6.S: Gauss's Law (Summary) (2024)

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    Key Terms

    area vector vector with magnitude equal to the area of a surface and direction perpendicular to the surface
    cylindrical symmetry system only varies with distance from the axis, not direction
    electric flux dot product of the electric field and the area through which it is passing
    flux quantity of something passing through a given area
    free electrons also called conduction electrons, these are the electrons in a conductor that are not bound to any particular atom, and hence are free to move around
    Gaussian surface any enclosed (usually imaginary) surface
    planar symmetry system only varies with distance from a plane
    spherical symmetry system only varies with the distance from the origin, not in direction

    Key Equations

    Definition of electric flux, for uniform electric field \(\displaystyle Φ=\vec{E}⋅\vec{A}→EAcosθ\)
    Electric flux through an open surface \(\displaystyle Φ=∫_S\vec{E}⋅\hat{n}dA=∫_S\vec{E}⋅d\vec{A}\)
    Electric flux through a closed surface \(\displaystyle Φ=∮_S\vec{E}⋅\hat{n}dA=∮_S\vec{E}⋅d\vec{A}\)
    Gauss’s law \(\displaystyle Φ=∮_S\vec{E}⋅\hat{n}dA=\frac{q_{enc}}{ε_0}\)
    Gauss’s Law for systems with symmetry \(\displaystyle Φ=∮_S\vec{E}⋅\hat{n}dA=E∮_SdA=EA=\frac{q_{enc}}{ε_0}\)
    The magnitude of the electric field just outside the surface of a conductor \(\displaystyle E=\frac{σ}{ε_0}\)

    Summary

    6.2 Electric Flux

    • The electric flux through a surface is proportional to the number of field lines crossing that surface. Note that this means the magnitude is proportional to the portion of the field perpendicular to the area.
    • The electric flux is obtained by evaluating the surface integral

    \(\displaystyle Φ=∮_S\vec{E}⋅\hat{n}dA=∮_S\vec{E}⋅d\vec{A}\),

    where the notation used here is for a closed surface S.

    6.3 Explaining Gauss’s Law

    • Gauss’s law relates the electric flux through a closed surface to the net charge within that surface,

    \(\displaystyle Φ=∮_S\vec{E}⋅\hat{n}dA=\frac{q_{enc}}{ε_0}\),

    • where qencqenc is the total charge inside the Gaussian surface S.
    • All surfaces that include the same amount of charge have the same number of field lines crossing it, regardless of the shape or size of the surface, as long as the surfaces enclose the same amount of charge.

    6.4 Applying Gauss’s Law

    • For a charge distribution with certain spatial symmetries (spherical, cylindrical, and planar), we can find a Gaussian surface over which \(\displaystyle \vec{E}⋅\hat{n}=E\), where E is constant over the surface. The electric field is then determined with Gauss’s law.
    • For spherical symmetry, the Gaussian surface is also a sphere, and Gauss’s law simplifies to \(\displaystyle 4πr^2E=\frac{q_{enc}}{ε_0}\).
    • For cylindrical symmetry, we use a cylindrical Gaussian surface, and find that Gauss’s law simplifies to \(\displaystyle 2πrLE=\frac{q_{enc}}{ε_0}\).
    • For planar symmetry, a convenient Gaussian surface is a box penetrating the plane, with two faces parallel to the plane and the remainder perpendicular, resulting in Gauss’s law being \(\displaystyle 2AE=\frac{q_{enc}}{ε_0}\).

    6.5 Conductors in Electrostatic Equilibrium

    • The electric field inside a conductor vanishes.
    • Any excess charge placed on a conductor resides entirely on the surface of the conductor.
    • The electric field is perpendicular to the surface of a conductor everywhere on that surface.
    • The magnitude of the electric field just above the surface of a conductor is given by \(\displaystyle E=\frac{σ}{ε_0}\).

    Contributors and Attributions

    Samuel J. Ling (Truman State University),Jeff Sanny (Loyola Marymount University), and Bill Moebswith many contributing authors. This work is licensed by OpenStax University Physics under aCreative Commons Attribution License (by 4.0).

    6.S: Gauss's Law (Summary) (2024)

    FAQs

    6.S: Gauss's Law (Summary)? ›

    6.3 Explaining Gauss's Law

    What is the summary of Gauss law? ›

    Gauss's law for electricity states that the electric flux Φ across any closed surface is proportional to the net electric charge q enclosed by the surface; that is, Φ = q/ε0, where ε0 is the electric permittivity of free space and has a value of 8.854 × 1012 square coulombs per newton per square metre.

    What is the conclusion of the Gauss law? ›

    Conclusion:-

    It asserts that the flux of the electric field out of any closed surface is proportional to the electric charge enclosed by the surface, regardless of how that charge is distributed, in its integral form.

    What describes Gauss's law? ›

    Gauss's Law is a fundamental law of electromagnetism that relates the electric flux through a closed surface to the electric charge enclosed within the surface. It is named after the German mathematician and physicist Carl Friedrich Gauss. The mathematical form of Gauss's Law is: ∮E⋅dA = Q/ε₀

    What is the state of Gauss's law in words? ›

    In words, Gauss's law states: The net electric flux through any hypothetical closed surface is equal to 1/ε0 times the net electric charge enclosed within that closed surface. The closed surface is also referred to as Gaussian surface.

    What is the simplify of Gauss's law? ›

    The electric field is then determined with Gauss's law. For spherical symmetry, the Gaussian surface is also a sphere, and Gauss's law simplifies to 4πr2E=qencε0. For cylindrical symmetry, we use a cylindrical Gaussian surface, and find that Gauss's law simplifies to 2πrLE=qencε0.

    What is Gauss theorem in short? ›

    Gauss Theorem in electrostatics

    Gauss Law states that the total electric flux out of a closed surface is equal to the charge enclosed divided by the permittivity. The electric flux in an area is defined as the electric field multiplied by the area of the surface projected in a plane and perpendicular to the field.

    Why is the Gauss law important? ›

    Gauss law is used to quickly find electric field due to various arrangement like charged sheet, thin infinite straight charged wire., etc. It comes handy to find fields in cases where symmetry is strongly present.

    What are the points of Gauss law? ›

    (i) If a closed body not enclosing any charge is placed in either uniform on non-uniform electric field total flux linked with it will be zero. (ii) If a closed body encloses a charge q, total flux linked with the body is independent of the shape and size of the body and position of charge inside it.

    What is the essence of the Gauss law? ›

    Gauss's Law assumes a symmetrical charge distribution and it states that the net electric flux through a closed surface is equal to the integral of the electric field over that surface, which is also equivalent to the total charge enclosed by that surface divided by the permittivity of space.

    What is the characteristic of gauss law? ›

    Gauss' law tells us that the flux is equal to the charge Q, over the permittivity of free space, epsilon-zero. But flux is also equal to the electric field E multiplied by the area of the surface A. So EA equals Q over epsilon-zero. For a sphere, the surface area is given by 4pir^2, so we can plug that in for A.

    What is gauss law and its application? ›

    Gauss's theorem: It states that electric flux ϕE through any closed surface is equal to 1ε0 times the net charge 'q' enclosed by the surface. ϕε=∮E. dA=qε0. Two applications: (1) For finding out electric field due to different shapes i.e., infinite line of charge.

    What is Gauss's principle state? ›

    Complete answer:The competitive exclusion principle is also known as Gause's law. The law states that when two species compete for one limited resource, the population cannot coexist with the same population growth or same growth status.

    What is the correct statement of the gauss law? ›

    (i)-Gauss's law states that total electric flux out of a closed surface is equal to charge enclosed divided by permittivity. It does not depend on any shape and size. Statement (i) is correct.

    What can Gauss's law be used to find? ›

    Gauss'(s) Law is used to find the electric field for charge distributions which have a symmetry which we can exploit in calculating both sides of the equation: ∮ E · dA and qenc/ϵ0. Of course, we already know how to get the magnitude and direction of the electric field due to a point charge q.

    What is the important point about gauss law? ›

    (i) If a closed body not enclosing any charge is placed in either uniform on non-uniform electric field total flux linked with it will be zero. (ii) If a closed body encloses a charge q, total flux linked with the body is independent of the shape and size of the body and position of charge inside it.

    What is the Gauss theorem for dummies? ›

    Gauss's Law states that the electric flux passing through a closed surface is equal to the ratio of total charge enclosed by that surface to the permittivity of free space. This means that the electric flux passing through a closed surface is independent to shape or area of the surface.

    What is the gauss law in electrostatics and explain its important? ›

    Gauss Law is a basic law which is applicable for all closed surfaces. It is an important law because it allows the assessment of the amount of enclosed electric charge. It also does the mapping of the field on a surface outside the charge distribution.

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