91Ó°ÊÓ

Gauss's Law is a fundamental principle in electromagnetism. It relates the electric flux passing through a closed surface to the charge enclosed within that surface. Gauss's Law can be mathematically expressed as \[ \text{Flux} = \frac{Q_{\text{enc}}}{\text{É›}_0} \] where \( Q_{\text{enc}} \) is the enclosed charge and \( \text{É›}_0 \) is the permittivity of free space.
Gauss's Law is especially useful in situations where the symmetry of the system allows for easy calculation of electric fields. This law helps us understand how charges generate electric fields and simplifies the process of calculating electric flux in complex situations involving symmetrical shapes.

A point charge refers to an idealized model of a charged object where all its charge is concentrated at a single point in space. This simplification helps in solving many electrostatics problems.
In many practical problems, an object can be approximated as a point charge if its size is very small compared to the distances involved. In the given exercise, the \( 1.8 \, \mu \text{C} \) charge is considered a point charge located at the center of a cubic Gaussian surface. Understanding point charges helps in applying Gauss's Law effectively because it simplifies complex charge distributions.

Electric flux measures the quantity of the electric field passing through a given area. It is defined as \[ \text{Flux} = \mathbf{E} \cdot \mathbf{A} \] where \( \mathbf{E} \) is the electric field and \( \mathbf{A} \) is the area vector.
In this exercise, the electric flux is calculated using Gauss's Law because the point charge is enclosed by a Gaussian surface. Given the charge at the center of the cube, applying Gauss's Law simplifies the complex field calculations to a simple division: \[ \text{Flux} = \frac{Q_{\text{enc}}}{\text{É›}_0} \]. After substituting the given values, we have \( Q_{\text{enc}} = 1.8 \times 10^{-6} \text{C} \) and \( \text{É›}_0 = 8.85 \times 10^{-12} \text{F/m} \).

The permittivity of free space, denoted by \( \text{É›}_0 \) , is a physical constant that describes how electric fields interact with the vacuum. Its value is approximately \( 8.85 \times 10^{-12} \text{F/m} \).
This constant appears in many fundamental equations of electromagnetism, like Gauss's Law and Coulomb’s Law.
To calculate the electric flux through a Gaussian surface, knowing \( \text{É›}_0 \) is essential. It helps relate the enclosed charge to the resulting electric flux. In our exercise, replacing \( \text{É›}_0 \) with its value gives us a straightforward way to find the net electric flux: \( \text{Flux} = \frac{1.8 \times 10^{-6} \text{C}}{8.85 \times 10^{-12} \text{F/m}} \approx 2.03 \times 10^5 \text{N} \cdot \text{m}^2/\text{C} \).