## Tuesday, August 26, 2008

### IMPEDANCE SERIES PART 11, Lee week 14

August 27, 2008 Educational Radio Net, PSRG 14th session

This session is the 11th in the impedance series. Given that impedance is the combination of reactance and resistance and, further, that reactance is an alternating current phenomenon it is clear that we must have some elemental definitions under our belts to fully appreciate the subject. This multi-part narrative series is an attempt to elevate participants to an intuitive level of electrical understanding without using any serious mathematics as well as provide some review for those of us who have not spent a lot of time on fundamentals lately.

Thus far we have talked about electrical current, voltage, power, resistance, Ohm's Law, DC or direct current, AC or alternating current, Joule’s law, and Kirchoff’s 2 circuit laws. This 11th part of the series will introduce the idea of capacitance and subsequent parts of the series will introduce inductance, reactance, and, finally, impedance as the combination of resistance and reactance. All discussion material will be reviewed continually and be available on the blog.

Let's review what has been covered up to this point in the series.

Part 1 developed the idea of electrical current consisting of moving charge and defined the ampere as 1 coulomb of charge moving past a fixed point in 1 second. One coulomb was defined as a collection of charge numbering 6.24 x 10^18 electrons.

Part 2 developed the notion of mechanical "work" and considered objects at different "potential" levels in a gravitational field. The concept of "voltage", also known as electrical potential difference, and the relationship of voltage to current follows closely with the idea of a mechanical weight being moved between different levels. In both cases work is being done and energy is being manipulated in various ways.

Part 3 capitalized on Bob's lightning series to review electrical current in the context of a charged cloud redistributing charge in the form of lightning where modest amounts of charge make a large impression if moved rapidly.

Part 4 developed the notion of potential difference and ended with a definition of voltage. If you move 1 coulomb of charge from point A to point B in an electric field such that 1 joule of work is done then the potential difference between points A and B is defined as 1 volt. Another way to state this is that 1 joule of energy is required to push 1 coulomb through a potential difference of 1 volt.

Part 5 developed the notion of power by using a mechanical analogy. Power is the relationship between energy and time. Specifically power is the change in energy as in work done divided by the change in time to do the work. Conversely, energy is power multiplied by time.

Part 6 developed the notion of resistance by using a simple circuit to compare how well various materials conduct electrical current. We looked at a simple series circuit with fixed voltage, one D cell battery, a fuse, an ammeter, a switch, and a pair of DUT terminals as in Device Under Test. Substituting various materials across the DUT terminals yielded different measurements on the ammeter and we ranked these materials based upon their "conductance". Finally, we learned that resistance and conductance are reciprocals and that high conductance equals low resistance and vice versa.

Part 7 developed the notion of Ohm's Law by using a simple series circuit to illustrate the relationship of voltage, current, and resistance. Ohm's Law states that electrical current through a resistive device is directly proportional to the voltage across the device so, for example, doubling the voltage across the device will double the current through the device. This relationship stated in math terms is I (which is the symbol for current) equals E (the symbol for voltage) divided by R (the symbol for resistance).

Part 8 developed the notion of direct current and alternating current by using a sand filled tube with a scribed fiducial mark. By assuming that the sand particles represented electrons we could watch the action at the mark and deduce if the current, or moving electrons, was AC or DC.

Part 9 contrasted direct current and sinusoidal alternating current by measuring the temperature of a resistor when subjected to the same maximum voltage from each waveform. The conclusion was that equal values of DC voltage and AC rms voltage, if impressed across a resistor in turn, will produce the same heating effect, or work, in that resistor hence are equivalent. Heat produced as a consequence of current through a resistance is called Joule heating. Energy losses such as this are sometimes called Johnson losses as well.

Part 10 reviewed Ohm’s law and restated the concepts from part 9 in a manner called Joule’s law wherein energy is associated with time to define power and a variable substitution from Ohm’s law produces the familiar P = (E^2)/R formulation. Additionally, the very important Kirchoff’s voltage and current laws were introduced.

Part 11, tonight’s edition, will introduce the concept of capacitance.

Ok, let’s launch into the very sophisticated idea of electrical capacitance.

First, let’s set up the scenario as follows. Imagine a small vacuum isolated metal sphere, marble sized for example, and on this sphere we have managed to deposit some negative electrical charge which we will call -q. This negative charge is, of course, an excess of electrons.

Secondly, and in like manner to the first scenario, imagine another isolated metal sphere which is identical to the first except that the surface charge is exactly opposite to the first and which we will call +q. This positive charge is represented by a deficit of electrons.

Thirdly, imagine that these two spheres are located at an infinite distance from one another or, at least, far enough apart such that each does not know that the other exists. We can refine this distance idea a bit by remembering that each of these spheres is really an isotropic (all directions) radiator of electrical field lines representing the radial coulomb force produced by the surface charge. To satisfy the requirement that each of the spheres does not know that the other exists is to say that the coulomb force field lines of each sphere are not distorted by the other sphere.

So, we have two vacuum isolated metal spheres which are of opposite charge and each is unaware of the other. Now, using a voltmeter and placing a probe on each sphere we can measure some potential difference between the spheres as a result of the surface charge on each sphere. We hypothetically assume here that the voltmeter does not disturb the system in any meaningful way and that there are zero losses.

This next operation defies all intuition in that bringing the spheres closer together reduces the potential difference, or voltage, between the spheres as shown on the voltmeter. This might seem odd given that the charge on each sphere remains unchanged. The closer they are to one another the less the voltage indicated on the voltmeter. Move the spheres farther apart and the voltage increases. The only physical variable in this discussion is the separation distance of the spheres. Move the spheres back to their original positions and the initial voltage measurement is repeated. The explanation is that system energy is stored (concentrated might be a better word) in the intervening electrical field between spheres when they approach each other and returned (or disbursed ) to the system when they recede.

Typically "discrete" capacitors are constructed from parallel metal plates with attached wires and are separated by some insulating substance as simple as air to as complicated as electrolytic paste. This insulating material is known as the "dielectric". The capacitive effect is related directly to how large the plates are in surface area and related inversely to their separation. So, the larger the surface area the larger the capacitive effect and the greater the separation the less the capacitive effect. Additionally, a vacuum has a dielectric constant of 1. Air is very similar and is normally considered as 1 as well. Other insulating materials produce a dielectric constant generally larger than 1 so may be used to multiply the capacitive effect.

The interesting thing about parallel plates is that they represent an open circuit. There is no connection one to another. We know that capacitors can clearly block direct current in a circuit but pass changing direct current, a phenomenon known as a transient, and easily pass alternating current signals. The key to understanding this action is provided by the "displacement" current idea wherein an excess or deficit of charge will induce the opposite polarity across the capacitor separation gap. Like charges repel and unlike charges attract so anything that produces charge drift in one plate will force a redistribution of charge on the opposite plate and any moving charge constitutes an electrical current. The effect is that something went through the capacitor when, in fact, nothing did. In the jargon the charge was "displaced" by coulomb forces within the dielectric space.

In reality all objects are associated, one to another, by a distributed capacitive effect. Parallel wires, elements of a vacuum tube, a resistor to ground, wires through a panel, one circuit trace to another, etc. All are influenced by each other to the enhancement or detriment of a system depending on circumstances.

This concludes the set up discussion of capacitance. We will need one more session to complete the introduction of this fascinating circuit element. Are there any questions or comments?

Terminology

Voltage source: a perfect source of voltage with zero internal resistance. Such a source will stubbornly maintain its output voltage regardless of load. Real world devices always have some internal resistance hence depart from the "perfect" in varying degrees. Voltage regulated power supplies look like voltage sources within their operating limits. Automobile batteries look like voltage sources since their internal resistance is very small and they can provide very large cranking amperes without seriously changing the terminal voltage.

Current source: a perfect source of current with very high internal resistance. Such a source will supply a fixed current regardless of load. The internal resistance is so high that reasonable changes of loading resistance in the external circuit will not appreciably change the current. Dry cell batteries make better current sources than voltage sources since the internal resistance increases as the battery is discharged. For example... you cannot crank a car engine with a AA battery but you could use a AA to dribble a small and nearly constant current through an LED for a long time.

This week’s challenge question goes as follows: you plan to install a 910 ohm resistor in a circuit and you know that the direct current through this resistor will be 125 milliamperes. How many watts of power will the resistor dissipate? How many joules per second?

This is N7KC for the Wednesday night Educational Radio Net

## Wednesday, August 20, 2008

### MULTIBAND ANTENNAS WITH TRAPS, Bob, week 13

Multiband Antennas
Multiband antennas are a good choice for many whether working HF or VHF/UHF. They allow you to cover several bands while using a relatively small amount of space and a single feedline. There are many, many different ways to create a multiband antenna, some are quite esoteric, some are ingenious in their simplicity, and some are downright odd. The latest QST describes a multiband antenna that is switched from band to band using air pressure to open and close relays which change the length of the antenna. (Thanks to Lee for reminding me of this article.) Today we are going to look at a particular technique for creating multiband antennas and that is using traps.

Multiband Antenna Basics
Let's go all the way back to antenna basics. One of the most important things you want in an antenna is for it to be resonant for the band where you want to operate. For a single band this is usually easy. You just figure out the length you need for your center frequency, remembering that the electrical length is different than the physical length, cut it to size, check the SWR and you are ready to go. The only concern might be if you have a wide band and a sharp Q. In this case you might get farther away from your center frequency than you want to be when you go to the extremes of the band. This would get you far away from resonance and would result in a high SWR. For this discussion we aren't going to concern ourselves with this kind of in-band resonance issue.

For a multiband antenna, the antenna must be capable of being resonant on more than one band; that is, it must have the capability to have multiple electrical lengths. In the case of the air-relay antenna, the relays change the length by switching open or closed. So it doesn't necessarily have to be resonant on more than one band at the same time, but it usually is.

What Are Traps?
Traps are electronic circuits designed to allow or stop the flow of AC current depending on the frequency. In a sense they are like the relays mentioned above but nothing is physically opened or closed, or changed in any way to change the length of the antenna. Instead, the trap is designed to be resonant at a certain frequency such that it has near infinite impedance. So at that frequency it looks like an open circuit and thus like the end of the antenna. Much like the simple antenna the trap is designed to be "close enough" to resonance over the part of the band where you will operate so that it will still be effective for whatever frequency in the band you choose. Things get more interesting when you send a signal at a frequency of a different band. To find out what happens there, let's go a little deeper into what makes up a trap.

Anatomy of a Trap
So we know that a trap resonates with infinite impedance at a given frequency. Just how does it do that? A typical trap is what is called a parallel LC circuit, also known as a tank circuit. Now that is a mouthful and it once again jumps ahead a bit of Lee's series. I won't go into this in any detail now. We can return to it after Lee has laid the foundation. I will just say that the L is inductance and the C is capacitance and to say that this is a parallel circuit element is to say that the inductor and the capacitor are connected in parallel. This allows current flowing through that part of the circuit to flow through either the inductor or the capacitor. Because of the nature of this circuit, when you are at the resonant frequency, the inductor and capacitor interact in a way to stop all current flow. This gives you the "end" of your antenna for the resonant frequency.

When you go off frequency, the LC circuit still modifies the signal, it doesn't just become a conductor the way closing a relay does. It turns out that as you lower the frequency, the LC circuit is inductive and increases the electrical length. If you raise the frequency, the LC circuit becomes capacitive and decreases the electrical length. Just how much it alters the frequency depends on the actual values you choose for the inductor and the capacitor. By choosing certain values you can decrease the physical length by quite a bit creating a more compact antenna. Nothing comes for free though and by making a physically smaller antenna, even though you have resonance, you will have a weaker field than if you had a full size dipole for that frequency.

Multiple Traps and Variations
Putting one trap on each side of a dipole will give you a two band antenna. You can add another trap and length of conductor to the end of that dipole to add another band. There are also other ways to use traps in combination with other techniques to increase the number of bands. I don't have any examples prepared but maybe some of our experienced hams can tell us of their own experiences using traps to make multiband antennas.

References
Coaxial Traps for Multiband Antennas, the True Equivalent Circuit

## Sunday, August 10, 2008

### VHF Propagation by Jim K7WA, Week 12

VHF PROPAGATION
August 13, 2008 - Educational Radio Net, Session 12

Introduction:
This will be a discussion of propagation effects which we may experience on the VHF bands between 50 mHz and 440 mHz. Propagation on these bands can be local, near distant or far distant; contacts as far as Japan and even Europe have occurred recently on the six meter band. This discussion will not include F layer propagation in the ionosphere which is common on the high frequency bands and sometimes six meters - this subject deserves a session of its own. Tonight's presentation is intended as an introduction, much has been written and there is much to discover about propagation and I invite you to follow-up with questions and referral to the references listed below.

Most simply put, propagation is how a radio wave gets from the transmitting antenna to the receiving antenna. Like light waves, radio waves usually travel in straight lines. Propagation effects occur when the radio wave is bent or reflected by an object or some other medium on its journey from the transmitter to the receiver. If our primary purpose is reliable local communication, either simplex or through a repeater, we may experience propagation effects as a problem - for example interference on a repeater from a distant station, or noisy or broken-up signals over a normally reliable path. On the other hand, if we are trying to contact a distant station we will want to take advantage of propagation effects to extend the range of our signals. While most long-distance VHF communication takes place on SSB and CW, FM transmissions are also affected by propagation.

The Space Wave:
The ARRL Antenna Book defines the Space Wave as the dominant factor in local communication at 50 mHz and higher - this is what we commonly call "line of sight" propagation extending approximately 50 to 100 miles to the radio horizon. Distance covered by the Space Wave is limited by the curvature of the earth, the height of antennas at both ends of the path and obstacles, such as hills, that may exist in the signal path.

We have all experienced flutter and multi-path fading on our signals. These effects occur when a radio wave is reflected by the ground or some other object, resulting in some of the signal taking a slightly longer path to the receiving antenna than the rest of the signal. The differences of path length affect the phase of the received signal, sometimes interfering with itself in a way that reduces the signal strength. Moving the receiving antenna a short distance usually remedies this condition.

VHF operators often make use of mountains and other large objects as passive reflectors to extend the range of their transmissions. Other examples are reflecting signals off airplanes, orbiting objects, meteor scatter and moonbounce which will be discussed below.

Tropospheric Propagation:
Radio waves do not simply disappear or shoot off into space once they reach the radio horizon. Everything on earth and in the regions of space up to at least 100 miles is a potential forward-scattering medium. Scattering is the process that causes some of the signal to propagate beyond the horizon. Tropospheric effects occur in the lower atmosphere where boundaries between warm and cool air affect radio wave propagation. These effects are variable, but extensions of the minimum operating range occur almost daily. Locally these effects provide propagation north into British Columbia and Alaska, and south into Oregon. Tropospheric ducting sometimes occurs between Hawaii and southern California producing strong signals on the VHF bands.

Sporadic-E Propagation is caused by clouds or patches of abnormally intense ionization in the E layer of the Ionosphere at an altitude of approximately 60 miles. These clouds produce very effective propagation of radio waves above 28 mHz, sometimes as high as 144 mHz. Because the clouds may be small, propagation is often limited to an isolated geographic area. The clouds may also move, providing coverage to different areas during the opening. Single-hop Sporadic-E propagation is typically about 1300 miles although double and multi-hop propagation sometimes occurs extending the distance. Sporadic-E propagation occurs most commonly during the late spring and summer.

Auroral Propagation:
Auroral propagation is the result of charged particles from the sun interacting with gas molecules in the upper atmosphere. Because of the earth's magnetic field, auroras occur around the north and south magnetic poles. Sometimes the sun will emit an unusually large amount of charged particles toward the earth creating an aurora which may or may not be visible in the northern sky. During a strong event, radio waves will reflect off the aurora. Stations aim their antennas north and can make contacts to the east, west, and sometimes even south of their locations. Aurora reflected signals have a unique "swishy" sound to them making voice modes difficult to copy; CW is usually the most effective mode for aurora propagation.

Meteor Scatter and Moonbounce Propagation:
Meteor Scatter propagation utilizes the ionized trails of meteors entering the atmosphere to reflect radio signals from one location to another. There are so many small meteors entering the atmosphere that some commercial systems use this propagation mode to relay data on a regular basis. Amateurs use Meteor Scatter propagation on the 50 and 144 mHz bands for paths up to about 1000 miles. Voice and CW work with Meteor Scatter, but the advent of Digital modes has made this type of propagation much easier and more popular. Meteor trails dissipate quickly, sometimes a "ping" only lasts a second or so, other times a trail may last longer. High-speed digital signals can communicate much more data in a shorter time than voice or CW. K1JT has developed software (WSJT, Weak Signal JT) which enables relatively modest stations to enjoy Meteor Scatter digital communications.

Moonbounce propagation uses the moon to reflect radio signals back to earth. This usually requires large antennas and high power to overcome the considerable path loss to the moon and back. However, many amateurs are using K1JT's software to work moonbounce with relatively modest stations.

Contacts using Meteor-Scatter, moonbounce, and similar modes usually require scheduling and coordination and are rarely made on a spontaneous basis. Nevertheless, for many amateurs the challenge of these modes provides a great deal of fascination and interest.

Conclusion:
In this presentation I have tried to cover some of the common propagation effects we experience on the VHF frequencies between 50 mHz and 440 mHz. While you may encounter some of them on FM, most of the "weak signal" activity occurs on SSB and CW around the established VHF Calling Frequencies (50.125 mHz, 144.200 mHz, and 432.100 mHz). A basic multi-mode radio and simple antennas are all that's required to experiment with propagation on VHF bands. The references below provide much more information.

The ARRL Antenna Book, ARRL
The Shortwave Propagation Handbook, Cowan Publishing Corp. (CQ Magazine)

ARRL Technical Information Service
Propagation: http://www.arrl.org/tis/info/propagation.html
Meteor Scatter and Moonbounce: http://www.arrl.og/tis/info/moon.html

Pacific Northwest VHF Society: http://www.pnwvhfs.org

## Wednesday, August 6, 2008

### IMPEDANCE SERIES PART 10, Lee week 11

August 6, 2008 Educational Radio Net, PSRG 11th session

This session is the 10th in the impedance series. Given that impedance is the combination of reactance and resistance and, further, that reactance is an alternating current phenomenon it is clear that we must have some elemental definitions under our belts to fully appreciate the subject. This multi-part narrative series is an attempt to elevate participants to an intuitive level of electrical understanding without using any serious mathematics as well as provide some review for those of us who have not spent a lot of time on fundamentals lately.

Thus far we have talked about electrical current, voltage, power, resistance, Ohm's Law, DC or direct current, and AC or alternating current. Subsequent parts of the series will introduce capacitance, inductance, then reactance, and, finally, impedance as the combination of resistance and reactance. All discussion material will be reviewed continually and be available on the blog.

Let's review what has been covered up to this point in the series.

Part 1 developed the idea of electrical current consisting of moving charge and defined the ampere as 1 coulomb of charge moving past a fixed point in 1 second. One coulomb was defined as a collection of charge numbering 6.24 x 10^18 electrons.

Part 2 developed the notion of mechanical "work" and considered objects at different "potential" levels in a gravitational field. The concept of "voltage", also known as electrical potential difference, and the relationship of voltage to current follows closely with the idea of a mechanical weight being moved between different levels. In both cases work is being done and energy is being manipulated in various ways.

Part 3 capitalized on Bob's lightning series to review electrical current in the context of a charged cloud redistributing charge in the form of lightning where modest amounts of charge make a large impression if moved rapidly.

Part 4 developed the notion of potential difference and ended with a definition of voltage. If you move 1 coulomb of charge from point A to point B in an electric field such that 1 joule of work is done then the potential difference between points A and B is defined as 1 volt. Another way to state this is that 1 joule of energy is required to push 1 coulomb through a potential difference of 1 volt.

Part 5 developed the notion of power by using a mechanical analogy. Power is the relationship between energy and time. Specifically power is the change in energy as in work done divided by the change in time to do the work. Conversely, energy is power multiplied by time.

Part 6 developed the notion of resistance by using a simple circuit to compare how well various materials conduct electrical current. We looked at a simple series circuit with fixed voltage, one D cell battery, a fuse, an ammeter, a switch, and a pair of DUT terminals as in Device Under Test. Substituting various materials across the DUT terminals yielded different measurements on the ammeter and we ranked these materials based upon their "conductance". Finally, we learned that resistance and conductance are reciprocals and that high conductance equals low resistance and vice versa.

Part 7 developed the notion of Ohm's Law by using a simple series circuit to illustrate the relationship of voltage, current, and resistance. Ohm's Law states that electrical current through a resistive device is directly proportional to the voltage across the device so, for example, doubling the voltage across the device will double the current through the device. This relationship stated in math terms is I (which is the symbol for current) equals E (the symbol for voltage) divided by R (the symbol for resistance).

Part 8 developed the notion of direct current and alternating current by using a sand filled tube with a scribed fiducial mark. By assuming that the sand particles represented electrons we could watch the action at the mark and deduce if the current, or moving electrons, was AC or DC.

Part 9 contrasted direct current and sinusoidal alternating current by measuring the temperature of a resistor when subjected to the same maximum voltage from each waveform. The conclusion was that equal values of DC voltage and AC rms voltage, if impressed across a resistor in turn, will produce the same heating effect, or work, in that resistor hence are equivalent. Heat produced as a consequence of current through a resistance is called Joule heating. Energy losses such as this are sometimes called Johnson losses as well.

Part 10, tonight's edition, will review Joules Law and introduce Kirchoff's Laws.

Ok, let's review Joule's Law by first looking closely at Ohm's Law.

The idea of Ohm's Law was covered in part 7 and was shown to be the linear, as in straight line, relationship of current, voltage, and resistance. According to Ohm's Law, the current through... let's say a resistive circuit, is directly proportional to the applied voltage and inversely proportional to the circuit resistance. This is normally formulated as I = E/R or current equals voltage divided by resistance. Linear means that if you double the voltage then you double the current and inversely linear means that if you double the resistance you halve the current. There are no curved lines in this definition. If plotted on rectangular graph paper then nothing but straight lines will result. The I (current), E (voltage), and R (resistance) represent 3 variables so simple algebraic manipulation of I = E/R leads to two other formulations as in E = IR and R = E/I. Notice in particular that there is no mention of power in Ohm's Law.

Joule's Law was mentioned briefly in part 9 and is important enough to warrant a closer look. Part 4 showed that it requires some "work" as in energy to push some charge through a potential difference. In fact 1 joule of energy is required to push 1 coulomb of charge through a potential difference of 1 volt. Notice that there is no mention of time in this relationship of energy, charge, and voltage. Let's add the time element by moving the 1 coulomb of charge through the 1 volt potential difference in 1 second. Energy moved per unit time is power so, given that we have moved 1 coulomb through 1 volt in 1 second and, further, that 1 coulomb per second is 1 ampere, we can legitimately say that power is the product of volts times amperes or va. This is the essence of Joule's Law.

Now we have enough information to connect Ohm's Law and Joule's Law as follows. From Ohm's Law we know that I = E/R. From the development of Joule's Law we now know that power with symbol "P" is the multiple of volts times amperes or V times I. Since I = E/R (from Ohm's Law) substitute E/R for the "I" in Joule's Law and you end up with the familiar P = (E^2)/R. The also familiar P = (I^2)R is formed along these same lines. Note in particular that both of these expressions for power have a "squared" variable and as a result are not linear since they will plot as a curved line. Note also that you cannot simply manipulate Ohm's Law to produce a power expression without knowing something about the Joule's Law relationship and making the appropriate substitution.

Now on to Kirchoff's Laws. Kirchoff formulated two fundamental ideas in circuit theory which are commonly used to analyze circuit behavior. His first idea deals with electrical current and simply says that current into a junction equals the total current leaving that junction. Imagine this as a traffic "roundabout" where the number of cars entering the roundabout equals the total number of cars leaving the roundabout.

Kirchoff's second idea says that the algebraic sum of voltages in a closed loop is zero. Algebraic sum simply means that sign is important. So, imagine a battery with several series resistors connected to it in a closed loop. Each resistor will "drop" some voltage and the sum of the individual resistor voltage drops will obviously equal the battery voltage. So the sum around the loop is zero volts.

So, in summary at this point in the impedance series, we have looked at three very powerful formulations namely Ohm's Law, Joule's Law, and Kirchoff's Laws. Careful application of these circuit rules will allow one to deduce all sorts of things electrical and give you a leg up on the license advancement endeavor. We have not ventured far into the AC world at this point so let me assert that these very important rules work just fine regardless of the type of circuit involved.

This concludes the set up discussion of Ohm's Law as it relates to power. Are there any questions or comments?

Now let's look at last week's challenge question.

This question was intended to demonstrate scale. We have been using a coulomb of sand to represent a coulomb of electrons. We know that a coulomb is 6.24 x 10^18 electrons. Given that a cubic centimeter of sand contains about 3000 sand particles figure out how many cubic centimeters, cubic yards, and cubic miles would be required to contain 6.24 x 10^18 sand particles. What did this tell you about the real size of the electron?

The cubic centimeters answer is simply 6.24 x 10^18 divided by 3000 or 2.08 x 10^15 cc.
Then compute cc's per cubic yard: (36 inches x 2.54 cm/in)^3 = 7.65 x 10^5 So, (2.08 x 10^15)/(7.65x10^5) = 2.72 x 10^9 cubic yards.

Then compute cubic yards per cubic mile as (5280/3)^3 = 5.45 x 10^9
So, 2.72 x 10^9 cubic yards divided by 5.45 x 10^9 cubic yards per cubic mile = .499 mi^3

Here is another challenge question for those interested.
Suppose that we have a 75 ampere-hour capacity automobile battery. This means that we could expect to produce 75 amperes for 1 hour or 1 ampere for 75 hours. The standard discharge time for measuring purposes is normally 20 hours so we could expect to produce 3.75 amperes for 20 hours. Knowing that 1 ampere is 1 coulomb per second how many coulombs will have been transferred by the time the battery is formally exhausted?