3: ELECTRIC FIELD

CHAPTER 3

APPENDIX: SPARKS

3.11 Sparks in air

If air were a good conductor, your charged tape would very quickly become discharged. Like other gases, air is an excellent insulator, consisting mostly of neutral nitrogen and oxygen molecules. There are no moveable charged particles-all the electrons are bound to the gas molecules. But during a spark, electrons are ripped out of molecules ("ionization"), leaving free electrons and positive N2+ and W ions, all of which can move and transfer significant amounts of charge from one place to another through the air. Ionized air is a rather good conductor. The bluish light that you see is a side effect of the spark; the light is emitted whenever Eree electrons recombine with positive ions.

3.11.1 Conduction through ionized air

Consider two metal balls, one charged positively and the other charged negatively. If the air between them is not ionized, the air is an insulator, and the charges stay on the balls. (The balls polarize each other.)

We could transfer charge from one ball to the other by connecting a metal wire between the balls, in which case the free electrons in the wire (and balls) shift away ftom the negative ball and toward the positive ball, makin the negative ball less negative and the positive ball less positivrThis conduction of charge through the wire (and balls) lasts for a short time, until the system reaches static equilibrium. (Usually we don't draw 'Y' and '~-" charges on diagrams in regions where the net charge is zero, such as inside a wire, but in this chapter we will sometimes show individual charges in order to discuss details at the atomic level.)

A wire I meter long contains around 1023 free electrons, but a typical amount of charge on a ball is only about 1010 electron charges (about 10-9 coulomb; see your results for HWI-I). About how far must the free-electron sea shift inside the wire in order to neutralize the positive ball?

This is an extraordinarily short distance: remember that an atom has a radius of about 10-10 m

The net effect is as though some electrons on the negative ball had "jumped" to the positive bill. But is that what actually happens? Where exactly did the electrons come from that were added to the positive ball?

Suppose we could somehow manage to ionize the air so that there are free electrons and ions (we'll explain later how we can do this). Then charged particles in the ionized air would be E= to move, making the ionized air a conductor, and allowing a net flow of charge.

The only essential difference is that in the wire only the free electrons move, whereas in ionized air both the free electrons and the positive ions move (m opposite directions, of course). Air molecules on average are much farther apart than the atoms in a solid or a liquid. Neverthehm, we will see later that the average distance an electron travels before colliding with an air molecule is only about a thousandth of a millimeter, so electrons don't "jump" directly from one ball to the other.

During the spark, does the net charge of the positive ball change? The negative ball? If so. how?

Chris said, "During a spark, protons are pulled out of the positive boll and jump across to the negative ball." Give at least two reasons why this analysis cannot be correct.

3.11.2 Why does a spark last a short time?

Once the air becomes ionized, with plenty of mobile electrons and ions, what happens next? Electrons drift toward the positive ball, and the positive ions drift (muich more slowly) toward the negative ball. Since the electrons move so much more quickly, the ionized air conducts in a manner that is rather similar to electric conduction in a metal. There is a "sea!' of free electrons that drifts through the air. There is a kind of "start-stop" motion of an individual electron, in that it is accelerated until it collides with a molecule and comes nearly to a stop, then accelerates again. This accelerated "start-stop" motion has some average speed, called the "drift speed." We will discuss the similar process inside metals at the end of Chapter 5.

What happens when an electron collides with the positive metal ball? How is the metal affected?

What happens when an ion collides with the negative metal ball? How is the metal affected?

What happens to the ion?

We'll see later why it is that as long as the electric field is larger than about 3x 106 N/C throughout a region of air, the air remains heavily ionized, but if the electric field becomes too small, the ions and electrons recombine with each other, and the gas is no longer a conductor. Given what happens to the two metal balls, explain briefly why the spark lasts only a very short time:

However, if you continually resupply the metal balls with charge by connecting the balls to a battery or power supply, you may be able to maintain a steady sparL A dramatic example is the very bright spark maintained between two carbon electrodes in a searchlight or commercial movie projector (a "carbon arc" light). A gentler example is the continua= glow seen in neon lights, which contain neon at low pressure.

3.11.3 Production of visible light

What causes the light that we see emitted by a spark? Occasionally a free clemon comes near enough to a positive ion (not necessarily the original ion) to be attracted and nu=bine with the ion to form a neutral molecule. When the free electron and the ion recombine, thc~y lose energy in the form of light. This energy corresponds to the binding energy of the electron in the molecule.

It is important to understand that the emission of light goes on simultaneously with charge conduction through the ionized air. We will see later that as long as the applied electric field is big enough, neutral molecules are continually being ionized, at the same time dig some free electrons and ions are recombining to make neutral mokculcs (with the emission of light)- Abhough die light that we see is the most obvious aspect of a spark, the light can be considered so be cssentially a mere side effect of conduction in a gas.

We have now seen the main aspects of sparks in ionized gas. We'll mention that sparks are often referred to "gas discharges," and that a heavily ionized gas whose net charge is zero is called a "plasma." Plasma physics is an important area of contemporary research.

There remains the big question, how does the gas get ionized in the first place? Tba is the subject of the remaining sections.

3.11.4 How does the air get ionized?

It is observed experimentally that an electric field of about 3XI06 N/C is sufficiest to ionize air and make it be a conductor. It is natural to guess that this is the magnitude of electric field that is sufficient to yank an electron out of an air molecule, creating a positive ion and a free electron. We can check this guess. Knowing what you know about atoms, estimate about how big an electric field would be needed to rip an outer electron out of a neutral atom:

The enormous difference between the experimentally observed field of 3406 NIC and the value you calculated for l4onize an atom means that we can rule out the possibilij~r of direct atomic ionization by the applied electric field. This is an excellent example of one of the strengths of science: it is often possible to rule out a proposed explanation for a process, even if we can't figure out the real explanation.

Having ruled out direct atomic ionization by the applied electric field, what does happen? It takes a dramatic event to ionize air, because it requires a very large force to rip electrons out of air molecules. A fast-moving charged particle that collides with an atom or molecule can knock out an electron, leaving a singly-ionized ion behind. As it happens, there are fast-moving charged particles passing through your body at this very moment, ionizing some atoms and molecules in your body! Some of these charged particles are -muons" produced by cosmic rays in nuclear reactions at the top of the atmosphere. Others are electrons, positrons, or alpha particles (helium nuclei) emitted by radioactive isotopes present in trace quantities in materials in and around you.

3: ELECTRIC FIELD

A comment: This process is the cause of some potentially dangerous events. A neutral atom in your DNA that suddenly turns into an ios can cause biochemical havoc, and genetic mutations. The passage of fast-moving charged particies through the tiny and very sensitive components in a computer chip can change a bit with potestially disastrous consequences. To guard against such disasters, both DNA and computer circuits have checks built into them to try to compensate for damage caused by the unavoidable passage of high-speed charged particles

Invaders from outer space...

When charged particles from outside our solar system ("cosmic rays~- consisting mostly of very high energy protons) strike the nuclei of molecules in the earth's upper atmosphere, the nuclear reactions produce a series of other particles, of which only positive and negative "muons" W and g7) and neutral neutrinos are able to penetrate the atmosphere and reach the surface of the earth. (Unlike most particles, muons and neutrinos do not interact strongly with the- nuclei of air molecules.) The neutral neutrinos do almost nothing because, lacki~iW electric charge, they don't have any electric interactions with matter, and unlike neutrons they also happen to have extremely weak non-electric interactions with matter- The interactions of neutrinos with matter are so unimaginably weak that most of them go right through the entire earth, coming out the other side with no change

Muons are essentially just massive electrons, behaving in almost every way just like electrons except for having a mass about 200 times the electron mass. A positive or negative muon at rest has an average lifetime of about 2 microseconds before splitting up ("decaying") into a positron (if e) or electron (if jr) and two neutral neutrinos. It is surprising that these muons actually numige to reach the earth's surEwm Even if moving at nearly the speed of light, a muon would be expected to travel only about (3XlO8 m/s)(2x 10-6 s) = 300 meters before decaying, which is far too short a distance to reach the ground from the top of the atmosphere where the muon was produced.

But as predicted by Einstein's Special Theory 4 Relativity, time seems to elapse slowly for a fast-moving object, so that in the reference frame of the muon it takes kss than 2 microseconds to reach the ground, and many muons plunge through your room every minute. Another way of thinking about this is that from a fast-moving rderence frame, lengths in the direction of motion seem shortened (relativistic "length contraction") and to the fast-moving muon the long distance from the top of the atmosphere to the ground looks quite short.

As these rapidly moving charged moons; plunge downward through the air. they have enough energy to occasionally knock electrons out of air mokcules, producing free electrons and positive ions. Each muon leaves a trail of slightly ionized air. Muons produced by cosmic rays are not the only cause of ionization in the air. Materials around you contain trace quantities of radioactive nuclei which can emit high-speed electrons, positrons, or alpha particles (helium nuclei, He+2). These rapidly moving charged particles can also, ionize the air.

If the ionization were extensive, air would be a good conductor at all times, but the number of free electrons and iow produced by cosmic-ray unions and natural radioactivity is much too small to make the air conduct to a significant extenL However, in the next section we will see that the small amount of ionization produced in this way can act as a trigger for large-scale ionization.

3.11.5 A chain reaction: Many free electrons

If there happens to be a sirgle free electron in the air (due to muons or natural radioactivity), it can be accelerated away from its associated positive ion by an applied electric field (due to nearby charged objects such as the charged metal balls shown earlier):

If the electron gets going fast enough before colliding with an air molecule, it can knock another electron out of that molecule, so now there are two free electrons (and two ions):

These two free electrons can lmock out two more electrons, so the number of free electrons grows rapidly: 2, 4, 8, 16, 32, 64, etc. This is an avalanche, or "chain reaction."

The air becomes significantly ionized with a large number of mobile charged particles and is now a rather good conducton.

Note that the air is still neutral overall, just as the interior of a metal wire is neutral overall.

3.11.6 Electric field required to trigger a chain reaction

The few free electrons due to cosmic rays and radioactivity frequently bump into neutral air molecules, but these collisions won't start a chain reaction unless the electrons are traveling fast enough to rip electrons out of the molecules. Ut's see if we can predict from fundamental principles and the structure of matter how big an electric field would be required to accelerate the free electrons to high enough speeds to trigger a chain reaction. A key piece of experimental data is that it takes 2.4xlO-18joule of energy to pull an outer electron out of a nitrogen molecule.

Consider a region of air that is only very slightly ionized by muons and radioactivity:

If we apply an electric field E in this region, the applied electric field exerts the same electric force F = eE on free electrons and (singly-ionized) ions. Which particles accelerate more, the electrons or the ions? Why?

As a result, the electrons attain much higher speeds and run into neutral air molecules much more frequently than the ions do. In a short time interval At it is the electrons which are largely responsible for knocking electrons out of neutral air molecules and producing a chain reaction.

Kinetic energy gained by an accelerating electron

A free electron, accelerated by an electric force F = eE, travels some average distance d before running into a neutral air molecule. This distance d is called the "mean free patlf' of the electron:

The amount of work done on the electron, Fd = eEd, is equal to the increase in kinetic energy A(ICE) of the electron. If we assume on average that the electron has negligible kinetic energy to start with, we have

If cEd is less than the amount of energy required to ionize the molecule (knock an electron out of the neutral molecule), the incoming electron just bounces off:

But if eEd is greater than the ionization energy (2.4XIO-18joule for a nitrogen molecule), after the collision there are two electrons where there had been one:

Each of these electrons can now be accelerated on average through a distance d, and both may ionize molecules, leading to there being 4 free electrons, then 8, 16, 32, 64, etc.-a chain reaction. (After a typical ionizing collision both electrons are moving rather slowly, so each time they are accelerated nearly from rest, and the collision energy is always about equal to eEd.)

Ile critical condition is that the energy eEd be sufficient to ionize a neutral molecule. This depends on how big an electric field E is applied by the nearby charged objects, but also on the mean free path d of an electron in air. LeCs try to estimate d.

Mean free path

It is not difficult to estimate the mean free path d of an electron inair. The calculation depends on a simple geometrical insight. Draw a cylinder along the direction of motion of the electron, with length d and cross sectional area A equal to the cross-sectional area of an air molecule:

The geometrical significance of this cylinder is that if the path of the electron comes within one molecular radius of a molecule, the electron will hit the molecule, so any molecule that is found partly inside the cylinder will be hit.

By definition, d is the average distance the electron will travel before colliding with a molecule, so the cylinder drawn above should on average contain about one molecule. As you may know, at 44 standard temperature and pressurer (STP, one atmosphere at 0' Celsius) one mole of air (6xlO23 molecules) has a volume of 22.4 titers (22.4x 103 CM3). You also know something about the sizes of atoms.

Using the fact that the cylinder in the diagram above should contain approximately one molecule, calculate approximately the mean fi-ee path d of an electron in air:

Estimate of Ebreakdown

As we mentioned earlier, the amount of energy Fd (= eEd) required to singly ionize a nitrogen molecule (that is, knock one outer electron out of the molecule) has been experimentally measured to be 2.4xlG-18 joule. Since you have calculated an approximate value for d, you can now calculate an approximate value for the magnitude of electric field Ebreakdown required to cause massive ionization (breakdown) in air:

ExperimentaUy, we observe that Ebreakdown is about 3xlO6 N/C. It is satisfying that we predict the right order of magnitude (we're off by a factor of 5), and we get an intuitive feeling for the issues. To make an accurate prediction involves difficult calculations because of the statistical nature of the process. In our simple calculations we assumed that every electron goes one mean free path and picks up an amount of energy eEd. However, occasionally an electron happens to go much farther than d, and a smaller electric field is sufficient to accelerate such an electron to a high enough energy to ionize the molecule that it finally hits. Nor is it necessary that every electron ionize an atom in order to create a chain reaction. That is why our simple calculation has over-estimated the magnitude of electric field required to cause a spark.

3.11.7 Drift speed

We can now figure out the average "drift speeiT' of free electrons in a spark. Knowing that the kinetic energy acquired in one mean free path is 2.4x 10- 18 joule, determine the typical speed vf-,nal of an electron when it hits a molecule. If it loses most of this speed in the collision, the average speed is vfinal/2, and this speed is called the "drift speed- of the electrons. Estimate the drift speed:

3.11.8 Propagation of ionization

The magnitude of the electric field is not uniform in the region between the two charged metal balls, and it is largest near the balls (Do you see why?). Suppose the field near the surface of the negative ball is bigger than 3x106 N/C, so that the air near the surface of this ball becomes strongly ionized. How can the ionization extend into other regions where E is smaller?

Positive ions are attracted toward the negative ball, and electrons are repelled into regions of the air where the electric field was less than 3x106 N/C. These electrons increase the electric field in the un-ionized region, and their large number facilitates a chain reaction, so the spark penetrates into the air farther and farther from the ball. It is possible to form an unbroken column of heavily ionized air leading from one ball to the other, even though initially the field was stronger than 3x106 NIC only near the negatively-charged ball. Once the air is heavily ionized, an electric field significantly less than 3x 106 N/C is sufficient to keep the spark going for a while. Recombination of electrons and ions along this column makes the column glow, and you see a spark connecting the two balls.

Something very similar happens in a lightning storm. A column of ionized air propagates from the negatively- charged bottom of a cloud downward toward the ground. The speed of propagation from cloud to ground is somewhat slower than the drift speed you calculated, because the propagation occurs in steps, with pauses between steps due to the statistical nature of triggering a chain reaction in the neighboring air. Once the column of ionized air reaches all the way to the ground, a large current runs through this ionized column. The visible lightning flash is of course due to the recombination of ions and electrons in this column. The large current heats the column explosively, and the rapidly expanding gas makes the noise heard as thunder.

3.12 Homework problem

HW3A-1 In dry weather, if you shuffle across the carpet wearing rubber-soled shoes, and then bring your finger near a metal object such as a doorknob, you will probably get a shock and see a spark. Explain this in detail, with appropriate diagrams. Construct as complete an explanation as you can, including the creation of the spark, the role of the doorknob, and the very short duration of the spark.

(Note that rubber is an insulator, so charge can't move through the soles of your shoes. In this problem, you must explain in detail how and where an electric field big enough to trigger a spark can arise. Can you think of any experiments you can do, if it's dry enough, to test part of your proposed mechanism?)

3.13 Solutions to starred exercises

3.11.1a: In a fraction 101u11OZ6 = JO-Z of the 1-meter Wngth of the wire are enough electrons

(1010 of them) to neutralize the positive ball. 5o the free-electrom sea shifts about

10-1,3 meterl

3.11.1b: The electrons that are Initially NEAR the positive ball go onto the ball, not the ones that were way at the other end of the wirel

3.11.1c: Yes -- the positive ball becomes less positive. and the negative ball becomes less negative. Free electrons near the positive ball are pulled onto the ball, and positive Ions near the negative ball are pulled onto it.

3.11.1d: First, protons cannot be pulled out of the nucki of the metal atom this would take a HUGE amount of energyl

5econd, no single particle cam travel the whole distance between the balls in a single jump. Each electron travels only a very 5=11 distance between collisions with air molecules or Ions. (All of the free electrons in the ionized air move toward the positive ball. Just like the shift of the free electron sea in a metal. All of the positive ions move toward the negative ball.)

3.11.2a: The electron fills an electron deficiency in the po5itively-charged metal ball, making the ball le55 positively charged.

3.11.2b: The ion can pick up an excess electron from the negatively-chareed metal ball, and the Ion becomes an ordinary neutral air molecule. The negatively-charged ball is now less negatively charged.

3.11.2c: The excess charge on both metal balls is quickly reduced to an amount that is not -sufficient to make a large enough electric field to keep the air heavily ionized, and the spark goes out.

3.11.4a: If you could not do this calculation, think again for a moment before looking at the solution on the following page. You really do know enough to make this calculation. In particular, remember that the radius of an atom is about 10-10 m.

We have been t;oW that; a spherical charge acts like a point; charge. Think of an out~er electron as being atuacted by the rest of the at;om, vhlich has a charge +e- The elearic field made by t;hls +c charge that act;e on t;hc outer electron is

where r = 10-10 m is the approximate- radius of an at;om. We have t;o apply at; least that; big a fieW in order t;o pull an outer elearon out of an at;om.

3.1l.roa: The magnitude of the electric force qE is t;hc same on an electron (charge -e) and on a singly-ionized ion (charge +e), but the Ion Is MUCH more massive t;han the elearon. 5o the electrons accelerat;c a lot; more.

3.11.6b:

3.11.6c:

3.11.7a: