Electric Field at the Center of a Regular Pentagon

To determine the electric field at the center of a regular pentagon due to identical point charges at its corners, we can follow these steps:

Step 1: Understand the Problem
- We have identical point charges of magnitude Q placed at the corners of a regular pentagon.
- Our goal is to calculate the electric field at the center of the pentagon.

Step 2: Analyze the Symmetry
- The regular pentagon possesses rotational symmetry.
- All charges are equidistant from the center of the pentagon.
- Due to the symmetry, the electric fields created by each charge will cancel each other out along the axis passing through the center of the pentagon.
- Hence, the electric field at the center of the pentagon will be zero along this axis.

Step 3: Consider the Electric Field Components
- We need to analyze the electric field components along the axis perpendicular to the line connecting the center of the pentagon with any of its corners.
- Let's consider a single charge at one of the corners and denote it as Q1.
- We need to calculate the electric field created by Q1 at the center of the pentagon and then find the components along the x and y axes.

Step 4: Calculate the Electric Field Components
- The electric field created by a point charge at a distance r is given by Coulomb's law: E = kQ/r^2, where k is the electrostatic constant.
- The distance between the charge at the corner and the center of the pentagon is the length of the side of the pentagon, denoted as a.
- The magnitude of the electric field created by Q1 at the center of the pentagon is E1 = kQ/a^2.
- Considering the symmetry, the x-component of the electric field created by Q1 is zero.
- The y-component of the electric field created by Q1 is also zero due to symmetry.

Step 5: Summing Up the Electric Fields
- As mentioned earlier, the electric fields created by each charge at the corners of the pentagon will cancel each other out along the axis passing through the center of the pentagon.
- Since the electric field components along both the x and y axes are zero, the net electric field at the center of the pentagon is also zero.

Conclusion
The electric field at the center of a regular pentagon due to identical point charges at its corners is zero. This is because the electric fields created by each charge cancel each other out along the axis passing through the center of the pentagon. The symmetry of the pentagon ensures that the electric field components along both the x and y axes sum up to zero, resulting in a net electric field of zero at the center.

Answer is zero if any regular shape with same sigh of charges has zero electric field at center