Gauss's law for electricity

Gauss's law for electricity is one of Maxwell's four equations and a fundamental principle of electromagnetism. This law describes how electric charges create an electric field. It provides a way to calculate the distribution of electric charge in a given volume when the electric field is known, or vice versa.

Gauss's law states that the total electric flux through a closed surface is equal to the total charge enclosed by that surface divided by the electric constant (also called the permittivity of free space). Mathematically, this can be expressed as the integral form of Gauss's law:

∮ E · dA = Q_enclosed / ε₀

Here:

• ∮ E · dA is the electric flux through a closed surface.
• Q_enclosed is the total charge enclosed within the surface.
• ε₀ is the permittivity of free space.

Let us understand these concepts in more detail:

Understanding electric current

The electric flux through a surface is a measure of the number of electric field lines that pass through that surface. It is a way of describing the size and strength of the electric field in a given area. In simple terms, if you imagine the electric field as a flow of invisible lines, then the electric flux can be thought of as how many of these lines pass through a given area. It is calculated using the integral:

Φ_E = ∫ E · dA

The dot product E · dA means that we are looking at the component of the electric field that is perpendicular to the field dA.

Understanding closed surface

A closed surface is one that completely encloses a volume, such as the surface of a sphere or cube. These surfaces are important in Gauss's law because they allow us to enclose the charge and use the law to determine properties of the electric field, such as its strength or distribution.

The sphere is shown as a closed surface in the figure. Electric field lines can pass through this surface, and Gauss's law helps us relate the flow of these lines to any charge present inside the sphere.

Permeability of free space

The permittivity of free space, denoted by ε₀, is a constant that describes how electric fields interact with the vacuum. It is a proportionality factor that appears in many equations of electromagnetism, including Gauss's law. Its value is approximately:

ε₀ ≈ 8.85 × 10⁻¹² F/m (farads per meter)

Application of Gauss's law

Gauss's law is particularly useful when dealing with problems that have a high degree of symmetry, such as spherical, cylindrical or plane symmetry. In such cases, it can greatly simplify the process of finding the electric field.

For example, consider a point charge q. The electric field of a point charge is radial and decreases with the square of the distance from the charge. To find the electric field using Gauss's law, we can use a spherical closed surface with the charge at the center.

Given that the electric field E is radial and uniform over the surface, the flux is simply:

Φ_E = E × 4πr²

From Gauss's law, Φ_E = q / ε₀, hence:

E × 4πr² = q / ε₀

By simplification we get:

E = q / (4πε₀r²)

This equation shows that the electric field decreases as the square of the distance from the point charge.

Examples of the use of Gauss's law

Example 1: Uniformly charged sphere

Consider a uniformly charged sphere of radius R and total charge Q We want to find the electric field both inside and outside the sphere.

Out of area

For a point located at a distance r outside the sphere (where r > R) the sphere can be treated as a point charge. Using a spherical Gaussian surface:

E × 4πr² = Q / ε₀

Solving this gives:

E = Q / (4πε₀r²)

Inside the area

For a point located inside the sphere at a distance r (where r < R), the enclosed charge is proportional to the volume of the sphere of radius r.

Q_enclosed = (Q / (4/3)πR³) × (4/3)πr³ = Q × (r³/R³)

Use of Gauss's law:

E × 4πr² = (Q × (r³/R³)) / ε₀

Solving for E gives:

E = (Q × r) / (4πε₀R³)

This indicates that the electric field inside a uniformly charged sphere varies linearly with distance from the centre.

Example 2: Infinite plane sheet of charge

Consider an infinite plane sheet with uniform surface charge density σ.

Due to symmetry, the electric field must be perpendicular to the surface and uniform in magnitude. We use a cylindrical Gaussian surface, which extends the same distance above and below the sheet.

Electric flux passes through the upper and lower surfaces:

Φ_E = E × 2A = σA / ε₀

Solving for E, we get:

E = σ / (2ε₀)

This shows that the electric field is constant and does not depend on the distance from the plane.

Conclusion

Gauss's law is a powerful tool in electromagnetism, simplifying the calculation of electric fields for various charge distributions. It sheds light on the relationship between electric fields and the charges that create them through the concept of electric flux and closed surfaces. By understanding and applying Gauss's law, we can gain deep insight into the behavior of electric fields in various contexts.

This law is not only a cornerstone of electromagnetic theory, but also a bridge to understanding a wide range of phenomena in both classical and modern physics, making it an essential concept for both students and practitioners.

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