Gauss's law: statement and proof Explained with Examples
Gauss's law: statement and proof is a core Electrostatics concept in Physics. This guide explains what it is, walks through a fully worked example, and lists the key equations you need — with a short quiz to test yourself.
Key equations and worked example
Two 5 µC charges, opposite signs, 10 cm apart: F = k·q₁q₂/r² = (9×10⁹)(5×10⁻⁶)²/(0.1)² ≈ 22.5 N attractive. Flip one sign and the same 22.5 N becomes repulsive — the field lines redraw themselves instantly. The field midway between opposite charges is E ≈ 2kq/r², pointing from + to −.
- <code>Coulomb's law: F = k·q₁·q₂ / r²</code>
- <code>Electric field: E = F/q = k·Q / r²</code>
- <code>Potential: V = k·Q / r</code>
- <code>Field at conductor surface: E = σ/ε₀</code>
Gauss's law: statement and proof in detail
Gauss's law: statement and proof is one of the central ideas in Electrostatics, and it appears in Physics curricula under Gauss's law and applications. It is worth learning deeply because it connects to so many other topics in this section.
Static charges exert Coulomb forces along the line joining them — like charges repel, opposites attract. The electric field E = F/q maps the force a test charge would feel; field lines leave positive charges and enter negative ones, never crossing. Conductors in equilibrium carry charge only on their surface, with zero field inside.
For exams, the pattern is predictable: first a definition or statement of the result, then a direct numerical application of one of the equations above, then a "why" question — why the formula takes that form, or what changes when a variable is doubled or halved. The worked example and quiz below cover exactly that progression.
Quick self-check:
- Q: Doubling the distance between two charges changes the force by what factor?<br />A: It drops to one-quarter — Coulomb's law is inverse-square.
- Q: Which way do electric field lines point?<br />A: Away from positive charges and toward negative charges; they never cross.
- Q: Why is the electric field zero inside a charged hollow conductor?<br />A: Free charges repel to the outer surface; Gauss's law then gives zero enclosed charge, so zero field inside.