Gauss’s Law 📗

Gauss’s law links the total electric flux flowing out of a closed surface to the total charge tucked away inside that surface. It turns tough field problems into quick symmetry games—perfect for fast problem-solving! 💡

1. Electric Flux 🌀

  • The tiny flux through an area patch \(\Delta S\) is
    \(\displaystyle \Delta\phi \;=\;\mathbf E\!\cdot\!\Delta\mathbf S\) :contentReference[oaicite:0]{index=0}
  • For a sphere of radius \(r\) surrounding a point charge \(q\), every patch sits the same distance from the charge, so the fluxes add to
    \(\displaystyle \phi = \frac{q}{\varepsilon_0}\) :contentReference[oaicite:1]{index=1}

2. Gauss’s Law 📐

\(\displaystyle \boxed{\phi_E \;=\; \frac{q}{\varepsilon_0}}\) :contentReference[oaicite:2]{index=2}

If the net flux through any closed surface is zero, then no net charge hides inside—handy detective work! 🕵️‍♂️ :contentReference[oaicite:3]{index=3}

Why it always works 🌟

  1. Valid for every closed surface—big, small, or quirky-shaped. :contentReference[oaicite:4]{index=4}
  2. \(q\) counts all charges inside the surface, wherever they sit. :contentReference[oaicite:5]{index=5}
  3. The field in the flux includes contributions from charges both inside and outside, but \(q\) records only the inside ones. :contentReference[oaicite:6]{index=6}
  4. Don’t let a Gaussian surface slice through a point charge—fields explode right there! Continuous charge sheets are fine. :contentReference[oaicite:7]{index=7}
  5. Pick a Gaussian surface that matches the problem’s symmetry (spherical, cylindrical, planar) to make the math snap together quickly. 🏆 :contentReference[oaicite:8]{index=8}
  6. Gauss’s law whispers “inverse-square!”—any failure would signal a break from the \(1/r^{2}\) nature of Coulomb’s law. :contentReference[oaicite:9]{index=9}

3. Worked Examples 🔍

Example A — Non-uniform Field inside a Cube 🎲

A cube of side \(a = 0.1\;\text{m}\) sits in a field \(E_x = a\,x^{1/2}\), \(E_y = E_z = 0\) with \(a = 800\;\text{N C}^{-1}\text{m}^{1/2}\).

  • Total flux through the cube: \(1.05\;\text{N·m}^2\text{/C}\).
  • Charge enclosed: \(q = 9.27\times10^{-12}\;\text{C}\).

:contentReference[oaicite:10]{index=10}

Example B — Piece-wise Uniform Field through a Cylinder 🥤

The field is \(+200\,\hat{i}\;\text{N/C}\) for \(x>0\) and \(-200\,\hat{i}\;\text{N/C}\) for \(x<0\). A cylinder (radius \(0.05\;\text{m}\), length \(0.20\;\text{m}\)) straddles the origin.

  • Flux through each flat face: \(1.57\;\text{N·m}^2\text{/C}\).
  • Flux through the curved side: \(0\).
  • Net flux: \(3.14\;\text{N·m}^2\text{/C}\).
  • Charge enclosed: \(q = 2.78\times10^{-11}\;\text{C}\).

:contentReference[oaicite:11]{index=11}

4. High-Yield NEET Nuggets ✅

  • Memorize \( \phi_E = q/\varepsilon_0 \) and what each symbol means.
  • Zero net flux ↔ zero enclosed charge—fast consistency check.
  • Use symmetry to choose smart Gaussian surfaces and skip heavy integrals.
  • Gauss’s law works for every closed surface, whatever its shape.
  • The law rests on the \(1/r^{2}\) dependence of Coulomb’s force—fundamental insight.

Keep these ideas fresh, practice applying Gauss’s law to spheres, cylinders, and planes, and watch your problem-solving speed soar! 🚀