Understanding Electric Flux Through a Gaussian Surface

Gauss's Law is a fundamental principle in electrostatics that helps us understand the electric flux through a closed surface. This law states that the net electric flux through any closed surface is equal to the charge enclosed by the surface divided by the permittivity of free space. Let's explore how this works with a specific case involving a point charge.

Problem Statement

Consider a point charge of 2 μC located at the center of a cubic Gaussian surface with an edge length of 9.0 cm.

Gauss's Law and Its Application

According to Gauss's Law, the net electric flux ((Phi_E)) through a closed surface is given by the formula:

(Phi_E frac{Q_{text{enc}}}{varepsilon_0})

where: (Phi_E) is the electric flux through the surface, (Q_{text{enc}}) is the charge enclosed by the surface, (varepsilon_0) is the vacuum permittivity, approximately (8.85 times 10^{-12} frac{text{C}^2}{text{N} cdot text{m}^2}.)

Solving the Problem

In this case, the enclosed charge (Q_{text{enc}}) is given as 2.0 μC, which is equivalent to:

(2.0 times 10^{-6} text{C})

Substitute the Values

Now, we substitute this value into the formula for Gauss's Law:

(Phi_E frac{2.0 times 10^{-6} text{C}}{8.85 times 10^{-12} frac{text{C}^2}{text{N} cdot text{m}^2}})

Calculations

Performing the calculation, we get:

(Phi_E approx 226000 frac{text{N} cdot text{m}^2}{text{C}})

This value represents the total electric flux through the cubic surface due to the point charge at its center.

General Principle

The calculation we performed is a specific application of a more general principle. According to Gauss's Theorem of Static Electricity, the net flux through the closed surface of any shape and size is given by (Q/varepsilon_0), where:

(Q) is the net charge enclosed by the surface, (varepsilon_0) is the electric permittivity of the space.

Therefore, for a point charge of (2 mutext{C} 2 times 10^{-6} text{C}) and (varepsilon_0 8.85 times 10^{-12} frac{text{F}}{text{m}}), the net flux is:

(Phi_E frac{2 times 10^{-6} text{C}}{8.85 times 10^{-12} frac{text{F}}{text{m}}} 2.26 times 10^5 frac{text{N} cdot text{m}^2}{text{C}})

This shows that the size and shape of the Gaussian surface do not matter; the total electric flux passing through its surroundings is solely dependent on the charge enclosed and the permittivity of free space.

Conclusion

Gauss's Law is a powerful tool for understanding the behavior of electric fields and the flux through closed surfaces. By applying this principle, we can solve a wide range of problems, including the one we just discussed. Understanding these concepts and their implications is crucial for anyone studying or working with electrostatics.