Fundamentals Of Electrostatics - Online Article
For centuries people pondered lightning, lodestones, static electricity and other electromagnetic phenomenon without developing even rudimentary explanations or models. (I am summarily dismissing "Zeus throwing bolts" as an explanation!) It is hard to imagine anything as dramatic as lightning going completely unexplained in our modern technical world but, to be fair, electricity is usually invisible and, without electronic gadgets, it is rather elusive. Fortunately, today’s electronics experimenter has no shortage of electronic gadgets and electricity is available "on tap" at the nearest electrical outlet! Even a poorly financed experimenter can acquire precision equipment that would have served proudly in a modern laboratory only a few decades ago and junk yards have mountains of scrap electronic components that would have been worth a fortune to that same lab. The concepts are simple and materials are at hand, so it is time to begin the adventure into electronics.
Electrical interactions are surprisingly simple and a few basic laws of electricity model the real world with stunning precision. Quantum physics has shown that the theories of Faraday, Maxwell, and other scientists of the nineteenth century were "only" excellent approximations of a much more bizarre set of phenomena but the fundamental mathematical relationships they discovered are the workhorses of modern technology; virtually all modern technological advancement flows from the extreme predictability of electricity and magnetism.
Lets start our discussion with a quantum concept: As far as anyone knows there is a minimum quantity of charge - an amount of charge that cannot be subdivided under normal circumstances. This is the amount of charge that an electron and proton possess. (The Standard Model of physics says the proton's charge is actually made from three quarks, two of which contributes 2/3 of the charge and another that contributes -1/3 of the charge for a sum of 1. Whether quarks "really" have these fractional charges is debatable. These quarks cannot be seen directly and it takes high energy atom "smashers" to see evidence of their individual existence and the individual charges. Quarks "must" combine in such a way to create integer values of charge when forming matter and may be considered to be "virtual" particles that don't "exist" by themselves. This is curious stuff but lets just say that you won't run into fractionally charged particles at your workbench!) This packet of charge has a magnitude denoted by e and comes in two "polarities" denoted by the plus and minus signs (positive and negative). All electrical charges are made up of these tiny packets and charges with the same polarity physically repel each other and opposite charges attract each other. The electron charge is defined as the negative charge, thanks to Benjamin Franklin. (It might be nice if the electron were considered positive but changing now would make changing from English to Metric look like a cake walk!) A little more that 6,240,000,000,000,000,000 electrons are needed to make the common unit of charge called the coulomb. The unit of current, the ampere, is defined as the flow of one coulomb per second. Clearly, the charge on one electron is rather small. If the electric charge on the electron were represented by a drop of water, one coulomb would fill a lake more than 30 miles across and over 100 feet deep. The current flowing in a one amp flashlight bulb is one such lake per second! Despite this incredible flow rate, an electron entering a wire at the battery might take the good part of a minute to reach the bulb. Obviously, there are an inconceivable number of free electrons in the wire! As you might surmise, most electronic circuits deal with astronomical numbers of electrons and the discrete nature of the charge carriers is insignificant.
Since there are 6.24 x 1018 electrons in a coulomb, the charge on a single electron is 1/(6.24 x 1018 ) coulombs or 1.6 x 10-19 coulombs.
R.A. Millikan and associates are credited with being the first to accurately measure this charge using an ingenious apparatus in what is now known as the Millikan oil-drop experiment:
Tiny drops of oil from an atomizer were injected between two metal plates with an adjustable voltage between them. The resulting electric field would attract one polarity of charge to the top plate and if the voltage was just right, the force of gravity could be perfectly balanced, freezing the particle in mid-air. At this point, gravity times the drop’s mass equals the charge times the electric field. Millikan discovered that the random charge on the droplets was always an integer multiple of 1.6 x 10 -19 coulombs. The methods used to calibrate the apparatus are detailed in many physics texts and are well worth reading! Pay special attention to the technique used to determine the drop’s mass and ponder whether you would have had confidence in the measurement!
Well, the force between two tiny particles with charges q1 and q2 is given by Coulomb’s Law:
Charges But wait! Something is wrong! If the force is proportional to the product of two charges then why does the charged plastic rod pick up neutral particles? Since one of the charge terms is zero shouldn’t the force be zero? The answer is not obvious and may leave one wondering if the equation has any "real world" relevance. When the charged rod is held near a small object, a charge is induced in the object. If the object is a conductor, the like charges will be repelled to the far side of the object and the opposite charges will be attracted to the side nearest the charged rod. Since the charges are segregated at different distances, their contribution to the total force will be different (note the 1/r 2 term). The charge carriers in an insulating material are not free to move about as in a metal but the charge can redistribute on a microscopic level resulting in a somewhat weaker attraction to the charged rod. If a conductive object is touching a conductive surface, it can accumulate a net charge since like charges can leave the object entirely and opposite charges can accumulate and the resulting attraction can be quite strong. Try attracting little pieces of aluminum foil sitting on a metal cookie sheet. Make sure the foil pieces aren't laying flat by wrinkling or bending them a bit. They should fly off the metal sheet with enthusiasm!
Coulomb’s Law assumes infinitely small particles and induced charge cannot form since there is nowhere for the charge to go! We can illustrate the phenomenon with a simple experiment:
Lab Assignment
This experiment will require the construction of a unique differential electroscope. The traditional electroscope consists of two metal leaves hanging freely from a wire like a sheet draped over a clothes line. The leaves are usually placed inside a glass jar to block the wind. The wire protrudes through an insulated top to allow for the deposition of charge. But this unique differential electroscope has two isolated supports for the metal leaves (one leaf per support) allowing a different charge to be applied to each leaf. For the purposes of this experiment, construction can be quite simple requiring only a small block of Styrofoam, two large nails and a couple of strips of aluminum foil about 4 inches long and 1/2 inch wide. Push the two nails through the foam about 1 1/2 inches apart forming two hangers for the foil strips. Bend the ends of the foil strips around the nails so that the strips hang freely, easily swinging back and forth. Make sure they are hanging straight down, unlike the photo where they are slightly attracted to each other. The foam can hold a charge and it isn't always easy to tell if the leaves are charged or just bent. If you have trouble, try connecting an earth ground to both nails before adjusting the leaves to make sure there is no charge. They need to be parallel or induced charge will occur across the length.
The Electric Field
The acceleration term in Newton’s equation, F = MA, is also the gravitational field intensity when calculating the effects of gravity on masses. In fact, Einstein argued that acceleration and gravity were indistinguishable! The concept of a gravitational field is familiar to most people so the concept of an electric field is readily described. We can define an electric field as we would define a gravitational field - the gravitational field strength is the ratio of an objects weight to its mass. The electric field is defined as the ratio of the force to the charge causing the force. The equivalent equation to F = MA is F = qE where E is the electric field intensity. Electric fields act just like gravitational fields with one interesting exception! The force on a charge due to an electric field can have two polarities but gravity only attracts (so far as we know).
Here is an Exercise
Suppose that the distance between an electron and the proton in a hydrogen atom is 5 x 10-11 meters. What is the force attracting the electron? This is an easy problem! (Although you might have to look back a few paragraphs.) You know the charges of the particles (the smallest charge that exists, 1.6 x 10-19), the distance and the dielectric constant of the space in between (9 x109). See if you can calculate about 9 x 10-8 newtons.
The concept of a "force field" is quite familiar to us earth-bound humans. We spend our entire lives in the earth’s gravitational field which seems perfectly uniform and constant. The lines of force are clearly "down" - the direction the fine china heads when it slips out of our hands. A line of force is simply an imaginary line that traces the path an object would take due to the influence of a field. For diagramming purposes, it is useful to associate a line with a certain amount of force so that more lines indicate more force. A person standing on the moon would be "skewered" by fewer lines of force than a person standing on the earth since the moon’s gravitational field is weaker than earth’s. The force on an object is proportional to the line density - when the lines are far apart, the force is weak and when they are closely spaced, the force is high. Remember that the line density can be hard to judge since only a few lines are usually drawn; these lines are just a schematic representation of a continuous field. Lines of force are most useful for showing the general shape of a field - only hinting at the field strength at any particular point. One could draw so many lines that they blur together giving shades of gray, darkening in areas of greater force, but it would be difficult to determine the direction of the force if the individual lines could not be seen. The following figure shows typical lines of force for an attracting body and a simulation of how the lines of force might look if thousands were drawn. Remember, as an object moves along a line of force it usually encounters a changing level of force.
The lines of electric force are defined to point in the direction a positive charge would move so electrons have lines of force pointing at them and protons have similar lines of force pointing away. When several charges are in close proximity, the line of force at a particular point will be the vector sum of the contributions from all of the charges:
Unlike the gravitational field we know, electric fields have two polarities and can work together or oppose each other as the diagram shows. Notice how a positive particle precisely between the two positive charges would experience no force since the two fields cancel. But if the charge is slightly above or below the center line, it is squeezed out like a watermelon seed between wet fingers! Just like gravity, the electric field is a "force field" - the field intensity at any point is equal to the amount of mechanical force that would be applied to a charge of one coulomb positioned at the point. The units are newtons per coulomb. The electric field surrounding a charged Lucite rod can be detected by holding the rod near the electrode of a sensitive electroscope. Without the rod making contact, the leaves will spread due to the induced charge caused by the field.
Voltage
When you climb on the roof of your house the earth’s gravity has the potential to do you great harm. In fact, your body has potential energy which is proportional to your height off the ground. If you slip, you will accelerate and accumulate kinetic energy until something stops you (hopefully, a soft pile of hay). The scientific term, work, is defined as force multiplied by distance. So gravity does work on you as you fall - the force is your weight (mass times gravity) and the distance is the height of the roof. The total amount of work done on you during your fall determines how hard you hit! Since gravity is uniform, you can gauge your safety by peeking over the edge of the roof to see how high you are. But if gravity were not uniform, you would need to multiply the height by the local gravity to determine how much potential energy your body has (which is equal to the kinetic energy it will have when it hits the ground).
When dealing with charges and electric fields, this product (meters times newtons per coulomb) is referred to as "voltage". In a uniform electric field, the voltage between two points is the product of the magnitude of the electric field and the distance between the points. A charged particle will "fall" or accelerate in the electric field just as a body will accelerate in a gravitational field. The farther it falls, the more kinetic energy it accumulates. So the voltage between two points tells us how much energy will be imparted to a unit charge moving between them. Multiply the voltage by the amount of charge to get the energy (in newton-meters which are called joules).
For example, a single electron with a charge of 1.6 x 10-19 coulombs will pick up 1.6 x 10-16 joules moving between two points with 1000 volts of potential difference:
1.6 x 10-19 coul x 1000 volts = 1.6 x 10-16 joules
Notice that we did not need to know the physical dimensions or the magnitude of the electric field -these quantities are already combined in the "voltage". It is a handy combination since much of electronics involves the effects of potential differences regardless of the distances involved or the resulting electric fields. In your car, wiring carries the battery’s 12 volt potential difference to various motors, lamps and assorted gadgets in different locations and the electric field within the car is a complex three-dimensional structure. But the concept of voltage lets us greatly simplify the description of the functioning of the auto’s wiring without regard to the path the charges follow.
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