Derivation of Coulumb's Law from Gauss's Law

  • Coulumb's law can be derived from Gauss's law.
  • Consider electric field of a single isolated positive charge of magnitude q as shown below in the figure.

  • Field of a positive charge is in radially outward direction everywhere and magnitude of electric field intensity is same for all points at a distance r from the charge.
  • We can assume Gaussian surface to be a sphere of radius r enclosing the charge q.
  • From Gauss's law

    since E is constant at all points on the surface therefore,

    surface area of the sphere is A=4πr2
  • Now force acting on point charge q' at distance r from point charge q is

    This is nothing but the mathematical statement of Coulumb's law.
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