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.<br /><br />There are two cases to consider:<br /><br />**1. Inside the sphere (r < R), where R is the radius of the sphere:**<br /><br />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.<br /><br />Gauss's Law states:<br /><br />∮E⋅dA = Q_enclosed / ε₀<br /><br />Where:<br /><br />* E is the electric field<br />* dA is a differential area element on the Gaussian surface<br />* Q_enclosed is the charge enclosed by the Gaussian surface<br />* ε₀ is the permittivity of free space<br /><br />Since E is constant and radial, the integral simplifies to:<br /><br />E * 4πr² = Q_enclosed / ε₀<br /><br />The enclosed charge Q_enclosed can be calculated using the volume charge density:<br /><br />Q_enclosed = ρ * (4/3)πr³<br /><br />Substituting this into Gauss's Law:<br /><br />E * 4πr² = (ρ * (4/3)πr³) / ε₀<br /><br />Solving for E:<br /><br />E = (ρ * r) / (3ε₀) for r < R<br /><br />**2. Outside the sphere (r ≥ R):**<br /><br />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.<br /><br />The total charge Q_total of the sphere is:<br /><br />Q_total = ρ * (4/3)πR³<br /><br />Applying Gauss's Law:<br /><br />E * 4πr² = Q_total / ε₀<br /><br />E * 4πr² = (ρ * (4/3)πR³) / ε₀<br /><br />Solving for E:<br /><br />E = (ρ * R³) / (3ε₀r²) for r ≥ R<br /><br /><br />**Final Answer:**<br /><br />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:<br /><br />* **E = (ρ * r) / (3ε₀) for r < R**<br />* **E = (ρ * R³) / (3ε₀r²) for r ≥ R**<br />