Applications of Gauss' Theorem > Electric Field due to a Uniformly Charged Sphere

A uniformly charged sphere is a sphere in which charge q is distributed evenly throughout its volume (non conducting/insulating) or over its surface (conducting), with uniform volume charge density ρ = \[\frac{q}{\frac{4}{3}\pi R^3}.\]

Gauss's Law states:

This law is applied to a spherical Gaussian surface to determine the electric field at three distinct regions — outside, at the surface, and inside the sphere.

Assumptions:

• Sphere has radius R, total charge q, uniformly distributed.
• Charge density ρ is constant throughout the volume (for non-conducting case).
• Gaussian surface chosen is a concentric sphere of radius r.

Case 1: Outside the Sphere (r > R)

Entire charge q is enclosed by the Gaussian surface, so it behaves as a point charge at the centre.

Key Point: Field outside varies as 1/r², identical to a point charge — independent of how charge is distributed inside.

Case 2: At the Surface (r = R)

Key Point: This is the maximum field value for a uniformly charged insulating sphere.

Case 3: Inside the Sphere (r < R) — Non-Conducting Only

Only the charge enclosed within radius r contributes.

Key Point: E ∝ r — field increases linearly from zero at centre to maximum at surface.

• From r = 0 to r = R: straight line through origin (linear increase).
• At r = R: peak value E_max = \[\frac {ρR}{3ε_0}\]​.
• For r > R: smooth curve decaying as 1/r².
• Note: For a spherical shell, field inside is zero throughout (flat line at E = 0 till r = R, then 1/r² fall).

Think of the sphere's charge like water pressure inside a filled balloon: pressure (analogous to field contribution) builds up gradually as you move outward from the centre toward the surface, then dissipates in open air following an inverse-square-like spread — helping visualize why E grows steadily inside but fades farther outside.