II:

This part asks you to:

1. State Kirchhoff's Current Law (possibly in the context of electrostatics) and show that it is a consequence of the conservation of electric charge.
2. State Kirchhoff’s Voltage Law (in terms of electrostatics) and show that it arises from the fact that the electric field is conservative (i.e., it derives from a potential).

Kirchhoff’s laws are essential for analyzing electrical circuits, and in the context of electrostatics:

• Kirchhoff's Current Law (KCL) reflects the conservation of electric charge, which states that the total current entering a junction equals the total current leaving the junction.
• Kirchhoff's Voltage Law (KVL) applies to closed loops in electrostatics, where the integral of the electric field around a closed path is zero, implying that the electric field is conservative and can be derived from a potential.

How to proceed with the solution:

1. For Part I, the electric field at the center of the system can be computed by considering the contributions from both discs and using superposition.
2. For Part II, we can derive Kirchhoff’s laws by examining the conservation of charge and the conservative nature of the electric field.

Would you like me to continue solving this problem and go into more detail for each part?