4. Calculer le champ électrostatique ofee par sphère bleine chargée volumiquement avec une densité volumique de charge c const l'espace situé à la distance

The question asks to calculate the electric field produced by a uniformly charged solid sphere with a constant volume charge density ρ (rho, not c) at a distance *r* from the center of the sphere.There are two cases to consider:**1. Inside the sphere (r < R), where R is the radius of the sphere:**We use Gauss's Law. We choose a spherical Gaussian surface with radius *r* concentric with the charged sphere. The electric field is radial and has the same magnitude everywhere on the Gaussian surface.Gauss's Law states:∮E⋅dA = Q_enclosed / ε₀Where:* E is the electric field* dA is a differential area element on the Gaussian surface* Q_enclosed is the charge enclosed by the Gaussian surface* ε₀ is the permittivity of free spaceSince E is constant and radial, the integral simplifies to:E * 4πr² = Q_enclosed / ε₀The enclosed charge Q_enclosed can be calculated using the volume charge density:Q_enclosed = ρ * (4/3)πr³Substituting this into Gauss's Law:E * 4πr² = (ρ * (4/3)πr³) / ε₀Solving for E:E = (ρ * r) / (3ε₀) for r < R**2. Outside the sphere (r ≥ R):**Again, we use Gauss's Law with a spherical Gaussian surface of radius *r* concentric with the charged sphere. This time, the entire charge of the sphere is enclosed within the Gaussian surface.The total charge Q_total of the sphere is:Q_total = ρ * (4/3)πR³Applying Gauss's Law:E * 4πr² = Q_total / ε₀E * 4πr² = (ρ * (4/3)πR³) / ε₀Solving for E:E = (ρ * R³) / (3ε₀r²) for r ≥ R**Final Answer:**The electric field E at a distance *r* from the center of a uniformly charged solid sphere with volume charge density ρ and radius R is:* **E = (ρ * r) / (3ε₀) for r < R*** **E = (ρ * R³) / (3ε₀r²) for r ≥ R**