Gauss's Law

Gauss's Law relates the electric flux through a closed surface to the charge enclosed within the surface, divided by the permittivity.

Definition

• Introduced by Carl Friedrich Gauss in 1867.
• Describes the static electric field from a distribution of electric charges.
• Total electric charge enclosed by a closed surface is proportional to the electric flux through the surface.
• Positive charges produce a positive electric field.

Applications of Gauss's Law

• Useful for solving complex electrostatic problems with symmetrical shapes such as cylindrical, spherical, or planar.
• Simplifies calculations of electric fields which otherwise require extensive integration.

Steps for Using Gauss's Law

1. Choose a Gaussian surface that simplifies the evaluation of the electric field.
2. Utilize symmetry to ease calculations.
3. The Gaussian surface may be inside or outside the real surface.

Electric Field Calculations

Due to Infinite Wire

• Use cylindrical Gaussian surface due to wire's symmetry.
• Electric field (E) is radial; flux through the ends of the cylinder is zero due to perpendicular area vectors.

Due to Infinite Plate Sheet

• The electric field is uniform and outward in a cylindrical Gaussian surface.

Due to Thin Spherical Shell

• Outside the spherical shell: Electric field calculated based on symmetry.
• Inside the spherical shell: Different consideration due to absence of enclosed charge.

Types of Symmetry in Gauss's Law

1. Spherical Symmetry

• Charge density only varies with distance, not direction.
• Example: Sphere with uniform or sectional charge densities.

2. Cylindrical Symmetry

• Charge density depends only on distance from the cylinder's axis.
• Electric field direction varies based on charge polarity (positive/negative).

3. Planar Symmetry

• Charges evenly distributed on a flat surface.
• Electric field components parallel to the charge plane cancel out; field is perpendicular to the plane.

Using Gauss's Law for Electric Fields

Spherical Shell

• Inside and outside electric fields determined by charge enclosed and permittivity.

Cylindrical Distribution

• Electric field magnitude at distance 's' from axis is proportional to charge per unit length (λ).

Planar Distribution

• Planar symmetry results in vertical electric fields at any point due to symmetrical charge cancellation.

Conclusion

• Gauss's Law links electrostatic field behavior with charge distribution.
• Excess charge resides on the surface of electrostatic conductors.
• Valid when conductors are not in motion (current free).