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# 2: Electrostatic Potential

All we have done so far is to define the potential difference between two points. We cannot define “the” potential at a point unless we arbitrarily assign some reference point as having a defined potential. It is not always necessary to do this, since we are often interested only in the potential differences between point, but in many circumstances it is customary to define the potential to be zero at an infinite distance from any charges of interest. We can then say what “the” potential is at some nearby point. Potential and potential difference are scalar quantities.

• 2.1: Introduction to Electrostatic Potentials
Imagine that some region of space, such as the room you are sitting in, is permeated by an electric field.  If you place a small positive test charge somewhere in the room, it will experience a force. If you try to move the charge from point A to point B against the direction of the electric field, you will have to do work. If work is required to move a positive charge from point A to point  B , there is said to be an electrical potential difference between A and B.
• 2.2: Potential Near Various Charged Bodies
The geometry of the system has a strong effect on the electric potential. Several geometries are discussed below.
• 2.3: Electron-volts
The electron-volt is a unit of energy or work. An electron-volt (eV) is the work required to move an electron through a potential difference of one volt. Alternatively, an electronvolt is equal to the kinetic energy acquired by an electron when it is accelerated through a potential difference of one volt.
• 2.4: A Point Charge and an Infinite Conducting Plane
An infinite plane metal plate is in the xy-plane. A point charge +Q is placed on the z-axis at a height h above the plate. Consequently, electrons will be attracted to the part of the plate immediately below the charge, so that the plate will carry a negative charge density σ which is greatest at the origin and which falls off with distance ρ from the origin.
• 2.5: A Point Charge and a Conducting Sphere
A point charge +Q is at a distance R from a metal sphere of radius a. We are going to try to calculate the surface charge density induced on the surface of the sphere, as a function of position on the surface. We shall bear in mind that the surface of the sphere is an equipotential surface, and we shall take the potential on the surface to be zero.
• 2.6: Two Semicylindrical Electrodes
This section requires that the reader should be familiar with functions of a complex variable and conformal transformations. For readers not familiar with these, this section can be skipped without prejudice to understanding following chapters. For readers who are familiar, this is a nice example of conformal transformations to solve a physical problem.