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Surface charge density is the amount of electric charge per unit area on a surface. It is denoted by the symbol \(\sigma\) and measured in coulombs per square meter (C/m²). In the exercise, the surface charge density is given as 0.10 \mu\mathrm{C} / \mathrm{m}^2. This value indicates the charge concentration on one side of the infinite non-conducting sheet. Understanding surface charge density is crucial because it directly affects the electric field generated by the sheet. The higher the surface charge density, the stronger the electric field produced. This concept forms the basis for calculating the electric field and potential differences in electrostatics.

Equipotential surfaces are imaginary surfaces on which the electric potential is the same at every point. This means that moving a charge along an equipotential surface won’t change its potential energy. When dealing with an infinite sheet, the equipotential surfaces are parallel planes. These surfaces help in visualizing how the electric field works, as the electric field lines are always perpendicular to equipotential surfaces. The spacing between these planes depends on the strength of the electric field and the potential difference. In the provided exercise, we were tasked with finding the distance between two such equipotential surfaces where the potential difference is 50 V.

The potential difference, often referred to as voltage, is the difference in electric potential energy per unit charge between two points in an electric field. It is measured in volts (V). In this exercise, the given potential difference is 50 V between the equipotential surfaces. To find the distance between these surfaces, we used the formula \[ \Delta V = E \cdot \Delta x \]. With \[ \Delta V = 50 \mathrm{V} \] and the calculated electric field \[E = 5.65 \times 10^3 \mathrm{N} / \mathrm{C}\], we rearranged this equation to solve for the distance \(\Delta x\) between the surfaces. The resulting calculation gives us the separation needed to achieve the specified potential difference.

The permittivity of free space, represented by \(\epsilon_0\), is a fundamental physical constant that characterizes the ability of the vacuum to permit electric field lines. Its approximate value is \(8.85 \times 10^{-12} \mathrm{C}^2/\mathrm{N} \cdot \mathrm{m}^2 \). This constant is essential in calculations involving electric fields and forces in a vacuum. In the exercise, we used \(\epsilon_0\) to calculate the electric field generated by the infinite sheet. The formula \[ E = \frac{\sigma}{2 \epsilon_0} \] shows that the electric field directly depends on the surface charge density and the permittivity of free space. Understanding \(\epsilon_0\)'s role allows us to comprehend how electric fields propagate through a vacuum and interact with charged surfaces.