State and derive formula for Gauss's law....
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State and derive formula for Gauss's law....
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In physics, Gauss's law, also known as Gauss's flux theorem, is a law relating the distribution of electric charge to the resulting electric field. The surface under consideration may be a closed one enclosing a volume such as a spherical surface.
Coulomb's law states that the electric field due to a stationary point change is:
Where,
er is the radial unit vector,
r is a radius, |r|,
∈ is the electric constant,
q is the charge of the particle, which is assumed to be located at the origin.
Using the expression from coulomb's law, we get the total field at r by using an integral to sum the field at r due to the infinitesimal charge at each other point s in space, to give
Where rho is the charge density. It we take the divergence of both sides of this equation with respect to r, and use the known theorem;
Where is the Dirac delta function, the result is,
Using the "sifting property" of the Dirac delta function, we arrive at: