Introduction to Charged Sphere

Overview

In this lecture, we thoroughly understood the concepts, formulas, and graphs of the electric field generated by a charged sphere, including cases of conducting, non-conducting, hollow, and solid spheres.

Types of Charged Spheres

• Spheres are of two types: hollow (shell) and solid.
• Both types can be conducting or non-conducting (dielectric/insulator).
• There are four total cases: hollow conducting, hollow non-conducting, solid conducting, solid non-conducting.

Electric Field: Hollow/Solid Conducting Sphere (Charge on Surface)

• In all these cases, the charge resides only on the surface.
• Charge density (σ): charge per unit area on the surface (Q/A).
• Electric field inside (r < R): E = 0 (from Gauss Law, Q_inside = 0).
• Electric field outside (r > R): ( E = \dfrac{1}{4\pi\epsilon_0} \dfrac{Q}{r^2} ).

Electric Field: Solid Non-conducting Sphere (Charge in Volume)

• Charge is uniformly distributed throughout the volume.
• Charge density (ρ): charge per unit volume (Q/V).
• Outside (r > R): ( E = \dfrac{1}{4\pi\epsilon_0} \dfrac{Q}{r^2} ).
• Inside (r < R): ( E = \dfrac{1}{4\pi\epsilon_0} \dfrac{Q}{R^3} r ), i.e., E ∝ r.

Non-uniform Charge Distribution (ρ = ρ₀r)

• If charge density varies with radius, then dq = ρ₀x·4πx²dx.
• Electric field inside: ( E = \dfrac{\rho_0}{4\epsilon_0} r^2 ), i.e., E ∝ r².
• The outside formula remains the same, but Q_calculated = ( \rho_0 \pi R^4 ).

Graphical Analysis of Electric Field

• Hollow/conducting sphere: E is zero inside, maximum at surface, outside ∝ 1/r².
• Solid non-conducting sphere: E increases inside with r, decreases outside as 1/r².
• Behaves like a point charge: from outside, all charge can be considered located at the center.

Key Terms & Definitions

• Gauss Law — Total flux through a closed surface = Q_inside/ε₀.
• Surface Charge Density (σ) — Charge per unit area on the surface.
• Volume Charge Density (ρ) — Charge per unit volume in the volume.
• Conductor — Charge always resides on the surface.
• Non-conductor/Dielectric — Charge can be distributed in the volume.

Action Items / Next Steps

• In the next lecture, solve problems based on non-uniform charge distribution.
• Memorize the previous formulas and graphs well.
• Read related NCERT topics and solved examples.