word_count: 300

Gauss's law combined with spherical symmetry yields elegant results for spheres and shells. The Gaussian surface is always a concentric sphere of radius r.

Uniformly charged solid non-conducting sphere (total charge Q, radius R): The volume charge density rho = 3Q/(4piR^3). For the Gaussian sphere at radius r, the enclosed charge is Q_enc = Q(r/R)^3 (for r < R) or Q (for r > R).

Outside (r > R): E * 4pir^2 = Q/epsilon_0, giving E = kQ/r^2. The sphere behaves exactly like a point charge at its center for all external points.

Inside (r < R): E * 4pir^2 = Q(r/R)^3/epsilon_0, giving E = kQr/R^3. The field increases linearly from zero at the center to kQ/R^2 at the surface. This is analogous to gravitational field inside Earth.

Uniformly charged thin spherical shell: Outside (r > R): E = kQ/r^2 (point charge behavior). Inside (r < R): E = 0 (no enclosed charge). The entire field is concentrated outside. This result is responsible for electrostatic shielding — a Faraday cage protects its interior from external electric fields.

Concentric shells: Apply Gauss's law in each region separately. In each conducting shell material, E = 0, which determines induced surface charges. For inner shell charge Q1 and outer shell charge Q2: the inner surface of the outer shell has charge -Q1 (induced), and its outer surface has Q2 + Q1 (by conservation). The field in any region depends only on the total charge enclosed by a Gaussian surface in that region.

Non-uniform charge density rho(r): Integrate Q_enc = integral of rho(r) * 4pir^2 * dr from 0 to r, then apply Gauss's law. Common JEE variation: rho = rho_0(1 - r/R) or rho = rho_0(r/R).