Field due to a uniformly charged infinite plane sheet
In this section, we analyze the behavior of the electric field generated by an infinite plane sheet that carries a uniform surface charge density, denoted by . The symmetry of the configuration implies that the electric field will have important properties.
Properties of the Electric Field
- Symmetry: The electric field does not depend on the coordinates parallel to the sheet (y and z), and is directed perpendicular (along the x-axis) to the plane.
- Gaussian Surface: We can use a Gaussian surface, typically a rectangular parallelepiped or cube, to calculate the electric field. The two opposite faces of the surface through which the field lines pass contribute to the total flux, while the others do not, due to the field being parallel to these faces.
- By Gauss's law, the total electric flux through the Gaussian surface is proportional to the charge enclosed within it. The enclosed charge is equal to the surface charge density multiplied by the area of the Gaussian surfaces.
Gauss's Law Calculation
Using Gauss's law = q/ε_0, we establish the following relationship:
- The flux through the surfaces is given as 2EA, where E is the electric field and A the area.
- The total charge within the surface is given by = σA, where σ is the charge density.
Thus, we derive that:
- Electric Field Equation: E = σ / (2ε_0)
Direction: The electric field points away from the sheet if the surface charge density is positive and towards the sheet if negative. Importantly, the electric field remains constant and does not vary with distance from the sheet, which is unique compared to point charges.
This section's conclusion emphasizes the electric field's independence from distance and its constant nature near uniformly charged infinite surfaces, which is pivotal in electrostatics.