ADVANCED ELECTROMAGNETISM
A Deep Dive into Non-Uniform Volume Charge Density & Gauss's Law
Problem Statement
Consider a non-conducting solid sphere of radius R with a non-uniform volume charge density
ρ. The density varies with the distance r from the center according to:
ρ(r) = ρ∞ (1 - r/R) for r ≤ R
Calculate the electric field E for a point located inside the sphere (r < R).
Step 1: Establishing Gauss's Law
By symmetry, the electric field is radial. We choose a spherical Gaussian surface of radius r (where
r < R). Gauss's Law states:
∮ E · dA = Q_enclosed / ε_0
For a sphere, the surface area is 4πr², thus:
E(4πr²) = Q_enclosed / ε_0
Step 2: Calculating Enclosed Charge (Integration)
Since charge density is a function of radius, we integrate over thin spherical shells of volume dV =
4πr'²dr':
Q_enclosed = ∫ ρ(r') dV = 4πρ∞ ∫ [ (r')² - (r')³/R ] dr'
Evaluating the integral from 0 to r:
A Deep Dive into Non-Uniform Volume Charge Density & Gauss's Law
Problem Statement
Consider a non-conducting solid sphere of radius R with a non-uniform volume charge density
ρ. The density varies with the distance r from the center according to:
ρ(r) = ρ∞ (1 - r/R) for r ≤ R
Calculate the electric field E for a point located inside the sphere (r < R).
Step 1: Establishing Gauss's Law
By symmetry, the electric field is radial. We choose a spherical Gaussian surface of radius r (where
r < R). Gauss's Law states:
∮ E · dA = Q_enclosed / ε_0
For a sphere, the surface area is 4πr², thus:
E(4πr²) = Q_enclosed / ε_0
Step 2: Calculating Enclosed Charge (Integration)
Since charge density is a function of radius, we integrate over thin spherical shells of volume dV =
4πr'²dr':
Q_enclosed = ∫ ρ(r') dV = 4πρ∞ ∫ [ (r')² - (r')³/R ] dr'
Evaluating the integral from 0 to r:
