Electric Field of a Plane of Charge

Symmetry and Field Patterns

When analyzing the electric field produced by a uniformly charged infinite plane (e.g., a sheet in the xz-plane at y = 0), symmetry considerations are crucial for determining the possible field configurations.

• Key Symmetries: The field must be invariant under translation along the x and z axes, rotation by 180° about an axis perpendicular to the plane (the y-axis), and reflection in the plane itself.

• Field Direction: The electric field produced by a positively charged infinite plane points perpendicularly away from the plane on both sides (along the y-axis).

• Consistent Field Patterns: Only field configurations that respect these symmetries are physically possible for an infinite plane of charge.

Example: Field B (as shown in the slides) is consistent with the symmetries of the plane, as it points uniformly away from the plane along the y-axis. Field A, which radiates outward in all directions, does not match the required symmetry.

Electric Field of Spherical and Cubical Charge Distributions

Symmetry of Charge Distributions

Charge distributions with different geometries (spherical vs. cubical) produce electric fields with distinct symmetry properties.

• Spherical Shell: A uniformly charged spherical shell centered at the origin has full rotational symmetry about any axis through the center and reflection symmetry in any plane containing the center.

• Cubical Shell: A uniformly charged cube centered at the origin (with faces parallel to the axes) has symmetry under rotation by 90° about the x, y, and z axes, and reflection in planes parallel to these axes.

• Field Consistency: The electric field produced by each distribution must share the same symmetries as the charge distribution itself.

Example: Field I (with field lines pointing along the axes) is consistent with the cube's symmetry, while a radially symmetric field is consistent with the sphere.

The Concept of Electric Flux

Definition and Physical Meaning

Electric flux quantifies the number of electric field lines passing through a given surface. In electrostatics, a Gaussian surface is an imaginary closed surface used to apply Gauss's law.

• Electric Flux (): The total electric field passing through a surface.

• Gaussian Surface: A closed surface (not necessarily physical) used to relate the net electric flux to the enclosed charge.

• Flux and Charge: The net electric flux through a closed surface is proportional to the net charge enclosed by that surface.

Examples:

• Outward flux through a closed surface around a net positive charge.

• Inward flux through a closed surface around a net negative charge.

• Zero net flux through a closed surface enclosing no net charge.

Ranking Electric Flux Through Gaussian Surfaces

Application of Gauss's Law

When multiple charges are arranged and different Gaussian surfaces are considered, the net electric flux through each surface depends only on the net charge enclosed.

• Gauss's Law: , where is the net charge enclosed.

• Ranking: Surfaces enclosing negative charge have negative flux; those enclosing positive charge have positive flux; those enclosing zero net charge have zero flux.

Example: For three spheres enclosing charges , , and , the sphere enclosing has the smallest (most negative) flux, and those enclosing have equal, positive flux.

Calculating Electric Flux – Uniform Field

Mathematical Expression

For a flat surface in a uniform electric field, the electric flux is given by the dot product of the electric field vector and the area vector.

• Formula:

• is the area vector, normal to the surface, with magnitude equal to the area .

• is the angle between and .

• If is perpendicular to the surface (), .

• If is parallel to the surface (), .

Example: For , , and (surface tilted from the -plane), the flux is:

Generalization: Non-Uniform Fields and Integrals

Integral Form of Electric Flux

For non-uniform electric fields or curved surfaces, the total flux is calculated by integrating over the surface:

• This integral sums the contributions from infinitesimal area elements .

• For closed surfaces, always points outward.

Summary Table: Symmetry and Electric Fields

Key Terms and Definitions

• Electric Field (): A vector field representing the force per unit charge at each point in space.

• Electric Flux (): A measure of the number of electric field lines passing through a surface.

• Gaussian Surface: An imaginary closed surface used in Gauss's law to relate electric flux to enclosed charge.

• Gauss's Law: The net electric flux through a closed surface equals the net charge enclosed divided by the permittivity of free space (): .