Deriving Electric Field Using Gauss's Law

To derive the electric field of a uniformly charged sphere using Gauss's Law, follow these steps:

1. Consider a sphere of radius R with a uniform charge distribution and total charge Q.
2. To find the electric field outside the sphere (at a distance r where r > R), choose a Gaussian surface that is a sphere of radius r centered at the same point as the charged sphere.
3. By symmetry, the electric field E will be radial and have the same magnitude at every point on the Gaussian surface.
4. According to Gauss's Law: ∮ E · dA = Q_enclosed / ε₀
5. Here, Q_enclosed is Q (the total charge of the sphere), and dA is the differential area element of the Gaussian surface.
6. The electric flux ∮ E · dA becomes E · 4πr² (since the surface area of a sphere is 4πr²).
7. Therefore, E · 4πr² = Q / ε₀.
8. Solving for E, we get E = Q / (4πr²ε₀).
9. Thus, the electric field outside a uniformly charged sphere is E = Q / (4πr²ε₀), and it behaves like the field of a point charge.