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Boundless Physics
Electric Charge and Field
Electric Flux and Gauss's Law
Physics Textbooks Boundless Physics Electric Charge and Field Electric Flux and Gauss's Law
Physics Textbooks Boundless Physics Electric Charge and Field
Physics Textbooks Boundless Physics
Physics Textbooks
Physics
Concept Version 6
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Gauss's Law

Gauss's law is a law relating the distribution of electric charge to the resulting electric field.

Learning Objective

  • Describe relationship between the Gauss's law and the Coulomb's law


Key Points

    • Gauss's law is one of the four Maxwell's equations which form the basis of classical electrodynamics.
    • Gauss's law can be used to derive Coulomb's law, and vice versa.
    • Gauss's law states that: The net outward normal electric flux through any closed surface is proportional to the total electric charge enclosed within that closed surface.

Terms

  • electric field

    A region of space around a charged particle, or between two voltages; it exerts a force on charged objects in its vicinity.

  • electric charge

    A quantum number that determines the electromagnetic interactions of some subatomic particles; by convention, the electron has an electric charge of -1 and the proton +1, and quarks have fractional charge.

  • electric displacement field

    A vector field that appears in Maxwell's equations.


Full Text

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 law was formulated by Carl Friedrich Gauss (see ) in 1835, but was not published until 1867. It is one of the four Maxwell's equations which form the basis of classical electrodynamics, the other three being Gauss's law for magnetism, Faraday's law of induction, and Ampère's law with Maxwell's correction.

Carl Friedrich Gauss

Carl Friedrich Gauss (1777–1855), painted by Christian Albrecht Jensen

Gauss's law can be used to derive Coulomb's law, and vice versa. Note that since Coulomb's law only applies to stationary charges, there is no reason to expect Gauss's law to hold for moving charges based on this derivation alone. In fact, Gauss's law does hold for moving charges, and in this respect Gauss's law is more general than Coulomb's law.

In words, Gauss's law states that: The net outward normal electric flux through any closed surface is proportional to the total electric charge enclosed within that closed surface.

The law can be expressed mathematically using vector calculus in integral form and differential form, both are equivalent since they are related by the divergence theorem, also called Gauss's theorem. Each of these forms in turn can also be expressed two 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.

Gauss's law has a close mathematical similarity with a number of laws in other areas of physics, such as Gauss's law for magnetism and Gauss's law for gravity. In fact, any "inverse-square law" can be formulated in a way similar to Gauss's law: For example, Gauss's law itself is essentially equivalent to the inverse-square Coulomb's law, and Gauss's law for gravity is essentially equivalent to the inverse-square Newton's law of gravity.

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