Electric Theory in a Nutshell

In this, and in all reading assignments, Discussion Questions and Activities are meant to be completed when they are reached in the reading, before continuing.  Getting the "correct" answer to Discussion Questions is not important.  Instead, the purpose of Discussion Questions is to address the issues, start you thinking about the material, and identify your preconceptions.  Completing these assignments before continuing with the reading will aid you greatly in the learning process.

Warning and Reassurance:  Due to the nature of the course, this reading assignment is very condensed and intensive.  Do not worry if you don't follow all of the calculation details; traditional physics courses spend 2-3 weeks on the material summarized here.  The key points you should remember are summarized at the end of the assignment and covered in the Reading Quiz.  Once you have a general feel for the definitions of electric field, voltage, and capacitance, the discussion of circuit electronics will be more meaningful.
 

At some point you may have learned the expression F = ma.  This is the second of Isaac Newton's famous three laws describing the motion of objects.  Newton's Second Law reveals that the acceleration of a mass is directly proportional to the force applied to it.  In other words, if the velocity of an object changes (either by its speed changing or its direction of motion changing), a force must be acting on the object.

Newton's Laws were first applied to mechanical forces, such as the force a horse exerts on the cart it pulls, and to gravity.  As science progressed in subsequent centuries, Newton's Laws remained true even as the number of known forces fluctuated.  Scientists currently believe that all accelerations can be attributed to just three fundamental forces:  gravity, electroweak, and strong nuclear.  At reasonable temperatures, the electroweak force behaves as two separate forces:  the weak nuclear force and the electromagnetic force.  We are held to the earth, and planets are held in their orbits, by gravity.  Atoms and molecules are held together by the electromagnetic force.  Nuclei stay together because of the strong force, and many nuclear decays are governed by the weak nuclear force.  Mechanical forces, such as tension in a wire, pushes and pulls on objects, and friction, are not separate forces but are primarily due to the electromagnetic force; as two objects move closer to each other, the electrons on the surfaces of the objects repel each other until the objects are separated.

Information systems do not involve objects as heavy as planets, nor do they involve nuclear interactions or nuclear decays.  Instead, most of the physical processes responsible for the operation of information systems take place at the atomic or molecular level.  Gravity and the nuclear forces therefore make negligible contributions to governing how information systems work.  Electromagnetism is the force that affects the workings of information systems, so it is the force we  consider in this course.

You may have previously studied electricity and magnetism as two separate phenomena.  But James Clerk Maxwell showed in the 1800s that electricity and magnetism were really two aspects of a single "electromagnetic" force.  Furthermore, he deduced that light is nothing but traveling electric and magnetic fields.  In this reading, we will consider only electric phenomena.

Take a piece of cellophane tape about a foot long and stick it to your desk.  Lift up one end of the tape carefully, then quickly pull up on that end to remove the tape from the desk.  This will charge the piece of tape.  If you do not see any attraction or repulsion of the tape in the following activities, try a new piece of tape or a different roll of tape.

Now bring the tape close to the desk without letting it touch the desk.  What happens?  What if you bring the tape close to a person?  Close to your chair?  Does this effect depend on which side of the tape is closest to the object?

Repeat the procedure and charge another piece of tape.  Does this piece behave the same as the other piece of tape?  What happens if you bring the two pieces of tape close (don't let them touch!)?  Do you think the two pieces have the same electric charge or different charges?  Why?  Does the effect increase or decrease as you decrease the distance between the two pieces of tape?

Now stick a new piece of tape to the desk and leave it there.  Stick another piece on top of it.  Quickly remove BOTH pieces, leaving them stuck together.  Does this double piece act like the original piece of tape?

Pull the two pieces of tape apart, and examine their behavior when you bring them close to the table.  Do they behave similarly or differently?  What happens when you bring them close to the original piece of tape?  Do they behave similarly or differently?  Describe what happens when you bring them close to each other.  Do you think the two pieces have the same electric charge or different charges?  Why?

When you pulled your pieces of tape off of the table, they (hopefully) became electrically charged.  And charged objects exert electric forces on each other.  Scientists in the 1700s such as Ben Franklin studied charged objects much in the same manner as you have just done.   I doubt that those early physicists used cellophane tape, but they were able to draw several important conclusions:
 

-	Electric charge comes in two varieties.
-	Oppositely charged objects attract each other.
-	Similarly charged objects repel each other.
-	The attraction or repulsion increases with a decrease in distance.

Charles Augustin Coulomb did more quantitative experiments with charged objects in the late 1700s.  He noticed certain properties of the electric force between two charges.
 

-	If either of the charges is doubled, the force doubles:  the electric force is directly proportional to each of the charges.
-	If the distance r between the charges is doubled, the force falls by a factor of 1/4:  the electric force is inversely proportional to the square of the distance between the charges.
-	The force on one charge q1 points away from the other charge q2 if q1 and q2 have the same sign.  If q1 and q2 are oppositely charged, the force on q1 points toward q2.

These properties are combined into the equation known as Coulomb's Law:

Mechanical forces, such as pushes and pulls, act only if objects are in contact.  The electrical force, however, acts over a distance.  In an effort to envision how one charge can push away another charge at a distance, scientists developed the concept of an electric field.  The electric field is defined as the force on a charge divided by that charge:

Coulomb's law for the force between two point charges can in principle be used to find the force on any collection of charges due to any other collection of charges.  But such a calculation would be quite complicated and could involve vector calculus.  Calculating the electric field at any point due to a collection of charges could be similarly difficult.  But if you take a known charge into an electric field, you can determine the field by measuring the force on the known charge.  Once you know the field in a region, you can then find the force on any charge you bring into that region by using the electric field equation.

The electric field can be visualized by using field lines.  By definition, electric field points in the direction of the force on a positive test charge.  Thus field lines will point away from the red positive charge as shown in the left side of the figure.  The field lines toward the blue negative charge in the right side of the picture.

Once you can draw electric field lines, the positions of the charges creating the field become unneccesary.  One source of electric fields we will encounter later in the semester is the parallel-plate capacitor.  A parallel-plate capacitor consists of two parallel plates with opposite charges.  If the plates are sufficiently wide and sufficiently close together, the charge on the plates will line up as shown below.  This gives rise to a uniform electric field between the plates pointing from the positive plate to the negative plate.

Since it is the electric field and the resulting properties of the capacitor that will be of interest, we don't usually include the charge distribution in diagrams.  We can represent a capacitor merely in terms of its field.

Want more information?  Try the Project Links Electric Field Module.
 
 

The electric force is one example of a force varying with distance.  The potential energy stored in the field of two point charges will change as you increase the distance between the charges from r₁ to r₂.  The amount by which it changes is found as follows:

One way to think about potential difference is as the potential for charges to move.  The parallel plate capacitor above has a potential difference between its plates.  If the two plates were connected by a conductor, charges would start moving from one plate to the other until the potential difference between the plates became zero.

Potential difference is closely related to electric field.  We can combine some of the definitions above and find

If the electric field is uniform over the distance, such as inside a parallel-plate capacitor, the relationship becomes

Want more information?  Try the Project Links Electric Potential Module.
 
 

Capacitors, Capacitance, and Permittivity

Since the field lines are parallel and evenly spaced, the electric field is uniform between the plates.  For infinitely long plates, the electric field has precisely the same value everywhere between the plates and is zero outside the plates.  For physical capacitors of finite length, the field lines at the edges spread out a bit, and the field is not exactly zero outside the plates.  But we will stick with the assumption that the plates are infinitely long, since that is an adequate approximation to real capacitors in most situations.

The positive charge on the left plate of this device is attracted to the negative charge on the right plate, and vice versa.  But the region in between the plates is made of an insulating material (like air), so the positive and negative charges cannot cross through and recombine.  Thus, once the charge is on the capacitor, it will stay there until a the plates are connected by a conducting material.

Moving charge onto the plates of a capacitor requires work, since additional positive charges will be repelled by the initial positive charge on the left plate.  This work is provided by a voltage source, such as a battery.  The maximum amount of charge a capacitor will aquire is proportional to the voltage:

Note that Q is NOT the total charge on both plates (that would be zero), but is the charge on a single plate.  A larger voltage can shove more charge onto the plates.  The constant of proportionality is called capacitance, C

Capacitance is measured in the unit of farads (1 farad = 1 Coulomb/Volt = 1 Coulomb²/Joule).  The capacitance of an object depends primarily on its geometry.  If the plates have a larger area, they can hold more charge.  If they are closer together, the attraction is stronger, so the plates can hold more charge.
For a parallel-plate capacitor such as the one we've been considering, the capacitance is

where A is the area of one plate, d is the distance between the plates, and e is called the permittivity.  Permittivity depends on the insulator used.  For a capacitor with nothing but a vacuum between the plates, the permitivity is e₀, called the permittivity of free space.  We have seen it before in Coulomb's Law, and its value is

The insulating material used affects the capacitance due to insulators' ability to become polarized.  In an insulator, electrons cannot move freely throughout the material, but individual molecules may rotate slightly in an electric field.  A polarized molecule is one in which the negative charge is primarily on one side of the molecule, while the positive charge is primarily on the other side.  Water is a good example of a molecule which polarizes easily.  In water, the two hydrogen atoms each give up an electron to the oxygen.  The atoms bond to create the shape illustrated below.

If the insulating layer in a capacitor can be polarized, more charge can be stored on the plates. Consider the diagram below.  When the left plate becomes positively charged, the polarized molecules will rotate to put their negative sides toward the positive plate.  This causes the strength of the electric field inside the capacitor to diminish, so it is easier to place more positive charge on the left plate (or to pull more negative charge off the plate).  Another way of looking at it is to consider the force exerted on the charge entering a plate.  A positive charge attempting to join the positive plate is repelled by the other positive charges already there.  It is drawn to the negative charges on the other plate, but they are farther away, so the repulsion of the nearby positive charges wins out.  If, however, the insulator is polarized, the attractive negative sides of the polarized molecules are closer to the positive plate than are the repelling positive sides.  Thus it takes less effort to bring more positive charge onto the positive plate.

A capacitor with a polarized insulator will store more charge and thus have a higher capacitance than one with an unpolarized insulator.  Thus easily polarized materials have a high permittivity e.
 

Capacitors and Energy

Capacitors store electric energy when charged.  The charges on the capacitor plates produce an electric field inside the capacitor.  Moving along electric field lines results in a change of electric potential:

If a conducting wire were to connect the two plates of a capacitor, charges would gain kinetic energy and flow from one plate to the other until both were discharged.  This kinetic energy has to come from somewhere, so the capacitor must store potential energy.  One might first assume that the energy stored is just the energy required to move a charge Q through a potential difference of V, or QV, but this is not quite right.  The first charge q to move from one plate to another does indeed lose potential energy equal to qV, but the next charge to move experiences a different change in potential, since there is now less charge on the plates.  The result is that the potential energy stored in a parallel plate capacitor U_E is given by the charge times the average voltage:

This energy can be expressed in terms of just the charge or just the voltage by using the definition of capacitance:

For a parallel-plate capacitor, the voltage is proportional to the electric field.  We can use that property, and the equation for capacitance to get

The energy stored in a capacitor depends on the capacitor's geometry as described by A and d.  We often want to discuss the energy in a generic electric field.  We do so by defining energy density, u, as the energy per unit volume:

This expression is independent of geometry and depends only on the electric field and the permittivity medium in which that electric field exists.  While we have derived this expression for a parallel-plate capacitor, it is applicable to any electric field.

Capacitors have many features advantageous for data storage.  They can be used to represent binary data, with a charged capacitor representing a 1, and an uncharged capacitor representing a 0.  They can store binary data, since a charged capacitor retains its charge after the removal of the voltage supply which charged it.  This data stored by capacitors can be easily changed, just by discharging and/or recharging the relevant capacitors.  The data can be read without being destroyed by checking the voltage across each capacitor.  And all of these steps can be performed in the time it takes charge to flow on or off a capacitor.

While the ideal capacitors discussed above may seem perfect for any type of data storage, real physical capacitors pose a few obstacles. It's true that capacitors can easily represent binary data, and that such data is easily erased and replaced, but the rest of the claims in the above paragraph are unduly optimistic.  Capacitors are not well-suited for long-term storage of data, as charge leaks off capacitors through air fairly quickly.  So the information stored in the capacitors must be continually refreshed.  This refreshing is performed every few nanoseconds when capacitors are used in Dynamic Random Access Memory, or DRAM.

Another optomistic statement from the first paragraph has to do with reading data from a capacitor.  In principle, the voltage across a capacitor can be checked with a minimum amount of charge escaping.  To do this, one needs to use a very high resistance in the measuring device.  This high resistance insures a low current and thus a small amount of charge flowing.  As we will see in the next section, however, using high resistance slows down the circuit considerably.  In practice, DRAM devices re-write the data after reading it every few nanoseconds.  This is why it is called Dynamic RAM.

Finally, the charging and re-charging of capacitors does not happen instantaneously in real circuits.  Real circuits always have some amount of resistance.  When resistors and capacitors are combined in a circuit, the current through the circuit will no longer increase or decrease instantaneously, but will exhibit a gradual rise or fall.  This is the topic of our next section.
 
 

Project Links Electric Field Module - go to http://links.math.rpi.edu/devmodules/electromagnetism/ElectricField/index.html and click on the "Coulomb's Law and the Definition of Electric Field" link on the left navigation bar.

Project Links Electric Potential Module - go to http://links.math.rpi.edu/devmodules/electromagnetism/electricpotential/index.html and click on the "Part 1" and "Part 2" Review links on the left navigation bar.  The rest of the module involves some vector calculus, but there are some good examples and applets you can look through.

The Cartoon Guide to Physics, by Larry Gonick and Art Huffman.  (Harper Perennial: New York), 1991.  This is a great user-friendly treatment of the basic concepts in phsyics.

How Computers Work, by Ron White.  Includes short discussion of electric memory.

Instant Physics., by Tony Rothman (Fawcett Books), 1995.  A very entertaining look at the fundamentals of physics and arduous path taken to arrive at these laws.

Any introductory physics text, such as Fundamentals of Physics by Halliday, Resnick and Walker.