(II) A manufacturer claims that a carpet will not generate...

(II) A manufacturer claims that a carpet will not generate more than $5.0 \mathrm{kV}$ of static electricity. What magnitude of charge would have to be transferred between a carpet and a shoe for there to be a $5.0-\mathrm{kV}$ potential difference between the shoe and the carpet, approximating the shoe and the carpet as large sheets of charge separated by a distance $d=1.0 \mathrm{~mm} ?$

Key Concepts

Electric Potential Difference

This concept involves the work done per unit charge when moving a charge between two points in an electric field. It is central to understanding how a voltage, such as a 5.0 kV difference, is established between two objects. The idea is that the potential difference reflects the energy difference per unit charge, which is key when analyzing systems like charged plates or surfaces.

Uniform Electric Field Between Parallel Plates

The approximation of the carpet and the shoe as large, parallel sheets of charge allows us to assume a uniform electric field in the region between them. In such a setting, the potential difference is directly related to the electric field strength multiplied by the separation distance. This concept is important because it simplifies the analysis of the system by reducing the problem to one-dimensional motion of charges in a constant field.

Parallel Plate Capacitor Model and Capacitance

A parallel plate capacitor consists of two conductive sheets separated by a small distance and is characterized by its capacitance, defined as the ratio of the charge stored to the potential difference (C = Q/V). With the relationship C = ??A/d for large plates (where ?? is the vacuum permittivity, A the area, and d the separation), one can relate the charge transfer required to achieve a specific voltage. This model provides the framework for understanding the behavior of systems approximated as two charged sheets.

Relationship Between Surface Charge Density and Electric Field

For large, parallel sheets of charge, the electric field can be approximated using the surface charge density (?), where the total field created by two oppositely charged sheets is proportional to ?/??. This relationship is crucial in linking the amount of charge transferred to the resulting electric field, and subsequently, to the potential difference between the surfaces.

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