Relation between E1 and E2 for a Charged Conductor and an Infinite Thin Plane Sheet

The electric field intensity near a point and outside the surface of a charged conductor of any shape is denoted by E1. On the other hand, the electric field intensity due to a uniformly charged infinite thin plane sheet is denoted by E2. Let's discuss the relation between E1 and E2 in detail.

1. Electric Field Intensity near a Point and Outside the Surface of a Charged Conductor (E1):
- When a conductor is charged, the excess charge resides on its outer surface. Inside the conductor, the electric field is zero.
- Near a point outside the surface of the conductor, the electric field intensity is directly proportional to the surface charge density (σ) and inversely proportional to the square of the distance (r) from the point to the center of the conductor.
- Mathematically, E1 ∝ σ/r^2, where E1 is the electric field intensity, σ is the surface charge density, and r is the distance from the point to the center of the conductor.

2. Electric Field Intensity due to a Uniformly Charged Infinite Thin Plane Sheet (E2):
- An infinite thin plane sheet has a uniform charge density (σ) spread uniformly across its surface.
- The electric field intensity due to a uniformly charged infinite thin plane sheet is constant and independent of the distance from the sheet.
- Mathematically, E2 = σ/2ε0, where E2 is the electric field intensity, σ is the charge density, and ε0 is the permittivity of free space.

3. Relation between E1 and E2:
- The relation between E1 and E2 can be derived by considering a Gaussian surface just outside the surface of the conductor.
- According to Gauss's law, the total electric flux through a closed surface is equal to the net charge enclosed by that surface divided by the permittivity of free space (ε0).
- Since the electric field inside a conductor is zero, the electric flux through the Gaussian surface is zero.
- As a result, the net charge enclosed by the Gaussian surface is zero.
- Therefore, the electric field intensity near a point and outside the surface of a charged conductor (E1) is equal to the electric field intensity due to a uniformly charged infinite thin plane sheet (E2).
- Hence, E1 = E2.

In conclusion, the relation between E1 and E2 is that they are equal. This is because the electric field intensity near a point and outside the surface of a charged conductor and the electric field intensity due to a uniformly charged infinite thin plane sheet have the same magnitude.