Gauss's law is a powerful tool for calculating electric fields in symmetric charge distributions. It simplifies complex problems by exploiting symmetry, allowing us to determine fields for spheres, cylinders, and planes.

Understanding charge densities is crucial when applying Gauss's law. Linear, surface, and volume charge densities help describe different types of charge distributions, making it easier to solve real-world electromagnetic problems.

Charge Distributions

Spherical Charge Distributions

• Spherical charge distributions have charges uniformly distributed over the surface of a sphere
• Can be a solid sphere with charge distributed throughout the volume or a thin spherical shell with charge only on the surface
• Electric field outside a spherical charge distribution is equivalent to a point charge at the center with total charge equal to the sphere's charge
• Electric field inside a solid sphere varies linearly with distance from the center, reaching zero at the center

Cylindrical Charge Distributions

• Cylindrical charge distributions have charges uniformly distributed over the surface of an infinitely long cylinder
• Can be a solid cylinder with charge distributed throughout the volume or a thin cylindrical shell with charge only on the surface
• Electric field outside a cylindrical charge distribution is proportional to the inverse of the distance from the axis
• Electric field inside a solid cylinder is proportional to the distance from the axis

Planar Charge Distributions

• Infinite plane of charge has a uniform charge distribution over an infinitely large flat surface
• Electric field is constant and perpendicular to the plane, with magnitude proportional to the surface charge density
• Direction of the electric field depends on the sign of the charge distribution (positive or negative)
• Non-uniform charge distributions have varying charge densities over the surface or volume
• Gauss's law can still be applied, but the electric field calculations become more complex due to the non-uniformity

Charge Densities

Linear Charge Density

• Linear charge density ($\lambda$) is the charge per unit length along a line or curve
• Measured in coulombs per meter (C/m)
• Used to describe the charge distribution in one-dimensional objects like wires or thin rods
• Electric field near a long, thin wire is proportional to the linear charge density and inversely proportional to the distance from the wire

Surface Charge Density

• Surface charge density ($\sigma$) is the charge per unit area on a surface
• Measured in coulombs per square meter (C/m²)
• Used to describe the charge distribution on two-dimensional surfaces like planes, spheres, or cylinders
• Electric field near a uniformly charged surface is proportional to the surface charge density and independent of the distance from the surface

Volume Charge Density

• Volume charge density ($\rho$) is the charge per unit volume within a three-dimensional object
• Measured in coulombs per cubic meter (C/m³)
• Used to describe the charge distribution in three-dimensional objects like solid spheres or cylinders
• Electric field inside a uniformly charged volume depends on the volume charge density and the distance from the center or axis of symmetry

Gauss's Law Applications

Shell Theorem

• Shell theorem states that a uniformly charged spherical shell attracts or repels external charges as if all its charge were concentrated at its center
• Applies to both gravitational and electric fields
• Implies that the electric field inside a uniformly charged spherical shell is zero
• Allows simplification of electric field calculations for spherically symmetric charge distributions

Spherical Charge Distributions

• Gauss's law can determine the electric field for uniformly charged spheres and spherical shells
• For a solid sphere with uniform volume charge density, the electric field inside the sphere is proportional to the distance from the center
• For a thin spherical shell with uniform surface charge density, the electric field inside the shell is zero, and the field outside is equivalent to a point charge at the center
• Gauss's law simplifies the calculation by exploiting the symmetry of the charge distribution

Cylindrical Charge Distributions

• Gauss's law can determine the electric field for uniformly charged infinite cylinders and cylindrical shells
• For a solid cylinder with uniform volume charge density, the electric field inside the cylinder is proportional to the distance from the axis
• For a thin cylindrical shell with uniform surface charge density, the electric field inside the shell is zero, and the field outside is inversely proportional to the distance from the axis
• Gauss's law simplifies the calculation by choosing a cylindrical Gaussian surface that aligns with the symmetry of the charge distribution

Infinite Plane of Charge

• Gauss's law can determine the electric field for a uniformly charged infinite plane
• Electric field is constant and perpendicular to the plane, with magnitude proportional to the surface charge density
• Direction of the electric field depends on the sign of the charge distribution (positive or negative)
• Gauss's law simplifies the calculation by choosing a Gaussian surface that passes through the plane, with the electric field parallel to the surface normal