Electric Field of a Non Uniformly Charged Sphere

Spherical symmetry occurs when charge density depends only on distance from the center, not direction. This means the electric field at any point is radially directed and invariant under rotation. For a non-uniformly charged sphere, this symmetry allows us to use Gaussian surfaces that match the charge distribution's geometry, simplifying electric field calculations significantly.

The net charge is found by integrating the charge in infinitesimal spherical shells throughout the sphere. Each shell's charge equals its charge density multiplied by its volume. For a point outside the sphere, only shells within the sphere contribute. For a point inside, only shells within the Gaussian surface contribute to the net charge calculation.

A Gaussian surface with the same symmetry as the charge distribution simplifies field calculations. When the Gaussian surface matches the charge distribution's spherical symmetry, the electric field is constant on the surface. This allows direct application of Gauss's law: the electric flux through the surface equals the enclosed charge divided by permittivity, yielding the field magnitude.

Outside the sphere, the net charge equals the total charge integrated over the entire sphere's radius. Inside the sphere, the net charge equals only the charge integrated up to the Gaussian surface's radius. The region between the sphere and external Gaussian surface contains no charge and doesn't contribute to the field outside.

Since charge density varies with distance from the center, the charge enclosed within each infinitesimal shell differs. This variation requires integration to determine total enclosed charge. The resulting electric field depends on this integrated charge distribution, making the field calculation more complex than for uniform charge distributions.

Infinitesimal shells allow us to account for non-uniform charge density by dividing the sphere into thin layers. Each shell's charge is calculated as density times volume. Integrating these shell charges over the sphere's radius gives the total enclosed charge, which is then substituted into the electric field equation to find the field at any point.

Gauss's law states that electric flux through any closed surface equals the net enclosed charge divided by permittivity. For a spherically symmetric non-uniformly charged sphere, the flux through a Gaussian surface depends on the integrated charge within it. This relationship allows direct calculation of electric field once the enclosed charge is determined through integration.