Electric Field and Gauss's Law

Intuition of Gauss's Law

🎈 Think of a Balloon

①Wrap a charge in a balloon — total electric field lines through it stay constant

②Changing the balloon's size or shape doesn't change the total flux!

③This is Gauss's law: flux through a closed surface = enclosed charge / ε₀

Gaussian Surface and Flux

Charge Q4 C

💡 Key Observations

①Larger charge → more flux (arrows) through the surface

②Positive: outgoing flux; negative: incoming flux

③Charges outside do not affect the total flux!

Gauss's Law Formula

Gauss's Law

∮ E · dA = Qε₀

Flux through closed surface = enclosed charge / vacuum permittivity

Spherical Symmetry

E × 4πr² = Qε₀ → E = Q4πε₀r² = kQr²

For spherical symmetry, Gauss's law derives Coulomb's law!

🔑 Power of Gauss's Law

①Solves highly symmetric problems very easily

②Spherical: point charge, uniform charged sphere

③Cylindrical: infinite line of charge, coaxial cylinders

Worked Examples

Example 1

A closed Gaussian surface encloses charges of +2Q and −Q. What is the total electric flux through it?

By Gauss’s law, the total flux depends only on the enclosed net charge.

Φ = Q_encε₀

Substitute the net enclosed charge Q_enc = 2Q − Q = Q.

Φ = 2Q - Qε₀ = Qε₀

▸ Q/ε₀

Total flux is set only by the net enclosed charge. Opposite-sign charges add (cancel).

Example 2

If the radius of a spherical Gaussian surface around a point charge Q is doubled, how does the total electric flux change?

Gauss’s law depends only on the enclosed charge, not the surface size or shape.

Φ = Qε₀ (independent of r)

The enclosed charge Q is unchanged, so the flux is unchanged.

Φ(2r) = Qε₀ = Φ(r)

▸ No change

Total flux is independent of surface size (only charge matters). But the field E on the surface drops as 1/r² — keep the two distinct.

Summary

Gauss's Law (Core)

∮ E · dA = Q_enc / ε₀

ε₀ = 8.85×10⁻¹² C²/(N·m²)

CSAT-style

In vacuum the field at distance r from a point charge Q is E. What is the field at distance 2r?

①2E

②E2

③E4

④4E

⑤E

▸ ③ E4

By Gauss’s law (spherical symmetry), E = kQ/r², inversely proportional to the square of distance.

E = kQr²

If r doubles, r² quadruples → E is 1/4.

E(2r) = kQ(2r)² = E4

🎯 Exam Points

①Gauss's law: ∮E·dA = Q/ε₀ (total flux through closed surface)

②Choose Gaussian surface using symmetry so E is constant on it

③Spherical sym → spherical surface → E = kQ/r²

④E inside a conductor = 0 (charges only on surface!)

⑤Gauss's law is mathematically equivalent to Coulomb's law

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