This is a physics problem involving an infinite charged plane with a circular hole, as shown in the diagram. The goal is to find the electric field at a point $P$, located at a height $z$ above the center of the hole. Here's how to approach the solution:

Step-by-Step Solution

1. Electric Field of an Infinite Charged Plane:

   • For an infinite plane with uniform surface charge density $\sigma$, the electric field at any point above or below the plane is given by: $E_{\text{plane}} = \frac{\sigma}{2\epsilon_0}$ where $\epsilon_0$ is the permittivity of free space. The direction of the electric field is perpendicular to the surface.

2. Effect of the Hole:

   • The hole in the plane can be treated as if there is a superposition of two fields:
     • The field due to the complete plane (without the hole).
     • The field due to a disk of radius $R$ with the same charge density $-\sigma$ (opposite to that of the plane) to cancel out the field in that region.

3. Electric Field Due to the Disk:

   • The electric field at a height $z$ above the center of a disk with radius $R$ and surface charge density $-\sigma$ is: $E_{\text{disk}} = \frac{\sigma}{2\epsilon_0} \left(1 - \frac{z}{\sqrt{z^2 + R^2}}\right)$ The negative sign indicates that the field is opposite to that of the infinite plane.

4. Net Electric Field at Point $P$:

   • The net electric field $E_{\text{net}}$ at point $P$ (above the hole) is the sum of the field due to the infinite plane and the field due to the disk: $E_{\text{net}} = E_{\text{plane}} + E_{\text{disk}}$ Substituting the expressions: $E_{\text{net}} = \frac{\sigma}{2\epsilon_0} + \frac{\sigma}{2\epsilon_0} \left(-1 + \frac{z}{\sqrt{z^2 + R^2}}\right)$ Simplify: $E_{\text{net}} = \frac{\sigma}{2\epsilon_0} \cdot \frac{z}{\sqrt{z^2 + R^2}}$

Therefore, the electric field at point $P$ is: $E_{\text{net}} = \frac{\sigma z}{2\epsilon_0 \sqrt{z^2 + R^2}}$

Summary

The electric field at height $z$ above the center of the hole in the infinite plane is: $E = \frac{\sigma z}{2\epsilon_0 \sqrt{z^2 + R^2}}$

This expression shows how the electric field varies with height $z$ and the radius of the hole $R$.

Would you like more details or have any questions?

Related Questions:

1. What happens to the electric field when $z$ is much larger than $R$?
2. How does the electric field behave as $z$ approaches zero?
3. If the surface charge density $\sigma$ were doubled, how would the electric field change?
4. How can we use Gauss's Law to derive the electric field of an infinite plane without a hole?
5. What is the potential difference between point $P$ and the surface of the plane?

Tip:

When dealing with superposition problems, breaking down the scenario into simpler, additive fields (like plane and disk) can simplify the computation significantly.