Lesson 13 Faraday’s E&M Fields
The field idea
13.1 Contents
13.2 Goals of this lesson:
- I’d like you to Understand:
How the local interaction of a field is what matters in an electromagnetic interaction among charges and currents
How to determine the shape of electric field from a configuration of electric charges
How to determine the shape of a magnetic field from a configuration of currents
How to calculate the potential energy and kinetic energies of charged objects in an electric field
- I’d like you to Appreciate:
How vector fields combine graphically
That electric fields store energy
How an electric field is defined from the electric force
How the voltage relates to the potential energy in a parallel plate capacitor
- I’d like you to become Familiar With:
How we can visualize fields in demonstrations with iron filings and “dielectric” specks of matter
13.3 A Tiny Bit More of Faraday
One can wonder how the study of electricity and magnetism might have evolved had Michael Faraday become the bookbinder that he was apprenticed to be. He might have led a fine life, surely a better one than his parents’, had a family, and maybe introduced some innovative techniques to the printer trade. Then Ampere, Biot, Volta, Green, Weber, Henry, Kirchhoff, Thomson, Davy, and the others would have continued their researches which would have required them to eventually break from their firmly entrenched training and biases. The field idea would have eventually emerged somehow and James Maxwell, whom we’ll meet in the next lesson, would surely have played a role. But the world would not have had Faraday’s extensive laboratory notebooks which he periodically published and so all of the experimentation he’d done by himself would have to come from probably many other scientists.
And maybe, had Charles Wheatstone stood his ground and not fled in the face of a particular critic in the audience on an otherwise regular Friday Evening Discourse, Faraday’s path to the field idea might have been different. Wheatstone was a scientist and engineer who was notoriously shy and allergic to controversy. He had been invited to give a talk at the Royal Institution Friday series on the evening of April 3rd in 1846. These public talks generally drew hundreds of everyday people drawn to learn the scientific wonders of the day. On that particular evening Wheatstone saw a fellow in the audience who had a reputation for heckling presenters and he turned on his heel and fled into the dark, springtime city streets leaving Faraday alone on the stage with nothing to say. What he did changed everything.
For a bit, he tried to summarize Wheatstone’s work, probably unsatisfactorily. And then he started to chat off the cuff, presenting a “vague impression of his mind.” Let’s remember what the known-knowns were at the time. His demonstrations had shown that “lines of force” were present around permanent magnets, wires carrying currents, and distributions of electrically charged objects. These lines of force were real enough to twist bits of iron and line them up and twist bits of pollen and likewise cause them – unassisted from any visible source – to reveal a smooth pattern. That was known. Faraday had ideas about atoms being the cause of the lines of force, as he had become the world’s expert on electrochemistry and electrolysis for which an atomic picture can be useful. But he wasn’t prepared to totally commit to that idea.
Furthermore, everyone shared Newton’s disdain of Action at a Distance in gravitation – but nobody could offer an alternative explanation and so it lingered like a relative who is a general family embarrassment.
However, Action at a Distance was threatening to raise its head in the electrostatics and magnetism game and Faraday was having none of it. Nobody was. Another known-known was that light had turned out to be a wave phenomenon after Thomas Young (at the Royal Academy also) demonstrated it. That led to a collective nervous breakdown in scientific Britain, driving Young from science altogether but eventually ginning up the idea of a “luminiferous ether” as the all-pervasive substance that supported the vibrations of light waves. What did light wave? Why the ether is what physically waved. That became the model of light and Faraday was having none of it.
“The velocity of light through space is about 190,000 miles in a second; the velocity of electricity is, by the experiments of Wheatstone, shown to be as great as this, if not greater: the light is supposed to be transmitted by vibrations through an ether which is,… infinite in elasticity; the electricity is transmitted through a small metallic wire, and is often viewed as transmitted by vibrations also. That the electric transference depends on the forces or powers of the matter of the wire can hardly be doubted…”
But the matter in a copper wire has weight, but the ether doesn’t. If both are made of particles, they are very, very different. But the same.
The bottom line for Faraday, which he then had to defend multiple times afterwards was that the lines of force were the reality. Not an ether. He subsequently dubbed the reality a “field” and that name has stuck, of course to this day.
“For suppose two bodies, A B, distant from each other and under mutual action, and therefore connected by lines of force, and let us fix our attention upon one resultant of force having an invariable direction as regards space; if one of the bodies move in the least degree right or left, or if its power be shifted fora moment within the mass (neither of these cases being difficult to realize if A and B be either electric or magnetic bodies), then an effect equivalent to a lateral disturbance will take place in the resultant upon which we are fixing our attention; for, either it will increase in force whilst the neighbouring resultants are diminishing, or it will fall in force as they are increasing.”
He’s basically saying that A creates a field between it and B…wiggle A and B, a finite time later, will wiggle. What’s between them, that propagates the wiggle is his field. And it’s real. And furthermore, the field is a property of space itself – it, not atoms (“centers of force”) – was the field’s home. No ether, just space suffused with interweaving waves of electricity, magnetism, and yes, gravity. His was a theory of everything, or more correctly, a picture of everything.
Two days later he wrote >I think it likely that I have made many mistakes in the preceding pages, for even to myself, my ideas on this point appear only as the shadow of a speculation, or as one of those impressions on the mind which are allowable for a time as guides to thought and research. He who labours in experimental inquiries knows how numerous these are, and how often their apparent fitness and beauty vanish before the progress and development of real natural truth.
Reaction was swift:
“Faraday’s achievements are due to his immense earnestness and great love for his subject and this very mistiness which serves to obscure the verity of matters may have its compensations by rendering the subject attractive and thus wooing a man to work at it with more fervour.” Tyndall
“…he could not understand Faraday, and if you look for exact knowledge in his theories you will be disappointed—flashes of wonderful insight you meet here and there, but he has no exact knowledge himself, and in conversation with him he readily confesses this.” Biot
Friends and colleagues condescendingly suggested that he should leave the mathematics to the professionals. Everyone respected his experimental skill, and like his successor, Tyndall above, his imagination. But this time he’d gone too far. Let’s see.
13.4 The Electric Field
The first field was the Magnetic Field, which, as we’ve seen, was forcefully suggested by iron filing patterns around a permanent magnet or steady current. But for our general introduction, the Electric Field is easier to understand, so that’s where we’ll start.
Faraday believed strongly in a unified nature in which the laws would be related and so given his conclusion about magnetic lines of force and his speculation about electric lines of force, he also imagined a Gravitational field in the same spirit…performing many failed experiments to try to detect a gravitational influence on currents and magnets. He was limited to guessing about electricity since a visual demonstration for charges analogous to the iron filings experiments was beyond his lab’s capability. Nonetheless, he speculated about the existence electric lines of force and Maxwell baked that idea into his theory of both Electric and Magnetic fields. We’ve already dealt with Coulomb’s law, which is the force of attraction or repulsion between two electrically charged objects. It’s so important, let me reprise it here (\(Q_1\) and \(Q_2\), separated by a distance, \(R\)):
\[F=k\frac{Q_1Q_2}{R^2}.\]
Remember that the \(Q\)’s have an algebraic sign, \(+\) for a positively charged object and \(-\) for a negatively charged object – and then the multiplication results in an overall sign for \(F\): if positive (like two positive or two negative \(Q\)’s) then the force is repulsive and if negative (where one \(Q\) is positive and the other is negative), then the force is attractive.
Into the middle 1800s, everyone assumed that whatever effects were felt by charges in Coulomb’s law, masses in Newton’s Gravitational law, and magnets were instantaneous and facilitated through motions in an ether.
Faraday felt otherwise: his fields would propagate between objects at a finite speed and were themselves “a thing” not requiring any intermediate substance. By contrast, Maxwell thought that his theory was a model of the ether – that the propagation of electric and magnetic fields were disturbances in it. It wasn’t until the 20th century that the ether idea was abandoned. This strange substance maybe one of the longest-believed, mathematically sophisticated (lots of still useful mathematics was developed in trying to describe the nature of the ether), and wholly false models in the history of physics! So we’ll describe Maxwell’s theory differently from how he would have. Ours is an ether-free-zone.
Figure 13.1: The metaphor of the Maxwellian electric field showing the reaction of the negative electron as due to the local field and not the distant \(Q\)
The modern idea of the field was conceived by Faraday – who was right! – is shown in the cartoon above which imagines the electrostatic attraction of an electron and a proton. A mathematical metaphor, if you will. In this view there are three aspects to the field:
- The source (“cause”). An electric charge, a magnetic pole or current, or mass create a field in its vicinity.
- The sink (“effect”). An electric charge, a magnetic pole or a current, or a mass detects a field in its vicinity.
- The disturbance (“field”). The intermediate space is filled by the field, which propagates at a finite velocity from source, to sink.
Of course, one person’s source is another persons effect. That is, each acts on the other through the field.
13.5 The Revolutionary Idea of a Field
We all experience visual examples of fields…familiar to anyone who’s looked at a weather map. It’s nothing more than a distribution of some quantity in space (and time) with a value – a number—associated with every point in space. If it’s weather then any map that shows the distribution of temperature is a perfect example of a Temperature Field. You could imagine a million little weather-people all armed with thermometers and GPS transmitters who patiently take the temperature of the air in front of them and report it back continuously to Weather Central which displays it on a map. You’d expect that the values of the temperatures would be continuously varying between any two correspondents and such continuity is an important feature of a field.
13.5.1 Scalar Field
Figure 13.2: A weather map from the National Oceanic and Atmospheric Administration (NOAA) which is colorized to show the regions of common temperature values.
The figure above shows such a map. Continuity is manifest in the weather map in that the colors are not speckled like a pointillist painting, but continuous (transitions between the colors are continuous also…look at the scale at the top). Largely, the blues, greens, and yellows are connected and the colors indicate a continuous change of temperature across the country. This assures that fields can be described by smooth, mathematical functions.
Another feature of a field, which will become important is that if you’re holding thermometers in each hand while in a swimming pool, you expect that the temperature of the right hand thermometer only depends on the actual temperature of the water in the vicinity of the right hand – not from the temperature across the pool, or down the street. Likewise, the temperature of the left hand thermometer only depends on the water near the left hand.
This is the idea of locality…that you can describe the effect by only the local conditions. In this way, the field is an intermediate carrier of some condition.
If we have a model that’s correct about whatever that condition is (heat propagation in water) we describe the cause (the distant pool heater) as creating the condition (the temperature field) which in turn, causes the effect (your thermometer reading). (The physical mechanism that creates the temperature field in this example is the direct infrared radiation from the fire. This falls on your skin can causes its molecules to vibrate. Your nerves and brain interpret this as warmth. In the example of thermometers in a pool (just below this), the molecules of the water near the thermometers are the physical mechanism. They have kinetic energy which is actually the physical measure of heat.)
13.5.2 Vector Field
Let’s take the field idea a step further: What’s the direction of 70 degrees Fahrenheit? That’s a nonsense question, right? Temperature, like speed or mass, is called a scalar quantity, not a vector quantity. But what about the distribution of wind on a weather map, such as a hurricane? There, as is the case for all vector quantities, you care a lot about the speed – the magnitude of a hurricane’s wind velocity (which is a scalar) – and its direction (which makes it a vector). In the case of a North American hurricane, that direction is counter clockwise. So if you’re on the east coast nervously watching a hurricane just coming ashore from the Atlantic Ocean it’s probably coming at you from the northeast, (What direction is the wind if the hurricane eye has passed by you?) So wind velocity is an example of another kind of field – a vector field.
Figure 13.3: A Euro model forecast of wind speeds on the Jersey Shore during the height of 2011 Hurricane Irene: www.wunderground.com/wundermap
While a complicated mathematical subject, vector fields are easy to think about if you keep the wind-velocity idea in your head. The figure above shows another weather map, this time a model for wind velocity over the NYC region during the 2011 Hurricane Irene.
Electric and Magnetic Fields are Vector Fields with magnitudes and directions both required in order to characterize them.
How the fields change in time depends on the physics being modeled (heat? sound? mechanical vibration? electromagnetism?). A model of the particular phenomenon would consist of a set of “field equations” which would be the calculation-machinery that would lead to predictions and encompass the physics of the particular fields in question. Like any function, you supply it with a space coordinate and a time and the function will then tell you the temperature…or the wind speed and direction at that time…or the electric field strength and direction. Of course the weather person has a computer do constant evaluation over a region of space during different times and the computer has helpfully converted the evaluated function into colors.
A model of a field is a mathematical function. Sometimes the field stands for something (like kinetic energies of air molecules?). But sometimes fields are real entities themselves.
We’ll need to understand field patterns for various configurations of electric charges and currents. Just like Faraday’s magnetic field of force picture, we can do something similar for electric fields. In the spirit that a picture is worth \(10^3\) words this figure is a picture of an electrode of a positive charge – a macroscopic analog of a point electric charge like a proton. It looks alive.
Figure 13.4: A photograph of little pollen bits which orient themselves in an electric field created by the charge in the center of the photograph.
The green lines are little specks (sometimes of pollen) that are themselves influenced by electricity and align in a clearly visible pattern. The pollen specks can become differentially electrically charged and so respond to the electric field in exact analogy to little iron filings responding to a magnetic field. “Something’s there”! That was the conclusion that Faraday became convinced of for magnetic fields. Here you should have the same feeling about the “reality” of an electric field revealing itself in that green arrangement above.
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Please answer Question 1 for points: Be there or be square
13.5.3 How To Detect An Electric Field
An electric charge needn’t be of a point or an elementary particle – indeed in Faraday and Maxwell’s time, such a notion was not even imagined. Rather their subjects were macroscopically sized objects like your finger when you’ve generated a spark from walking across the carpet -— or like the silly, charged vegetable in this figure.
Figure 13.5: In the left hand figure a little positive charge is brought near positively charged piece of, um…a vegetable… and is released. The two positive charges repel with equal and opposite forces and the amount of force that each feels we could calculate using Coulomb’s law. In the right hand figure, the Electric Field lines are drawn in radiating outward from the central broccoli.
In this figure I’ve imagined a large piece of charged vegetable and Faraday’s field lines emanating outward from it just like in the figure above of the little green dielectric pieces.
Dielectric materials
Dielectric materials do not conduct electricity but can themselves carry excess charges on their surfaces and allow their atoms to twist around in place to present all of one charge one direction and the other in the opposite direction. We say they can be “polarized.” Pollen was used in many such pictures but today we have plastics that do a better job.
We say that there’s an electric field created by the positive charge. How do I know that it’s actually there? I can’t see it or taste it or hear it.
We have to interact with a field in order to know that it’s there.
This is our first example where the measurer is an integral part of the definition of a physical phenomenon! That is, in order to “see” that an electric field is present, you must introduce another charge and watch what happens to it. That’s what’s pictured in the left-hand cartoon above.
The broccoli is sitting there minding its own business and we bring a little, tiny charge, \(+q\) and place it at that point shown. The straightforward interpretation of Coulomb’s law would say that the little \(+q\) would feel a force of repulsion
\[ F = k \frac{Q q}{R^2},\]
and by Newton’s rules, it would begin to accelerate away from it. Remember the mantra: if there’s an acceleration, there’s a force and if there’s an acceleration, there has to be a force.
If we carefully note the direction of \(q\)’s acceleration and its magnitude then we can declare that an otherwise invisible electric field value is non-zero at that point and is riiiight \(\to \) there. If the little test charge does not experience a force (and stays stationary), then either that region is field-free – or, there are multiple fields present that just happen to vectorially cancel one another at that point.
Let’s think about it in terms of the right hand figure above where we have a large positive charge, \(+Q\) and a smaller positive “test” charge \(+q\) with the presumed electric field lines drawn in. How do we know that a field is there and how do we characterize it with numbers? Well, we introduce little charges…little \(q\)’s…and we see whether they accelerate and how. You do this all of the time with your car radio and with your cell phone. The little \(q\)’s are the conduction charges (electrons) in the wire of the antennas that are built into all radios and phones. When there’s an electric field in the vicinity, these little charges feel a force and start to move and that motion is a current, which is suitably sampled and turned into Mom calling to find out where you are.
There is no alternative but to declare that:Electric and magnetic fields are real.
Notice that Coulomb’s Law depends on both the big charge (\(+Q\)) and the little charge. And we might say that the electric field is produced by \(+Q\), but if we changed \(Q\), we would change the field. Further, if we insert the litte \(+q\) “test charge” we also change the field.
This is an unusual definition for a physical thing. We presume it’s there, but in order to be sure we have to probe it with something…in this case, little \(q\). How it responds tells us about the field. The “little” adjective for \(q\) means that we want to interpret our results as the field generated by “big” \(Q\) and not the effects of little \(q\) added in. On the one hand, we are really never observing the unadulterated field of \(Q\). But on the, um, other-other hand, charges are really, really small. I’ve avoided quantitative examples in electricity until now. Let’s see just how much charge we’re talking about here before worrying too much about our inability to perfectly measure the pristine field of a charge.
Back to our charged broccoli. It’s pretty easy to imagine a little test charge in the presence of any sort of charged object that we’d ever produce in a lab. So our need to not disturb the field is pretty easy. Maybe you’ll see this demonstration in a class – if not, ask Mr Google for a video of charging a “pith ball.” You’ll find that a charge that you can reasonably put on a little ball is about a micro-Coulomb, \(1\times 10^{-6} \) C. So if we don’t want to disturb the field around such a little object, we’d have to use a test charge of much less than this…say 0.1% of that? If so, then the amount of charge that we’d get away with using as just a test would be \(0.001 \times 1\times 10^{-6} = 1\times 10^{-9} \) C. But that’s still a lot of electrons-worth of charge so if we detected our field with, say, conduction electrons in a wire? We could indeed get away with this.
You Do It: How many electrons?
For that total charge of \(10^{-9}\) Coulombs, how many electrons would that be?
Work it out!
If this is bothering to you, don’t worry. You’re correct to be bothered and when we talk about Quantum Mechanics we’ll dig deeper for an interpretation.
But this workable metaphor suggested in the “cause-disturbance-effect” figure above with the hand pushing on the electron leads to a convenient, if not subtle definition of the Electric Field…just take out the little \(q\) from the force equation and define that to be the electric field:
\[E = \frac{F}{q} =\frac{1}{q} \times k \frac{Qq}{R^2} = k \frac{Q}{R^2}\]
entirely as due to \(Q\). This separates out the field from the probe of the field, thereby giving the field a stand-alone existence, created by the charge distribution (here, on broccoli).
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This works, and is kind of clever. The force depends on the product \(Qq\) so if we put in some other little charge (or even a big one), say \(p=2q\) then the magnitude of the force that \(p\) would feel is \(F_p=2F_q\):
\[F_p = k \frac{Q2q}{R^2} = 2k \frac{Qq}{R^2} = 2 F_q.\]
But when I calculate the field (which is still due to \(Q\) in this narrative), I get: the same thing as when \(q\) was the guinea pig, by always dividing out the test charge amount:
\[E = \frac{F_q}{q} = \frac{F_p}{2q}= k \frac{Q}{R^2}\]
The vector nature of the force means that the electric field is also a vector.
\[\vec{E} = \frac{\vec{F}}{q}\]
The force lines are now replaced by the Electric Field Lines as shown in the right-hand figure above. The lines get farther apart – less dense – and that’s the visual way in which we interpret the field’s strength getting weaker and weaker as we move away from \(Q\).
We define the field lines to point away from the positive \(Q\). This is a convention and coincides with the sign of the force that a positive charge would feel due to that field.
By the way the signs work out for all of the possible combinations of signs of \(\pm Q\) combined with \(\pm q\): both positive or both negative (repel, so negative force) and if different, come together (attract). That means that the direction of the electric field lines created by a negative charge is the opposite of the positive. This figure shows how to think about that for two of our more popular elementary particles, the positively charged proton and the negatively charged electron.
Figure 13.6: Electric field lines point away from a postiively charged object and towards a negatively charged object. Here both objects (proton and electron) have the same magnitude of charge so the $ ec{E}$ field lines would have the same lengths.
Remembering that Faraday had the original idea and that he did not like the idea (nobody did!) of Action at a Distance, it’s a small step from this discussion for electric fields back to the discussion of Action at a Distance from Newton’s gravitational theory. Nobody liked it! But both time and Newton’s huge reputation meant that an instantaneous influence across space for two masses was pretty much the accepted norm. Faraday’s electric lines of force were not particularly well received and it was nearly a century before people were willing to overthrow the originally distasteful Action at a Distance for something more sophisticated.
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Please answer Question 2 for points: More squares
13.6 Electric Field for Other Configurations
The Electric Field for a concentrated charge at a point gives rise to the inverse-squared strength of the above discussion and the radially-outward distribution of the field lines. But what about other distributions of electric charge? Not everything is a point-charge…but all charge distributions can be constructed from point charges. Let’s see. If I bring the proton and the electron closer together we can see that the field lines start to overlap (of course they would overlap to infinity! But here it’s more obvious):
Figure 13.7: Now the outward electric field lines from the proton overlap with those of the inward electric field lines from the electron.
Now there are two charges of opposite sign where obviously it’s apparent that there is a much different force distribution from a single charge and of course, a different field shape. It’s not too hard to think about how this is constructed: just overlay the field of a positive charge (here green) with that of a negative charge (purple) and add the vectors.
Directly in the middle you can see that they would add constructively and point to the right. Just off the center, they would also add and the results (up or down) would tend to point toward the electron, but less so, since some of the vector addition would cancel in the up and down directions. The side photograph shows the result for two charges of opposite signs. You can see how the field lines connect as they twist the little pieces of dielectric to align. The sketch below shows this in an abstract way.
Figure 13.9: The lines of force…which we’re not calling electric field lines…are readily apparent. They’ve done the vector addition for you! We call such an arrangement of oppositely charged points an “electric dipole.”
13.6.1 Line of Charge
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Let’s look at charges of the same sign, lined up side by side. This figure shows two positive charged and in the middle a region where I’ve extended two of the field lines over one another as the dotted vectors.
Figure 13.10: Two positive charges, side by side.
Can you see that in that the horizontal components of the dotted vectors will cancel, right minus left, whereas the two vertical components would add and point up? Every single field line pair of the two charges will do exactly that, leaving the net field: up or down (above or below the line connecting the charge-centers).
We can also see that in the photograph. Notice that there’s essentially a void between them where the field lines essentially cancel near the center-line.
Now we have the building-blocks of a number of different electric field arrangements. Let’s toe the line.
Suppose we now align an infinite number of identical charges, side by side on a straight line extending to infinity in each direction. What would the electric field look like for that combination? I think you can “see” it in your mind’s eye. All of the adjacent field vectors would cancel any horizontal components, neighbor by neighbor. Meanwhile, all of the vertical components would add. The end result is an electric field configuration that would point radially outward from the line — which we can imagine now as a wire with excess charge. This figure shows the result in three different views.
Figure 13.12: The top view is a two-dimensional segment of an infinite line of positive charges after all of the canceling electric field lines are eliminated. The result is only electric field pointing up and down. But this can be a three-dimensional object and the bottom left sketch tries to make that point in a poorly-drawn perspective view, while the right hand sketch shows the configuration in a view in which the line of charges points out of the screen at you.
It’s not very hard to extrapolate to a sheet of charges from this single line of charge. You just have to imagine laying an infinite number of lines of charge, side by side. The result: a three dimensional electric field, uniformly perpendicular to the “sheet” and pointing above it and below it. Here’s a segment of such a sheet:
Figure 13.13: A sheet of positive charge.
Suppose we made our sheet not from positive charges, but from negative charges. Then the arrows would all point towards the sheet rather than away from it. Here’s how that would look:
Figure 13.14: The top left shows an edge view of the previous sheet of positive charge. The top right is the same thing, but now for a sheet of negative charge with the field lines pointing to the charge rather than away from it. The bottom figure shows the two made into a sandwich.
Now let’s make a sandwich of two such sheets which have been prepared with the same magnitude of charge, but with one positive and one negative. The “meat” part of the sandwich we’ll take to be empty space. If the negative sheet is on top and the positive sheet on the bottom then the vectors from the bottom point up (the orange solid vectors) between the sheets while the vectors from the top (negative) plate point up toward from it. Now we can look in three regions:
A. and C. The regions outside of the middle have the orange solid \(\vec{E}\) vectors from the positive plate pointing in the opposite directions from the blue dashed \(\vec{E}\) vectors from the negative plate: they cancel.
B. The meat part of the sandwich between the plates shows that the \(\vec{E}\) fields are aligned in that region.
Don’t take my word for it. The next photograph shows the field lines between two parallel plates in the same dielectric material that we showed point electric charges and the dipole.
What could cause this arrangement of charges, and hence this configuration of \(\vec{E}\) fields? This cartoon sets up that narrative. A simple battery does the trick:
Figure 13.16: A boutique electric field created by an arrangement of metal plates and a battery.
This combination of metal, wires, and a battery creates a beautifully uniform and functional designer-electric field. It’s confined to the region only between the plates (if the are large in area as compared with the spacing between them) and the field created is uniform throughout that volume. In circuits such a device is called a “capacitor” but in our QS&BB life, it’s a particle accelerator.
13.7 Energy In Electric Fields
Fields are the thing. We’ve established that a positive charge (our test charge) will be repelled by a positively charged vegetable (or, anything, fruit or vegetable), but now it’s really time to drop the language that suggests any direct contact between charges. What matters is only how a charge is influenced by the fields in its vicinity – without regard to how those fields came about. We could create simple field shapes or complex ones, of course by arranging charges or shaping conductors with excess charges in them. We’ll go for “simple” since we can do a lot with that.
Remember our picture of a point-charge’s \(\vec{E}\) fields pointed radially out from a positive charge and in toward a negative charge. Our drawings suggest that the field is diluted, or less intense the further away from the charge source you look and that sense is conveyed by the fact that there are fewer \(\vec{E}\) vectors piercing any equal area the further out you look. That’s the beauty of these field drawings since that visual impression is accurate.
That’s a relatively complicated geometry, but we just strung a bunch of charges together and created a uniform field. Above, no matter where we are inside the capacitor, the same number of arrows would pierce any area parallel to the plates. Notice that I’ve put a positive charge (still the broccoli) and a negative charge (my apple is back) in between the plates…what would happen to them?
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Well, we expect a force on each and here the force is given by a slight manipulation of \(\vec{E} = \frac{\vec{F}}{q}\). Now we can sing a familiar little song:
when there’s a force, there’s an acceleration.
When there’s an acceleration, the velocity changes.
If the velocity changes, then the kinetic energy changes.
Not very catchy, but you’ll see the point. Remember the definition of the electric field in terms of a charge that senses it and the force that it experiences:
\[\vec{E} = \frac{\vec{F}}{q}\]
which if we turn it around to represent the force:
\[\vec{F} = q\vec{E}.\]
This reminds us that a positive charge will experience a force that is oriented along the electric field direction. Let’s set up a fake-unit situation. I’ve brought the broccoli out of retirement and charged it positively to \(q_B=+2\). You’ve been wondering where the apple has been and so you’re relieved to see that it’s playing a new role, that of a negative charge of \(q_A=-4\) (forgetting units). So \(\vec{F_B}=+2\vec{E}\) and \(\vec{F_A}=-4\vec{E}\) where the negative sign obviously reminds us that this force is in the opposite direction of the other one (“positive” setting the direction).
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Please study Example 2: Forces on produce:
This is the important question: which charge will gain the most kinetic energy? Normally, we would have expected the mass of the accelerated object to matter. Does it here?
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Kinetic Energy of Electric Charges In An Electric Field
How much kinetic energy does an electric charge gain when it’s placed in an electric field? Assume that it starts from rest and placed in the center of the capacitor above. Also assume that they don’t interact with each other. Finally, assume that the kinetic energies are compared after they have both moved a distance, \(d\) from the center.
Here’s how to think about this: \[\vec{F} = q\vec{E}\] gives us a constant force. From Newton’s Second Law, we can calculate the constant acceleration that would result…in terms of the mass of each charge. From one of our kinematics equations (\(v^2 = 2ax\)) we can calculate the speed-squared. Then from there, it’s a simple move to calculate the kinetic energy, \(K=1/2mv^2\). Okay? Here we go:
String together Newton’s Second Law with our electric field force, and solve for the acceleration, \(a\) in terms of the mass of the positive charge, \(m_p\):
\[\begin{aligned} F &= m_pa = qE \nonumber \\ a & =\frac{qE}{m_p} \nonumber \end{aligned}\] Now find the speed squared from \(v^2 = 2ax\) after it’s gone through a distance, \(d\):
\[v^2 = 2ax = 2\frac{qE}{m_p}d\]
And now determine the kinetic energy,
\[\begin{aligned} K &= 1/2 mv^2 = \frac{1}{2}m_p\left( 2\frac{qE}{m_p} \right)d \nonumber \\ K & = qEd \label{parallelenergy} \end{aligned}\]
See how the mass cancelled? Since the positive charge is half in magnitude of the negative charge, then the kinetic energy of the negative charge will be twice that of the positive.
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Please study Example 3: Kinetic energies of produce:
This is an important conclusion. For our constant electric field, the kinetic energy acquired by electric charges depends only on the charge and not the mass of the object! For example an electron would get the same kinetic energy boost than the much heavier proton. Here, the kinetic energy of the charges comes from the force, but the force comes from the field. We conclude that fields carry energy…we could say that
Electric fields store energy and can do work on electric charges.
If a circumstance can create a kinetic energy, it’s reasonable to think of a potential energy that enables it and that’s the case here. For example, the term “voltage” comes in relating the work done on a charge in an electric field. When you deploy a battery in your flashlight, you’re arming it to supply energy to electrons to force them through the circuit and the higher the voltage rating, the more current you can supply.
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It’s useful to think about the energy that could be expended in moving a charge, and that’s just the electrical potential energy, \(U\). So the kinetic energy we found above, \(K = qEd\) can be more conventionally recast as: A particular arrangement of electric charge will change how \(U\) relates to \(E\):
\[U=qEd,\]
which is just a force (\(qE\)) times a distance (\(d\)), like we first learned.
“Voltage” is then the potential (energy) per unit charge with units of Volts (V), or Joules/Coulomb. So
\[ 1\text{ V} = 1 \text{ J/c}.\]
The form of the voltage, or \(U\) depends on the configuration of the electrified plates that create the field. For this simple arrangement of parallel flat plates,
\[\text{from } U=qEd, \text{ we get } V=Ed\]
which in turn leads us to the more practical measure of Electric Field of “Volts/meter,” which is pretty much universally used in engineering. (But from the original definition of the field, \(E=F/q\), “Newton’s per Coulomb” is another, perfectly acceptable unit for the field). Here are some typical electric fields that you might encounter.
| Source | Electric Field Strength, V/m Comments | ||
|---|---|---|---|
| atmosphere | 100-150 | near the surface of the Earth | |
| home background | <100 | typical home | |
| inside your grandparents’ TV tube | 40,000 | ||
| near an electric blanket | <1,000 | in typical use | |
| near a microwave oven | 600 | ||
| near a cell phone | <50 | <0.1 inside your body just below the surface | |
| below a huge 500~kV power distribution line | 10,000/m | some states restrict to 3-5 KV/m |
Let’s put this together.
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Please study Example 4: Energy in your parents’ old TV:
All of that voltage for just a tiny bit of kinetic energy in each electron? There are lots and lots of electrons!
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Please answer Question 3 for points: Charge trapped between plates
13.8 Electron Volts
All of these powers of 10 are really irritating. More importantly, they represent mistakes just waiting to happen. Not to fear! We have a very useful unit of energy that works very nicely for atomic physics, nuclear physics, and particle physics: the “electron-volt” aka, “eV.” Here’s how it works.
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Suppose we have another capacitor that has a 1 V battery connected to it. What’s the energy that a single electron (or proton, for that matter) would acquire as it’s accelerated through that 1 Volt difference? By now we know that it would be: \[ U = qV = (1.6\times 10^{-19}\text{ C})( 1 \text{ J/C}) = 1.6 \times 10^{-19} \text{ J}\]
Ah. Since the fundamental electric charge quantity figures into so many things, let’s keep it hanging around as a symbol and define the electron volt to be:
\[1 \text{eV} = 1.6 \times 10^{-19}\text{ J}.\]
We’ll see how this becomes useful and we’ll try to think in “electron volts” when we deal in energies appropriate to elementary particles.
13.9 Magnetic Fields
By this point, it’s no surprise that the field idea could be applied to magnetic configurations, but with a twist. We know of no magnetic charges which would be called “magnetic monopoles.” Not for lack of trying! Many theories of the beginning of the universe demand that they exist. But just as when you might chop a bar magnet in two, and then chop one of the pieces in two, and then again, and again…you will never find a separate North and South pole! Only N-S pole pairs seem to exist, all the way to the atomic level.
The field due to a bar magnet also follows the lines of force and start on the North pole and stop on the South.
Ampere’s guessed that magnetism was due to little circular currents and that’s compatible with the distribution of field lines as can be seen by comparing the iron filings from the bar magnet as sketched by Faraday in the figure at the right with those of the straight wire from his notebook in the previous lesson. Here is a photograph of a more modern attempt:
Figure 13.18: Iron filings arranging themselves around a wire carrying current out of the plane of the photograph
The concentric circles patiently mapped out by the little iron filings are an indirect map of the concentric circular magnetic field lines, which we’ll refer to as the magnetic field vector \(\vec{B}\). Some reorientation is required. So let’s introduce our first “Right-Hand-Rule.”
This idea of circular fields of influence suggested by (the mathematically illiterate Faraday) was not at all well received by the sophisticated and learned British Newton Fan Club. Forces were supposed to be straight. Ampere’s law (Ampere, the Frenchman) states that a current of magnitude \(I\) will produce a magnetic field in concentric circles around the wire in a direction that you can predict with your right hand: put your thumb in the direction of the current and unless you are built very strangely, your fingers will curl around the direction of the magnetic field, \(\vec{B}\). Here’s how to do it:
Figure 13.19: Demonstrating the right hand rule to show the direction of a magnetic field around a wire. Notice the inset of a compass needle confirming that direction, ala’ Oersted.
The value of the magnetic field diminishes the further one is from the wire, but unlike Coulomb’s Law, the rate of decrease is inversely proportional to the distance: \[B=k'\frac{I}{2\pi R}\] (\(k'\) is a constant that depends on the material outside of the wire.)
Remember Oersted’s discovery? A compass would align itself around a wire when brought near a current. A compass is nothing but a little bar magnet and his discovery was just the statement that the magnet aligns itself with the magnetic field with the north pole following the \(\vec{B}\) field direction. Of course that’s all a compass is doing as a navigational device, since there is a tiny magnetic field due to molten currents in the core of the earth. It’s following the Earth’s \(\vec{B}\) field pointing to the geographical North pole, which is the magnetic South pole.
Now let’s take the wire and bend it into a circle. The field is still concentric around the wire, but look at how its field manifests itself:
Figure 13.20: CAPTION
Inside of the circle, the field is concentrated where all of the field lines add together (imagine wrapping your fingers around the wire, all around the wire). The field rises out of the plane of the circle, traverses around and returns through the loop from below. It’s exactly the form of the field of a bar magnet. Now Ampere didn’t know this, but his imagination was such that he got it right.
If we make a tube of current circles like a Slinky, the field lines continue to add inside and there result is a useful circuit element called a Solenoid (also useful as a part of your car’s starter circuitry).
Figure 13.21: Many loops of current that are side by side concentrate the magnetic field on the inside and closely eliminate it on the outside.
The solenoidal magnetic field is nearly uniform inside of the “slinky” wire arrangement and in some sense is the magnetic field analog to the uniform electric field of a parallel plate capacitor.
We can do one more thing with the solenoid…wrap it around a circle like a donut, where the wires encircle the dough. This is called a toroid and is the last field configuration that we’ll need.
Figure 13.22: Iron filings tracing out the magnetic field of a toroid.
In a toroid the magnetic field created is essentially uniform and confined to be within the “donut dough” with no field in the “hole” or around the outside.
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Please answer Question 4 for points: Magnetic fields from currents
These are the field configurations that we’ll draw on periodically:
- Electric field due to a point charge.
- Electric field due to oppositely charged parallel plates.
- Magnetic field due to a current.
- Magnetic field due to a loop.
- Magnetic field due to a solenoid.
- Magnetic field due to a toroid.
Now let’s build the modern 20th century with Dr Maxwell in the next lesson.
13.10 What to Remember from Lesson 13
13.10.1 Fields
The notion of a field is critical for us. In essence, a field is a disturbance in space which links together a cause and an effect. In electricity, the cause might be an electrically charged object and the effect – some distance away – might be a force that a different electric charge might feel. The disturbance between them is the Electric Field. This field idea will receive corrections when we get to quantum mechanics, but this image is a pretty reliable one.
BTW. This worked so well for electricity that we certainly also think of a gravitational field – another “disturbance” where the source might be some mass like a planet or an apple and the effect would have to be some other mass. The force that the “effect” mass feels, unlike in electricity, is always attractive.
There are two kinds of fields, those in which one can assign just a number (with some units) to every point in space. The temperature field on TV is the perfect example and when direction doesn’t matter, we call these “scalar” fields.
The other kind of field is one in which not only can you assign a number to every point in space, but also a direction associated with that number. Again, the weather is the perfect example where the distribution of winds is such a field, called a “vector” field. The Electric and Magnetic fields are both vector fields, where the direction and the magnitudes are a little harder to get a feel for.
13.10.2 Electric Field
We know that the electric field is a real thing because not only does it make a cause and effect relationship happen, but between the two objects, one can tickle the space with another charge and watch it move. By carefully watching what direction a little test electric charge is moved and the force that it feels, one can map out the force field.
But the force that a test charge would feel depends on the charge of the source and the charge of the test field. It’s not useful to have a physical quantity’s value depend on the probe so…we divide it out leaving a quantity that is only related to the source itself. So for a point electric charge of \(Q\), the force that a little test \(q\) would feel is: \[ F=k\frac{Qq}{R^2} \nonumber \] So if we divide out the little \(q\) we create a quantity that we define to be the Electric Field, \(E\): \[ E=\frac{F}{q} = k \frac{Q}{R^2} \nonumber \] Now it doesn’t matter if the test probe is \(q=1\) or \(q=50\), the result for the value of the Electric Field is the same.
Many different real electronic components are constructed from engineered arrangements of electric charges. We’ll consider only:
- point electric charges, singly and in pairs
- lines of electric charges
- sheets of electric charges
- two parallel sheets of opposite electric charges
The shapes of the resulting electric fields are shown in the text.
13.11 Energy in Electric Fields
Here’s one of our mantras:
- If there is a force, then there’s an acceleration
- If there’s an acceleration, then there’s a change in velocity
- If there’s a change in velocity, then there’s a change in kinetic energy
So somehow when an electric charge is dropped into an electric field it will gain kinetic energy and above we learned that for a uniform electric field, \(E\), the kinetic energy that any charge \(Q\) would gain as it experiences the electric force through a distance \(x\) would be \[ K = U = QEx \nonumber \] (Notice that’s a force times a distance, which is work.) Where does this new energy come from? From the electric field itself. Electric fields store energy and can transmit energy.
It’s also useful to think of the fields ability to perform work on a charge…where I’ve intentionally appropriated the language we used in Lesson 7. The field is fixed and through some distance \(x\) the amount of energy gained depends on the charge that’s put into the field. So a big charge would experience a large kinetic energy and a small one, less. But what’s constant – what’s Potentially possible doesn’t depend on the charge, but only the value of the field and the distance. So we speak of a Potential Energy per unit charge and give it a name that you know: Volts (V)…which is also Joules/Coulomb. \[ V=\frac{U}{Q} = Ex \nonumber \] This in turn leads to a practical unit for the electric field of Volts/meter in addition to Newtons/Coulomb.
13.11.1 Electron Volts
Coulombs is an old unit that stuck around. It came from the time when people were fretting over electrically charged big objects. Now we know that these big objects had an enormous number of our fundamental atomic electric charge of \(e=1.6 \times 10^{-19}\) C. Those \(10^{-19}\)’s are just arithemetic mistakes waiting to happen, so we have invented another unit more applicable to atomic physics and chemistry: the electron-volt.
A single electron volt is the amount of energy that a fundamental charge, like a proton or electron, gains while being accelerated through a potential of 1 Volt. That number is: \[ 1 \text{ eV}=e \times 1 { V} = 1.6 \times 10^{-19} \text{ C} \times 1 \text{ J/C} = 1.6 \times 10^{-19} \text{ J} \]
13.12 Magnetic Fields
Just like there is a definition for the electric field, there is one for magnetic fields. These are a little harder to calculate and so we’ll leave it at the picture level, where the whole point of a magnetic field is that a current causes a magnetic field to be produced around the wire, according to the right hand rule: put your thumb in the direction of the current and your fingers will curl around in the direction that the magnetic field points – for your right hand only. A number of pictures of different magnetic configurations are above and the only ones that we’ll care about are:
- the magnetic field of a single wire
- the magnetic field of a current loop
- the magnetic field of a solenoidal current
- the magnetic field of a torridal current