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Gauss's Law is a fundamental principle in the study of electrostatics, helping us understand how electric fields are generated by charges. It states that the net electric flux through any closed surface is equal to the enclosed net charge divided by the permittivity of free space, denoted by \( \varepsilon_0 \).
Gauss's Law can be mathematically expressed as:

• \( \Phi_E = \frac{Q_{ ext{enclosed}}}{\varepsilon_0} \)

Here, \( \Phi_E \) represents the electric flux and \( Q_{\text{enclosed}} \) is the charge within the surface.
This law simplifies solving problems involving symmetrical charge distributions, like an infinite plane. For an infinite sheet with surface charge density \( \sigma \), the electric field is calculated using the specialized form of Gauss's Law:

• \( E = \frac{\sigma}{2\varepsilon_0} \)

This formula highlights how the electric field is proportional to the charge density and lessens with increasing permittivity.
Using Gauss's Law in the given exercise helps derive the electric field created by an infinite non-conducting sheet, ultimately aiding in calculating the distance between equipotential surfaces.

The electric field is a vector field surrounding charged particles, describing the force exerted per unit charge on other charges in its vicinity. It’s represented as \( E \) and measured in newtons per coulomb (\( \mathrm{N/C} \)).
An electric field due to a point charge is calculated using Coulomb's Law:

• \( E = \frac{k \cdot |q|}{r^2} \)

where \( k \) is Coulomb's constant, \( q \) is the point charge, and \( r \) is the distance from the charge.
However, for an infinite sheet, the electric field remains constant and is given by:

• \( E = \frac{\sigma}{2\varepsilon_0} \)

This consistent field means it doesn’t weaken with distance from the sheet, a unique feature in electrostatics.
In the exercise, this electric field calculation is used to determine how much distance corresponds to a potential difference of 50 V between two equipotential surfaces, demonstrating the relationship between field strength and potential difference.

Equipotential surfaces are imaginary surfaces on which every point holds the same electric potential. These surfaces simplify understanding how electric fields influence charges.
Here are some key aspects of equipotential surfaces:

• They are always perpendicular to electric field lines.
• No work is done by or against the electric field when moving a charge along such a surface.
• The spacing between these surfaces reduces as the field strength increases.

In the context of an infinite sheet, equipotential surfaces are parallel planes. The distance between these planes is dictated by the electric field and the potential difference they are separated by.
Using the relation \( V = E \cdot d \), where \( V \) is the potential difference and \( d \) is the distance between surfaces, you can find how far apart two equipotential surfaces are for a given potential difference. This method allows us to find that for an electric field of \( 5.65 \times 10^3 \mathrm{~N/C} \) and a potential difference of \( 50 \mathrm{~V} \), the surfaces are 8.85 mm apart, as calculated in the exercise.