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Gauss' Law

Gauss’s Law is the law relating the distribution of electric charge to the electric field which results. Gauss’s Law was first stated by Carl Friedrich Gauss in 1835. However, it was not published until 1867.¹ Gaus' law is one of the four Maxwell equations. These equations are the basis of classical electrodynamics. Gauss’s Law can be used to derive Coulomb’s Law.

Gauss’s Law States: The net outward normal electric flux through any closed surface is proportional to the total electric charge enclosed within that closed surface.²

Gauss’s law is an electrical analogue of Ampere’s law which deals with magnetism. Both of these forms are equivalent since they are related by Gauss’s theorem. Each of these forms can be expressed in two different ways; in terms of a relation between the electric field E and the total electric charge or in terms of the electric displacement field D and the free electric charge.

Integral form of Gauss’s Law stated using the electric field E

ΦE = Q / Ɛ0

Differential for of Gauss’s Law stated using the electric field E

∇ * E = ρ/Ɛo

Integral form of Gauss’s Law stated using the displacement field D

ΦD = Qfree

Differential form of Gauss’s Law stated using the displacement field D

ΦD = ∫∫D * dA



1. Bellone, Enrico (1980). A World on Paper: Studies on the Second Scientific Revolution.

2. Serway, Raymond A. (1996). Physics for Scientists and Engineers with Modern Physics, 4th edition. p. 687

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