1. Gauss's Law — Statement and Formula
Gauss's Law: The total electric flux through any closed surface is equal to the total charge enclosed by that surface divided by ε₀.
- The closed surface is called the Gaussian surface — it is an imaginary mathematical surface, not a physical one.
- Q_enc = total charge inside the surface (charges outside do not contribute to the net flux).
- The law holds for any closed surface, but is only useful when E is constant in magnitude and makes a constant angle with the surface — which happens only with high symmetry.
Strategy for Applying Gauss's Law
- Identify the symmetry of the charge distribution (spherical, cylindrical, planar).
- Choose a Gaussian surface matching that symmetry so E is constant over the surface.
- Write Φ = E × A (surface area of Gaussian surface).
- Set equal to Q_enc / ε₀ and solve for E.
2. Application 1 — Infinite Line Charge (Cylindrical Symmetry)
Setup: Infinite wire with uniform linear charge density λ (C/m).
Gaussian surface: Coaxial cylinder of radius r and length L.
Flux through curved surface = E × 2πrL (E is perpendicular to curved surface, parallel to flat ends).
Q_enc = λL
Direction: Radially outward (if λ > 0).
Dependence: E ∝ 1/r (falls off slower than a point charge).
Example: λ = 5 μC/m, r = 0.1 m:
E = 2 × 9×10⁹ × 5×10⁻⁶ / 0.1 = 9.0 × 10⁵ N/C
3. Application 2 — Infinite Plane Sheet (Planar Symmetry)
Setup: Infinite plane with uniform surface charge density σ (C/m²).
Gaussian surface: A pillbox (cylinder) straddling the sheet, each face of area A parallel to the sheet.
Flux = 2EA (from both flat faces; curved sides contribute zero).
Q_enc = σA
E is uniform — independent of distance from the sheet.
Between two oppositely charged parallel plates (conductor plates): E = σ/ε₀ (fields add); outside: E = 0 (fields cancel).
4. Application 3 — Uniformly Charged Spherical Shell
Setup: Thin shell of total charge Q, radius R.
| Region | Gaussian Surface | Q_enc | E |
|---|---|---|---|
| Outside (r > R) | Sphere radius r | Q | |
| Inside (r < R) | Sphere radius r | 0 | E = 0 |
| On surface (r = R) | — | Q | E = kQ/R² (maximum) |
The field inside a charged spherical shell is zero — a key result used in shielding and in defining potential inside the shell.
5. Application 4 — Uniformly Charged Solid Sphere
Setup: Solid insulating sphere of total charge Q, radius R, uniform volume charge density ρ.
| Region | Q_enc | E |
|---|---|---|
| Outside (r > R) | Q (entire sphere) | |
| Inside (r < R) | Q × (r/R)³ (fraction of volume) | |
| On surface (r = R) | Q | E = kQ/R² (maximum) |
Inside a solid sphere: E ∝ r (increases linearly from centre; zero at centre).
Outside: E ∝ 1/r² (falls off like a point charge).
At r = R, both expressions give the same E — no discontinuity.
Example: Q = 8 nC, R = 10 cm. Find E at r = 5 cm (inside):
E_inside = kQr/R³ = (9×10⁹ × 8×10⁻⁹ × 0.05) / (0.1)³ = 3.6 / 10⁻³ = 3.6 × 10³ N/C
At surface (r = R = 10 cm): E = kQ/R² = 9×10⁹ × 8×10⁻⁹ / (0.1)² = 7.2 × 10³ N/C ✓ (double, as expected)
6. Properties of Conductors in Electrostatic Equilibrium
- E = 0 inside a conductor — all free charges redistribute on the surface until internal field cancels.
- All charge resides on the surface — applying Gauss's Law to any surface inside the conductor: Q_enc = 0.
- E at surface = σ/ε₀, perpendicular to surface.
- Electrostatic shielding: Interior of a hollow conductor is shielded from external electric fields.
