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:

Φ𝐸 = ∮ₛ 𝐸⋅𝑑𝐴 = 𝑄ₑₙ꜀​​/𝜖₀

Where:

• Φ𝐸 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,
• 𝑄ₑₙ꜀​ is the total charge enclosed within the surface,
• 𝜖₀​ 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 – 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 𝜆λ (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 Φ𝐸 through a closed surface is equal to the charge enclosed 𝑄ₑₙ꜀​​ divided by the permittivity of free space 𝜖₀​:

Φ𝐸 = ∮ₛ 𝐸⋅𝑑𝐴 = 𝑄ₑₙ꜀​​/𝜖₀

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

𝑄ₑₙ꜀ = 𝜆𝐿

Evaluate the Electric Flux Φ𝐸​: The cylindrical Gaussian surface has a radius 𝑟r and length 𝐿L. The surface area of the cylinder’s side (ignoring the ends as the field lines are tangential there and contribute no flux) is 2𝜋𝑟𝐿. The electric field is constant and perpendicular over this surface, so:

Φ𝐸 = 𝐸⋅2𝜋𝑟𝐿

Apply Gauss’s Law: Setting the flux equal to the charge enclosed over 𝜖₀​ gives:𝐸⋅2𝜋𝑟𝐿=𝜆𝐿/ 𝜖₀ ​Simplifying, we find the electric field 𝐸 at a distance 𝑟 from the wire:

Conclusion

The resulting expression for the electric field 𝐸E due to an infinite wire shows that the field’s magnitude decreases with the radial distance 𝑟r from the wire, inversely proportional to 𝑟r. 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 of Gauss Law

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

Feature	Gauss’s Law	Ampere’s Law
Fundamental Principle	Relates 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 Formulation	Φ𝐸=∮𝑆𝐸⋅𝑑𝐴=𝑄ₑₙ꜀𝜖₀	∮𝐶𝐵⋅𝑑𝑙 = 𝜇₀𝐼ₑₙ꜀ (original form) or ∮𝐶𝐵⋅𝑑𝑙 = 𝜇₀𝐼ₑₙ꜀+𝜇₀𝜖₀𝑑Φ𝐸𝑑𝑡(Maxwell’s addition)
Type of Fields	Deals with electric fields (E).	Deals with magnetic fields (𝐵).
Key Variables	Electric flux (Φ𝐸), electric field (E), enclosed charge (𝑄ₑₙ꜀​).	Magnetic field (𝐵), total current enclosed (𝐼ₑₙ꜀​), rate of change of electric flux (used in Maxwell’s addition).
Conservation Laws	Based 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 Considerations	Effectively 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.