Guass 's Law

4. Applications of Gauss's Law

(A) Derivation of Coulumb'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.

    Derivation of Coulumb's Law from Gauss's law

  • 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 Coulomb's law.

(B) Electric field due to line charge

  • Consider a long thin uniformly charged wire and we have to find the electric field intensity due to the wire at any point at perpandicular distance from the wire.
  • If the wire is very long and we are at point far away from both its ends then field lines outside the wire are radial and would lie on a plane perpandicular to the wire.
  • Electric field intensity have same magnitude at all points which are at same distance from the line charge.
  • We can assume Gaussian surface to be a right circular cylinder of radius r and length l with its ends perpandicular to the wire as shown below in the figure.

    Cylinderical Gaussian surface for the electric field calculation for line charge

  • λ is the charge per unit length on the wire. Direction of E is perpendicular to the wire and components of E normal to end faces of cylinder makes no contribution to electric flux. Thus from Gauss's law

  • Now consider left hand side of Gauss's law

    Since at all points on the curved surface E is constant. Surface area of cylinder of radius r and length l is A=2πrl therefore,

  • Charge enclosed in cylinder is q=linear charge density x length l of cylinder,
    or, q=λl
    From Gauss's law

    Thus electric field intensity of a long positively charged wire does not depends on length of the wire but on the radial distance r of points from the wire.

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