Gauss's Law with Planar Symmetry
Planar symmetry is the third major type of symmetry we encounter when applying Gauss's Law. When a charge distribution has planar symmetry, the electric field at any point depends only on the perpendicular distance from the plane.
Key Properties of Planar Symmetry
- 1
The electric field points perpendicular to the plane (outward for positive charge, inward for negative charge).
- 2
For an infinite uniformly charged plane, the electric field magnitude is constant and does not depend on the distance from the plane!
- 3
A cylindrical Gaussian surface with its axis perpendicular to the plane is the natural choice for applying Gauss's Law.
Key Insight: Notice that the electric field strength is the same at all distances from the plane! This is different from point charges (1/r²) and line charges (1/r).
Example: Infinite Uniformly Charged Plane
Problem: Electric Field Due to an Infinite Charged Plane
Calculate the electric field at any point near an infinite plane with uniform surface charge density σ.
Given:
- Infinite plane with uniform surface charge density σ
- We want to find the electric field at any distance from the plane
Special Case: Parallel Charged Plates
Parallel Plate Capacitor: When two oppositely charged plates are placed parallel to each other, the electric field between them is uniform and given by E = σ/ε₀ (twice the field of a single plate).
Important Properties:
- • The electric field between the plates is uniform (same everywhere)
- • The field is directed from the positive plate to the negative plate
- • The voltage between the plates is V = Ed, where d is the separation
- • This arrangement forms a capacitor, which can store electric charge
Applications of Planar Symmetry
Capacitors
Parallel Plate Capacitors
These store electric charge and energy in electronic circuits. The uniform electric field between the plates is a direct application of Gauss's Law for planar symmetry.
Variable Capacitors
By changing the separation or overlap area of the plates, the capacitance can be varied. This is used in tuning circuits for radios and other electronic devices.