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Surface charge density is a measure of the amount of electric charge per unit area on a surface. In the context of this problem, the infinite nonconducting sheet is described by its surface charge density, denoted as \( \sigma \). This parameter tells us how much charge is distributed over a given area of the sheet. The units for surface charge density are usually expressed in coulombs per square meter (\( \mathrm{C/m^2} \)).
Understanding surface charge density is crucial when dealing with electric fields generated by charged surfaces. For example, if you know the surface charge density of an infinite sheet, you can calculate the electric field emanating from it using the expression for an infinite plane:

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

where \( \varepsilon_0 \) is the permittivity of free space, a constant that relates electric field and force.Knowing \( \sigma \) helps us determine how strong the electric field is, which in turn influences how charges interact with the field. This concept is foundational because it allows further exploration into electric potentials and the work required to move charges within the field.

Electric potential, often referred to simply as potential, is the work done by the electric field in bringing a unit positive charge from a reference point (usually where the potential is zero) to a specific point without producing an acceleration. In simpler terms, it tells us how much potential energy a charge would have at a certain point in space due to an electric field. It's measured in volts (\( \, \mathrm{V} \)).
To calculate the electric potential at a point \( P \), a distance \( d \) from the infinite sheet, we used the relationship tied to a uniform electric field:

• \( V = Ed \)

where \( E \) is the electric field and \( d \) is the distance moved from the reference point where the potential is zero.
For instance, if the electric potential is zero on the sheet, then at a point \( P \) at a distance \( d \), the potential \( V \) is calculated as:

• Substituting the electric field \( E = 0.328 \, \mathrm{N/C} \) and \( d = 0.0356 \, \mathrm{m} \) yields \( V = 0.0117 \, \mathrm{V} \).

Electric potential helps in understanding energy distribution within electric fields and is essential for calculating the subsequent work needed when charges are moved.

The concept of work done by an electric field bridges the gap between electric potential and energy. When a charge moves within an electric field, the field does work on the charge. This "work" is essentially the energy required to move a charge from one point to another without any change in kinetic energy.
The formula to determine the work done \( W \) by the electric field when moving a charge \( q \) over a distance \( d \) is:

• \( W = qEd \)

where \( q \) is the charge in coulombs, \( E \) is the electric field strength in newtons per coulomb (\( \mathrm{N/C} \)), and \( d \) is the distance in meters.
In our example, by substituting the values \( q = 1.60 \times 10^{-19} \, \mathrm{C} \), \( E = 0.328 \, \mathrm{N/C} \), and \( d = 0.0356 \, \mathrm{m} \), we find that the work done by the electric field is approximately:

• \( W = 1.87 \times 10^{-21} \, \mathrm{J} \)

This calculation shows how much energy is utilized when moving a charge through the electric field and ties back to the electric potential concept by indicating how energy varies at different positions in the field.