91Ó°ÊÓ

Electric flux is a fundamental concept in electromagnetism, representing the amount of electric field passing through a given surface area. Think of it as the number of field lines going through a surface. It's denoted by the Greek letter \( \Phi \) and measured in \( \text{N} \cdot \text{m}^2 / \text{C} \).
Imagine holding a net in a stream of water - the amount of water flowing through the net represents the flux. Similarly, electric flux captures how much electric field penetrates a surface.
This is crucial when dealing with closed surfaces, like a box, as it involves summing up the flux through each surface to determine the total flux enclosing a charge. According to Gauss's Law, this total electric flux (\( \Phi_{\text{total}} \)) around a closed surface is directly related to the charge \( Q \) inside the surface through the relationship: \\[ \Phi_{\text{total}} = \frac{Q}{\varepsilon_0}. \]

To find the charge enclosed within a closed surface using Gauss’s Law, the steps are systematic. First, calculate the total electric flux \( \Phi_{\text{total}} \) by summing the electric fluxes through all surfaces of the structure enclosing the charge.
For a rectangular box having six surfaces, the total flux is determined by: \\[ \Phi_{\text{total}} = \Phi_{1} + \Phi_{2} + \Phi_{3} + \Phi_{4} + \Phi_{5} + \Phi_{6}. \]From the exercise, substituting the given values, we sum: \\[ \Phi_{\text{total}} = 1500 + 2200 + 4600 - 1800 - 3500 - 5400 = -2400 ext{ N} \cdot \text{m}^2 / \text{C}. \]
The negative value indicates the flux exiting the box outweighs the entering flux - a hint that the enclosed charge is negative. By applying Gauss's Law, where \( \Phi_{\text{total}} = \frac{Q}{\varepsilon_0} \), isolate Q: \\[ Q = \Phi_{\text{total}} \cdot \varepsilon_0. \]Substitute the known values to find the charge \( Q \): \\[ Q = -2400 \cdot 8.854 \times 10^{-12} \approx -2.12 \times 10^{-8} \, \text{C}. \]This result shows a negative charge within the box.

Vacuum permittivity, symbolized by \( \varepsilon_0 \), is a constant crucial for calculations involving electric fields in a vacuum. Sometimes, it's referred to as the "electric constant."
Its value is approximately \( 8.854 \times 10^{-12} \, \text{C}^2/\text{N}\cdot\text{m}^2 \).
Think of \( \varepsilon_0 \) as the measure of the ability of a vacuum to "permit" electric field flow. It dictates how much electric field is generated per unit charge in empty space.
This constant appears in Gauss's Law, linking electric flux and enclosed charge: \[ \Phi = \frac{Q}{\varepsilon_0}. \] Thus, \( \varepsilon_0 \) serves as the proportion factor, ensuring that even in a vacuum, the presence of charge influences the field, offering a standard reference for field calculations.