Field inside uniformly charged sphere

Given: ρ = 3.2 μC/m³ = 3.2 × 10⁻⁶ C/m³, sphere radius R = 5 cm, field point r = 3 cm (inside, r < R).

Method (Gauss's law)

Take a concentric spherical Gaussian surface of radius r < R. Only the charge inside it counts:

q_enc = ρ (4/3)πr³

By symmetry E is radial and constant on the surface:

E (4πr²) = q_enc/ε0 = ρ (4/3)πr³ / ε0

E = ρ r / (3ε0)

Substitution

r = 3 cm = 0.03 m, ε0 = 8.854 × 10⁻¹² C²/N·m²

E = (3.2 × 10⁻⁶ × 0.03) / (3 × 8.854 × 10⁻¹²) E = 9.6 × 10⁻⁸ / 2.656 × 10⁻¹¹ E = 3.61 × 10³ N/C

Answer: E ≈ 3.6 × 10³ N/C (V/m), directed radially outward.