Gauss Law

Last Updated: July 16, 2024

Gauss Law

What Is Gauss Law

Gauss’s Law, one of the four Maxwell’s Equations, is a fundamental principle in electromagnetism formulated by the German mathematician and physicist Carl Friedrich Gauss. It describes the relationship between electric charges and the resulting electric field.

Gauss Law Formula

Gauss’s Law can be expressed mathematically as:

ฮฆ๐ธ = โˆฎโ‚› ๐ธโ‹…๐‘‘๐ด = ๐‘„โ‚‘โ‚™๊œ€โ€‹โ€‹/๐œ–โ‚€

Where:

  • ฮฆ๐ธ is the electric flux through a closed surface S,
  • E is the electric field vector,
  • dA is the vector area of an infinitesimal piece of the surface S,
  • ๐‘„โ‚‘โ‚™๊œ€โ€‹ is the total charge enclosed within the surface,
  • ๐œ–โ‚€โ€‹ is the permittivity of free space, a fundamental physical constant.

Conceptual Understanding

Gauss’s Law states that the total electric flux out of a closed surface is directly proportional to the charge enclosed by that surface. The law implies that:

  • The electric flux through a closed surface is dependent on the charge inside the surface, not on the charges outside it.
  • Fields created by charges inside the surface contribute to the net flux, while external fields, regardless of their strength, do not change the total flux through that surface.

Electric Field Due to Infinite Wire โ€“ Gauss Law Application

The calculation of the electric field due to an infinitely long charged wire is a classic application of Gauss’s Law. It exemplifies how symmetry considerations simplify the analysis of electric fields. Here, we use Gauss’s Law to determine the electric field generated by an infinite line of charge.

Step-by-Step Application of Gauss’s Law

Choose an Appropriate Gaussian Surface: For an infinite wire with uniform linear charge density ๐œ†ฮป (charge per unit length), the electric field is expected to be radial due to the cylindrical symmetry of the arrangement. Therefore, a cylindrical Gaussian surface is chosen, coaxial with the wire. This simplifies the calculation, as the electric field E will have the same magnitude at every point on the cylindrical surface and is always perpendicular to the surface.

Setup Gauss’s Law: Gauss’s Law states that the total electric flux ฮฆ๐ธ through a closed surface is equal to the charge enclosed ๐‘„โ‚‘โ‚™๊œ€โ€‹โ€‹ divided by the permittivity of free space ๐œ–โ‚€โ€‹:

ฮฆ๐ธ = โˆฎโ‚› ๐ธโ‹…๐‘‘๐ด = ๐‘„โ‚‘โ‚™๊œ€โ€‹โ€‹/๐œ–โ‚€

Calculate the Charge Enclosed: The charge enclosed by the Gaussian surface, assuming a length ๐ฟL of the wire within the cylinder, is:

๐‘„โ‚‘โ‚™๊œ€ = ๐œ†๐ฟ

Evaluate the Electric Flux ฮฆ๐ธโ€‹: The cylindrical Gaussian surface has a radius ๐‘Ÿr and length ๐ฟL. The surface area of the cylinderโ€™s side (ignoring the ends as the field lines are tangential there and contribute no flux) is 2๐œ‹๐‘Ÿ๐ฟ. The electric field is constant and perpendicular over this surface, so:

ฮฆ๐ธ = ๐ธโ‹…2๐œ‹๐‘Ÿ๐ฟ

Apply Gauss’s Law: Setting the flux equal to the charge enclosed over ๐œ–โ‚€โ€‹ gives:๐ธโ‹…2๐œ‹๐‘Ÿ๐ฟ=๐œ†๐ฟ/ ๐œ–โ‚€ โ€‹Simplifying, we find the electric field ๐ธ at a distance ๐‘Ÿ from the wire:

Conclusion

The resulting expression for the electric field ๐ธE due to an infinite wire shows that the field’s magnitude decreases with the radial distance ๐‘Ÿr from the wire, inversely proportional to ๐‘Ÿr. This result is intuitive for a linear charge distribution, where the influence of the charge decreases as one moves further away.

This application highlights the power of Gauss’s Law in using symmetry to simplify complex electromagnetic problems, making it possible to derive fields for different charge distributions effectively.

Usage ofย Gauss Law

USAGE-of-Gauss-Law-2
  1. Field Calculations for Symmetric Charge Distributions: Gauss’s Law is particularly effective for calculating the electric fields of charge distributions with high symmetry (spherical, cylindrical, or planar symmetry). By choosing Gaussian surfaces that conform to the symmetry of the charge distribution, the integration over the surface becomes straightforward.
  2. Electrostatic Shielding: Gauss’s Law explains the principle behind electrostatic shielding, where a conductive shell blocks external static electric fields from penetrating a cavity inside the shell, as the net charge enclosed by the inner surface of the shell is zero.
  3. Flux Calculations: The law is used to calculate the flux of electric fields across given surfaces in complex geometries, which is important in diverse applications like electric flux meters and sensors.

Difference between Gauss law and Ampere’s law

FeatureGauss’s LawAmpere’s Law
Fundamental PrincipleRelates the electric flux through a closed surface to the charge enclosed.Relates the magnetic field along a closed loop to the electric current passing through the loop.
Mathematical Formulationฮฆ๐ธ=โˆฎ๐‘†๐ธโ‹…๐‘‘๐ด=๐‘„โ‚‘โ‚™๊œ€๐œ–โ‚€โˆฎ๐ถ๐ตโ‹…๐‘‘๐‘™ = ๐œ‡โ‚€๐ผโ‚‘โ‚™๊œ€ (original form) or โˆฎ๐ถ๐ตโ‹…๐‘‘๐‘™ = ๐œ‡โ‚€๐ผโ‚‘โ‚™๊œ€+๐œ‡โ‚€๐œ–โ‚€๐‘‘ฮฆ๐ธ๐‘‘๐‘ก(Maxwell’s addition)
Type of FieldsDeals with electric fields (E).Deals with magnetic fields (๐ต).
Key VariablesElectric flux (ฮฆ๐ธ), electric field (E), enclosed charge (๐‘„โ‚‘โ‚™๊œ€โ€‹).Magnetic field (๐ต), total current enclosed (๐ผโ‚‘โ‚™๊œ€โ€‹), rate of change of electric flux (used in Maxwell’s addition).
Conservation LawsBased on the conservation of electric charge.Based on the conservation of magnetic flux, enhanced by Maxwell’s addition which accounts for changing electric fields.
Symmetry ConsiderationsEffectively applied where electric charges exhibit symmetrical arrangements, allowing for simplified calculations using Gaussian surfaces.Effectively used in scenarios where currents or magnetic fields exhibit cylindrical or other forms of symmetry, facilitating the use of Amperian loops.

FAQs

What does Gauss’s Law tell us about electric fields?

Gauss’s Law indicates that electric fields are generated by charges and that the total electric flux through a closed surface is directly proportional to the total charge enclosed within that surface. This law is crucial for determining the behavior of electric fields in various charge distributions.

Can Gauss’s Law be applied to any closed surface?

Yes, Gauss’s Law can be applied to any closed surface. However, the choice of surface (Gaussian surface) affects the ease with which the law can be used to calculate electric fields. Symmetrical charge distributions often allow for simpler calculations when an appropriately symmetrical Gaussian surface is chosen.

How does Gauss’s Law relate to electrostatic shielding?

Gauss’s Law explains the phenomenon of electrostatic shielding, where a conductive material can block external static electric fields. Since the net charge enclosed by a closed surface within a conductor is zero when in electrostatic equilibrium, no electric field exists inside the conductor, shielding anything inside it from external electric fields.

Is Gauss’s Law applicable to varying electric fields?

While Gauss’s Law is typically associated with static (unchanging) electric fields, it is also a part of Maxwell’s Equations, which describe all electromagnetic phenomena, including dynamic fields. However, when dealing with time-varying fields, additional terms from Maxwell’s equations might be needed to fully describe the situation.

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