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Surface charge density, denoted by \( \sigma \), is a measure of the amount of electric charge per unit area on a surface of a conductor. In our context, it plays a pivotal role in determining the electric field between two parallel plates. The charge is distributed evenly over the surface, and this uniform distribution is what we refer to as surface charge density.
For two large, parallel conducting plates with opposite charges, the electric field \( E \) between them can be directly related to surface charge density through the equation:

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

This shows that the electric field strength is constant between the plates and directly proportional to the surface charge density. Therefore, higher the surface charge density, stronger will be the electric field between the plates. In our scenario, with \( \sigma = 47.0 \times 10^{-9} \mathrm{~C/m}^{2} \), the calculated electric field is \( 5316 \mathrm{~N/C} \), providing an example of how surface charge density influences the electric field.

Potential difference, commonly referred to as voltage (\( V \)), between two points in an electric field is the work done to move a unit charge from one point to another. For parallel plates, the potential difference can be determined using the electric field and the distance separated by the two plates:

• \( V = E \cdot d \)

Here, \( d \) is the separation distance between the plates, which is \( 0.022 \) meters in this instance. By knowing \( E \), which we've calculated to be \( 5316 \mathrm{~N/C} \), the potential difference between the plates becomes \( 117 \mathrm{~V} \).
This demonstrates that the potential difference is directly proportional to both the electric field and the distance between the plates. Additionally, if the distance doubles, like in our exercise step 3, the potential difference will also double. Such properties of potential difference are crucial for understanding energy storage in capacitors and many electronic applications.

The permittivity of free space, symbolized as \( \varepsilon_0 \), is a constant that describes how electric fields interact with the vacuum (or free space). Its value is approximately \(8.85 \times 10^{-12} \mathrm{~C}^{2}/\mathrm{N} \cdot \mathrm{m}^{2}\).

This constant is essential when calculating the electric field between surfaces like our parallel plates configuration. Using the surface charge density \( \sigma \) and the permittivity of free space \( \varepsilon_0 \), the electric field \( E \) is given by:

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

This equation highlights the role of \( \varepsilon_0 \) in moderating the effect of surface charge density on the electric field. As a fundamental physical constant, \( \varepsilon_0 \) serves to bridge the concepts of electric field, charge, and geometric arrangement of conductors, underscoring its significance in electromagnetic theory.