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Gauss' Theorem

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Estimated time: 5 minutes
CISCE: Class 12

Introduction

Gauss' Theorem is one of the four fundamental laws of electromagnetism and a powerful mathematical tool for calculating electric fields in symmetric charge distributions. It connects the concept of electric flux to the charge enclosed within a closed surface, and is derived as a logical consequence of Coulomb's Law.

CISCE: Class 12

Gauss' Theorem

Statement:

The total electric flux through any closed surface is equal to \[\frac{1}{\varepsilon_0}\]​ times the net electric charge enclosed by that surface.

This holds true regardless of the shape or size of the closed surface, and irrespective of the position of the charge within it.

Mathematical Formulation

In free space (vacuum/air):

\[\Phi_E=\oint\vec{E}\cdot d\vec{A}=\frac{q}{\varepsilon_0}\]   (1)
In a dielectric medium (dielectric constant K):
\[\Phi_E=\oint\vec{E}\cdot d\vec{A}=\frac{q}{\varepsilon_0K}\]   (2)
Where:
  • \[\vec{E}\] = electric field vector
  • \[d\vec{A}\] = outward area vector element of the surface
  • q = net charge enclosed by the surface
  • ε0 = 8.85 × 10−12 C2/(N m2)
CISCE: Class 12

Key Points: Gauss Theorem

  • ΦE ​= q/ε0​ (vacuum); ΦE = q/(ε0K) (dielectric medium)
  • Flux depends only on net enclosed charge, not surface shape/size.
  • Useful for symmetric bodies: sphere, cylinder, infinite sheet.
  • Derived from Coulomb's Law — a generalized statement of it.
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