91Ó°ÊÓ

Electric flux is a concept that describes how an electric field passes through a surface. You can think of it as the number of electric field lines going through a given area.
For instance, if you imagine a flat surface and a constant electric field, the electric flux would depend on both the strength of the electric field and the size of the surface.
The formula to determine electric flux (\( \Phi \)) through a closed surface is given by Gauss's Law:

• \( \Phi = E \cdot A \cdot \cos(\theta) \)

Here, \( E \) represents the electric field, \( A \) is the area the field lines pass through, and \( \theta \) is the angle between the field lines and the perpendicular to the surface.
In the context of the exercise, the electric flux through the Gaussian surface is given as \(5.00 \mathrm{~N} \cdot \mathrm{m}^{2} / \mathrm{C}\), directly helping to find the charge enclosed.

Charge density represents the amount of charge per unit area or volume, depending on whether we deal with surface charge density or volume charge density respectively. In the context of our problem, we are dealing with surface charge density.
Surface charge density (\( \sigma \)) is described by the formula:

• \( \sigma = \frac{Q}{A_t} \)

where \( Q \) is the total charge, and \( A_t \) is the total area.
In our exercise, the total charge is 87.6 pC and the area is 29.2 cm². By converting these into standard SI units, we find the charge density necessary to determine the area of the sheet enclosed by the Gaussian surface.
This value of charge density helps us relate the enclosed charge in the Gaussian surface to the area we need to find.

The permittivity of free space, often denoted as \( \epsilon_0 \), is a constant that characterizes the ability of the vacuum to allow electric field lines to pass through. It plays a fundamental role in electrical and magnetic calculations, particularly in relation to Gauss's Law.
Its value is:

• \( \epsilon_0 = 8.85 \times 10^{-12} \mathrm{\ m}^{2} \mathrm{ C}^{-2} \mathrm{\ N}^{-1} \)

In our exercise, Gauss's Law states that the electric flux \( (\Phi) \) through any closed surface is equal to the charge enclosed \( (Q_{enc}) \) divided by the permittivity of free space.
Mathematically:

• \( \Phi = \frac{Q_{enc}}{\epsilon_0} \)

By rearranging this formula, we can calculate the charge enclosed by the Gaussian surface from the given electric flux, further leading us to find the enclosed area. The permittivity of free space is thus crucial in solving problems concerning electromagnetic fields and charge distributions.