Müschenbroek’s Wonderful Bottle

Abstract

In his second letter to Mr. Collinson, Franklin presented his one-fluid theory of electricity. Contrary to Du Fay’s two-fluid theory, Franklin suggested that all objects contain within them a certain quantity of the so-called “electrical fire”. While the overall quantity of this electrical fire is unchanged—it is a conserved quantity—its distribution need not be uniform everywhere. Objects having a (perhaps temporary) surplus are said to be “plus”; objects having a deficit are said to be “minus”.

Such charge separation commonly occurs with friction between different objects, as when glass is rubbled with silk (the glass becomes “plus” and the silk becomes “minus”), or amber with fur (the amber becomes “minus” and the fur becomes “plus”). In this way, Franklin was able to account for many electrical phenomena by introducing the concept of positive and negative electricity. At this point, you might pause to consider whether (or to what extent) Franklin’s theory of electricity is different than the modern view.

Now, in his third letter to Mr. Collinson, Franklin deploys his one-fluid theory so as to understand the working of “Mueschenbroek’s wonderful bottle”. Where might the bottle store its electricity when electrified? In the water? In the glass? In the central wire?

So wonderfully are these two states of Electricity, the plus and the minus, combined and balanced in this miraculous bottle!

—Benjamin Franklin

4.1 Introduction

In his second letter to Mr. Collinson, Franklin presented his one-fluid theory of electricity. Contrary to Du Fay’s two-fluid theory, Franklin suggested that all objects contain within them a certain quantity of the so-called “electrical fire”. While the overall quantity of this electrical fire is unchanged—it is a conserved quantity—its distribution need not be uniform everywhere. Objects having a (perhaps temporary) surplus are said to be “plus”; objects having a deficit are said to be “minus”. Such charge separation commonly occurs with friction between different objects, as when glass is rubbled with silk (the glass becomes “plus” and the silk becomes “minus”), or amber with fur (the amber becomes “minus” and the fur becomes “plus”). In this way, Franklin was able to account for many electrical phenomena by introducing the concept of positive and negative electricity. At this point, you might pause to consider whether (or to what extent) Franklin’s theory of electricity is different than the modern view.

Now, in his third letter to Mr. Collinson, Franklin deploys his one-fluid theory so as to understand the working of “Mueschenbroek’s wonderful bottle”. Where might the bottle store its electricity when electrified? In the water? In the glass? In the central wire? When reading through these letters, it helps to recognize that materials may be divided into two classes: conductors, such as iron and saltwater, which readily transport electricity through them; and insulators, such as wax and glass, which do not. In a subsequent letter Franklin employs these terms, but in the letter below Franklin uses the older terminology, referring to insulators as electrics per se and to conductors as non-electrics.

4.2 Reading: Franklin, Experiments and Observations on Electricity

Franklin, B., Experiments and Observations on Electricity, fourth ed., Henry, David, London, 1769. Letter III, and selections from Letters IV and XXXVIII.

4.2.1 Letter III

From Benj. Franklin, Esq; at Philadelphia, To Peter Collinson, Esq; F.R.S. London. Sept. 1, 1747.

Sir,

The necessary trouble of copying long letters, which, perhaps, when they come to your hands may contain nothing new, or worth your reading, (so quick is the progress made with you in Electricity) half discourages me from writing any more on that subject. Yet I cannot forbear adding a few observations on M. Muschenbroek’s wonderful bottle.

• 1. The non-electric contain’d in the bottle differs when electrised from a non-electric electrised out of the bottle, in this: that the electrical fire of the latter is accumulated on its surface, and forms an electrical atmosphere round it of considerable extent; but the electrical fire is crowded into the substance of the former, the glass confining it.¹

• 2. At the same time that the wire and top of the bottle, &c. is electrised positively or plus, the bottom of the bottle is electrised negatively or minus, in exact proportion: i.e. whatever quantity of electrical fire is thrown in at the top, an equal quantity goes out of the bottom.² To understand this, suppose the common quantity of electricity in each part of the bottle, before the operation begins, is equal to 20; and at every stroke of the tube, suppose a quantity equal to 1 is thrown in; then, after the first stroke, the quantity contain’d in the wire and upper part of the bottle will be 21, in the bottom 19. After the second, the upper part will have 22, the lower 18, and so on, till, after 20 strokes the upper part will have a quantity of electrical fire equal to 40, the lower part none: and then the operation ends: for no more can be thrown into the upper part, when no more can be driven out of the lower part. If you attempt to throw more in, it is spued back through the wire, or flies out in loud cracks through the sides of the bottle.

• 3. The equilibrium cannot be restored in the bottle by inward communication or contact of the parts; but it must be done by a communication formed without the bottle between the top and bottom, by some non-electric, touching or approaching both at the same time; in which case it is restored with a violence and quickness inexpressible; or, touching each alternately, in which case the equilibrium is restored by degrees.

• 4. As no more electrical fire can be thrown into the top of the bottle, when all is driven out of the bottom, so in a bottle not yet electrised, none can be thrown into the top, when none can get out at the bottom; which happens either when the bottom is too thick, or when the bottle is placed on an electric per se. Again, when the bottle is electrised, but little of the electrical fire can be drawn out from the top, by touching the wire, unless an equal quantity can at the same time get in at the bottom.³ Thus, place an electrised bottle on clean glass or dry wax, and you will not, by touching the wire, get out the fire from the top. Place it on a non-electric, and touch the wire, you will get it out in a short time; but soonest when you form a direct communication as above.

  So wonderfully are these two states of Electricity, the plus and minus, combined and balanced in this miraculous bottle! situated and related to each other in a manner that I can by no means comprehend! If it were possible that a bottle should in one part contain a quantity of air strongly comprest, and in another part a perfect vacuum, we know the equilibrium would be instantly restored within. But here we have a bottle containing at the same time a plenum of electrical fire, and a vacuum of the same fire; and yet the equilibrium cannot be restored between them but by a communication without! though the plenum presses violently to expand, and the hungry vacuum seems to attract as violently in order to be filled.

• 5. The shock to the nerves (or convulsion rather) is occasioned by the sudden passing of the fire through the body in its way from the top to the bottom of the bottle. The fire takes the shortest course, as Mr. Watson justly observes: But it does not appear from experiment that in order for a person to be shocked, a communication with the floor is necessary: for he that holds the bottle with one hand, and touches the wire with the other, will be shock’d as much, though his shoes be dry, or even standing on wax, as otherwise. And on the touch of the wire, (or of the gunbarrel, which is the same thing) the fire does not proceed from the touching finger to the wire, as is supposed, but from the wire to the finger, and passes through the body to the other hand, and so into the bottom of the bottle.

4.2.2 Experiments Confirming the Above

Experiment I.:

Place an electrised phial on wax; a small cork-ball suspended by a dry silk thread held in your hand, and brought near to the wire, will first be attracted, and then repelled: when in this state of repellency, sink your hand, that the ball may be brought towards the bottom of the bottle; it will be there instantly and strongly attracted, till it has parted with its fire.

If the bottle had a positive electrical atmosphere, as well as the wire, an electrified cork would be repelled from one as well as from the other.

Experiment II.:

(Fig. 4.1a) From a bent wire ($A$) sticking in the table, let a small linen thread ($b$) hang down within half an inch of the electrised phial ($C$). Touch the wire of the phial repeatedly with your finger, and at every touch you will see the thread instantly attracted by the bottle. (This is best done by a vinegar cruet, or some such belly’d bottle). As soon as you draw any fire out from the upper part, by touching the wire, the lower part of the bottle draws an equal quantity in by the thread.

Fig. 4.1

Apparatus for Franklin’s Experiments II (a) and III (b).—[K.K.]

Experiment III.:

(Fig. 4.1b) Fix a wire in the lead, with which the bottom of the bottle is armed ($d$) so as that bending upwards, its ring end may be level with the top or ring-end of the wire in the cork ($e$) and at 3 or 4 in. distance. Then electricise the bottle, and place it on wax. If a cork suspended by a silk thread ($f$) hang between these two wires, it will play incessantly from one to the other, till the bottle is no longer electrised; that is, it fetches and carries fire from the top to the bottom⁴ of the bottle, till the equilibrium is restored.

Experiment IV.:

(Fig. 4.2a) Place an electrised phial on wax; take a wire ($g$) in form of a $C$, the ends at such a distance when bent, as that the upper may touch the wire of the bottle, when the lower touches the bottom: stick the outer part on a stick of sealing-wax ($h$), which will serve as a handle; then apply the lower end to the bottom of the bottle, and gradually bring the upper end near the wire in the cork. The consequence is, spark follows spark till the equilibrium is restored. Touch the top first, and on approaching the bottom with the other end, you have a constant stream of fire from the wire entering the bottle. Touch the top and bottom together, and the equilibrium will instantly be restored; the crooked wire forming the communication.

Fig. 4.2

Apparatus for Franklin’s Experiments IV (a) and V (b).—[K.K.]

Experiment V.:

(Fig. 4.2b) Let a ring of thin lead, or paper, surround a bottle ($i$) even at some distance from or above the bottom. From that ring let a wire proceed up, till it touch the wire of the cork ($k$). A bottle so fixt cannot by any means be electrised: the equilibrium is never destroyed: for while the communication between the upper and lower parts of the bottle is continued by the outside wire, the fire only circulates: what is driven out at bottom, is constantly supplied from the top.⁵ Hence a bottle cannot be electrised that is foul or moist on the outside, if such moisture continue up to the cork or wire.

Experiment VI.:

Place a man on a cake of wax, and present him the wire of the electrified phial to touch, you standing on the floor, and holding it in your hand. As often as he touches it, he will be electrified plus; and anyone standing on the floor may draw a spark from him. The fire in this experiment passes out of the wire into him; and at the same time out of your hand into the bottom of the bottle.

Experiment VII.:

Give him the electrical phial to hold; and do you touch the wire; as often as you touch it he will be electrified minus, and may draw a spark from anyone standing on the floor. The fire now passes from the wire to you, and from him into the bottom of the bottle.

Experiment VIII.:

Lay two books on two glasses, back towards back, 2 or 3 in. distant. Set the electrified phial on one, and then touch the wire; that book will be electrified minus; the electrical fire being drawn out of it by the bottom of the bottle. Take off the bottle, and holding it in your hand, touch the other with the wire; that book will be electrified plus; the fire passing into it from the wire, and the bottle at the same time supplied from your hand. A suspended small cork-ball will play between these books till the equilibrium is restored.

Experiment IX.:

When a body is electrised plus, it will repel an electrified feather or small cork-ball. When minus (or when in the common state) it will attract them, but stronger when minus than when in the common state, the difference being greater.

Experiment X.:

Though, as in Experiment VI, a man standing on wax may be electrised a number of times by repeatedly touching the wire of an electrised bottle (held in the hand of one standing on the floor) he receiving the fire from the wire each time: yet holding it in his own hand, and touching the wire, though he draws a strong spark, and is violently shocked, no electricity remains in him; the fire only passing through him, from the upper to the lower part of the bottle. Observe, before the shock, to let some one on the floor touch him to restore the equilibrium in his body; for in taking hold of the bottom of the bottle, he sometimes becomes a little electrised minus, which will continue after the shock, as would also any plus Electricity, which he might have given him before the shock. For restoring the equilibrium in the bottle, does not at all affect the Electricity in the man through whom the fire passes; that Electricity is neither increased nor diminished.

Experiment XI.:

The passing of the electrical fire from the upper to the lower part⁶ of the bottle, to restore the equilibrium, is rendered strongly visible by the following pretty experiment. Take a book whose covering is filletted with gold; bend a wire of 8 or 10 in. long, in the form of (m) Fig. 4.3. Slip it on the end of the cover of the book, over the gold line, so as that the shoulder of it may press upon one end of the gold line, the ring up, but leaning towards the other end of the book. Lay the book on a glass or wax, and on the other end of the gold lines set the bottle electrised: then bend the springing wire, by pressing it with a stick of wax till its ring approaches the ring of the bottle wire, instantly there is a strong spark and stroke, and the whole line of gold, which completes the communication, between the top and bottom of the bottle, will appear a vivid flame, like the sharpest lightning. The closer the contact between the shoulder of the wire, and the gold at one end of the line, and between the bottom of the bottle and the gold at the other end, the better the experiment succeeds. The room should be darkened. If you would have the whole filletting round the cover appear in fire at once, let the bottle and wire touch the gold in the diagonally opposite corners.

Fig. 4.3

Apparatus for Franklin’s Experiment XI.—[K.K.]

I am, &c.

B. Franklin.

4.2.3 Letter IV

From Benj. Franklin, Esq; at Philadelphia, To Peter Collinson, Esq; F.R.S. London. Further Experiments and Observations in Electricity. 1748

Sir,…⁷

• 13. Glass, in like manner, has, within its substance, always the same quantity of electrical fire, and that a very great quantity in proportion to the mass of glass, as shall be shewn hereafter.

• 14. This quantity, proportional to the glass, it strongly and obstinately retains, and will have neither more nor less though it will suffer a change to be made in its parts and situation; i.e. we may take away part of it from one of the sides, provided we throw an equal quantity into the other.

• 15. Yet when the situation of the electrical fire is thus altered in the glass; when some has been taken from one side, and some added to the other, it will not be at rest or in its natural state, till it is restored to its original equality. And this restitution cannot be made through the substance of the glass, but must be done by a non-electric communication formed without, from surface to surface.

• 16. Thus, the whole force of the bottle, and power of giving a shock, is in the GLASS ITSELF; the non-electrics in contact with the two surfaces, serving only to give and receive to and from the several parts of the glass; that is, to give on one side, and take away from the other.

• 17. This was discovered here in the following manner: Purposing to analyze the electrified bottle, in order to find wherein its strength lay, we placed it on glass, and drew out the cork and wire which for that purpose had been loosely put in. Then taking the bottle in one hand, and bringing a finger of the other near its mouth, a strong spark came from the water, and the shock was as violent as if the wire had remained in it, which shewed that the force did not lie in the wire. Then to find if it resided in the water, being crouded into and condensed in it, as confined by the glass, which had been our former opinion, we electrified the bottle again, and placing it on glass, drew out the wire and cork as before; then taking up the bottle, we decanted all its water into an empty bottle, which likewise stood on glass; and taking up that other bottle, we expected, if the force resided in the water, to find a shock from it; but there was none. We judged then that it must either be lost in decanting, or remain in the first bottle. The latter we found to be true; for that bottle on trial gave the shock, though filled up as it stood with fresh unelectrified water from a tea-pot. To find, then, whether glass had this property merely as glass, or whether the form contributed anything to it; we took a pane of sash-glass, and laying it on the hand, placed a plate of lead on its upper surface; then electrified that plate, and bringing a finger to it, there was a spark and shock. we then took two plates of lead of equal dimensions, but less than the glass by 2 in. every way, and electrified the glass between them, by electrifying the uppermost lead; then separated the glass from the lead, in doing which, what little fire might be in the lead was taken out, and the glass being touched in the electrified parts with a finger, afforded only very small pricking sparks, but a great number of them might be taken from different places. Then dexterously placing it again between the leaden plates, and completing a circle between the two surfaces, a violent shock ensued. Which demonstrated the power to reside in glass as glass, and that the non-electrics in contact served only, like the armature of a loadstone, to unite the force of the several parts, and bring them at once to any point desired: it being the property of a non-electric, that the whole body instantly receives or gives what electrical fire is given to or taken from any one of its parts.

• 18. Upon this we made what we called an electrical-battery, consisting of eleven panes of large sash-glass, arm’d with thin leaden plates, pasted on each side, placed vertically, and supported at 2 in. distance on silk cords, with thick hooks of leaden wire, one from each side, standing upright, distant from each other, and convenient communications of wire and chain, from the giving side of one pane, to the receiving side of the other; that so the whole might be charged together, and with the same labour as one single pane; and another contrivance to bring the giving sides, after charging, in contact with one long wire, and the receivers with another, which two long wires would give the force of all the plates of glass at once through the body of any animal forming the circle with them.⁸ The plates may also be discharged separately, or any number together that is required. But this machine is not much used, as not perfectly answering our intention with regard to the ease of charging, for the reason given, Sect. 10.⁹ We made also of large glass panes, magical pictures, and self-moving animated wheels, presently to be described.

4.2.4 Letter XXXVIII

To Mr. Kinnersley, in answer to the foregoing. London, Feb. 20, 1762.

Sir,…¹⁰

You know I have always look’d upon and mentioned the equal repulsion in cases of positive and of negative electricity, as a phænomenon difficult to be explained. I have sometimes, too, been inclined, with you, to resolve all into attraction; but besides that attraction seems in itself as unintelligible as repulsion, there are some appearances of repulsion that I cannot so easily explain by attraction; this for one instance. When the pair of cork balls are suspended by flaxen threads, from the end of the prime conductor, if you bring a rubbed glass tube near the conductor, but without touching it, you see the balls separate, as being electrified positively;¹¹ and yet you have communicated no electricity to the conductor, for, if you had, it would have remained there, after withdrawing the tube; but the closing of the balls immediately thereupon, shows that the conductor has no more left in it than its natural quantity. Then again approaching the conductor with the rubbed tube, if, while the balls are separated, you touch with a finger that end of the conductor to which they hang, they will come together again, as being, with that part of the conductor, brought to the same state with your finger, i.e. the natural state. But the other end of the conductor, near which the tube is held, is not in that state, but in the negative state, as appears on removing the tube; for then part of the natural quantity left at the end near the balls, leaving that end to supply what is wanting at the other, the whole conductor is found to be equally in the negative state. Does not this indicate that the electricity of the rubbed tube had repelled the electric fluid, which was diffused in the conductor while in its natural state, and forced it to quit the end to which the tube was brought near, accumulating itself on the end to which the balls were suspended? I own I find it difficult to account for its quitting that end, on the approach of the rubbed tube, but on the supposition of repulsion; for, while the conductor was in the same state with the air, i.e. the natural state, it does not seem to me easy to suppose, that an attraction should suddenly take place between the air and the natural quantity of the electric fluid in the conductor, so as to draw it to, and accumulate it on the end opposite to that approached by the tube; since bodies, possessing only their natural quantity of that fluid, are not usually seen to attract each other, or to affect mutually the quantities of electricity each contains.

There are likewise appearances of repulsion in other parts of nature. Not to mention the violent force with which the particles of water, heated to a certain degree, separate from each other, or those of gunpowder, when touch’d with the smallest spark of fire, there is the seeming repulsion between the same poles of the magnet, a body containing a subtle moveable fluid, in many respects analagous to the electric fluid. If two magnets are so suspended by strings, as that their poles of the same denomination are opposite to each other, they will separate, and continue so; or if you lay a magnetic steel bar on a smooth table, and approach it with another parallel to it, the poles of both in the same position, the first will recede from the second, so as to avoid the contact, and may thus be push’d (or at least appear to be push’d) off the table. Can this be ascribed to the attraction of any surrounding body or matter drawing them asunder, or drawing the one away from the other? If not, and repulsion exists in nature, and in magnetism, why may it not exist in electricity? We should not, indeed, multiply causes in philosophy without necessity; and the greater simplicity of your hypothesis would recommend it to me, if I could see that all appearances might be solved by it.¹² But I find, or think I find, the two causes more convenient than one of them alone. Thus I would solve the circular motion of your horizontal stick, supported on a pivot, with two pins at their ends, pointing contrary ways, and moving in the same direction when electrified, whether positively or negatively: When positively, the air opposite the points being electrified positively, repels the points; when negatively, the air opposite the points being also, by their means, electrified negatively, attraction takes place between the electricity in the air behind the heads of the pins, and the negative pins, and so they are, in this case, drawn in the same direction that in the other they were driven. You see I am willing to meet you half way, a complaisance I have not met with in our brother Nollet, or any other hypothesis-maker, and therefore may value myself a little upon it, especially as they say I have some ability in defending even the wrong side of a question, when I think fit to take it in hand.

What you give as an established law of the electric fluid, “That quantities of different densities mutually attract each other, in order to restore the equilibrium,” is, I think, not well founded, or else not well express’d. Two large cork balls, suspended by silk strings, and both well and equally electrified, separate to a great distance. By bringing into contact with one of them, another ball of the same size, suspended likewise by silk, you will take from it half its electricity. It will then, indeed, hang at a less distance from the other, but the full and the half quantities will not appear to attract each other, that is, the balls will not come together. Indeed, I do not know any proof we have, that one quantity of electric fluid is attracted by another quantity of that fluid, whatever difference there may be in their densities. And, supposing in nature, a mutual attraction between two parcels of any kind of matter, it would be strange if this attraction should subsist strongly while those parcels were unequal, and cease when more matter of the same kind was added to the smallest parcel, so as to make it equal to the biggest. By all the laws of attraction in matter, that we are acquainted with, the attraction is stronger in proportion to the increase of the masses, and never in proportion to the difference of the masses. I should rather think the law would be, “That the electric fluid is attracted strongly by all other matter that we know of, while the parts of that fluid mutually repel each other.” Hence its being equally diffused (except in particular circumstances) throughout all other matter. But this you jokingly call “electrical orthodoxy.” It is so with some at present, but not with all; and, perhaps, it may not always be orthodoxy with any body. Opinions are continually varying, where we cannot have mathematical evidence of the nature of things; and they must vary. Nor is that variation without its use, since it occasions a more thorough discussion, whereby error is often dissipated, true knowledge is encreased, and its principles become better understood and more firmly established.

4.3 Study Questions

Ques. 4.1. How, exactly, does Mueschenbroek’s bottle store electrification?

• a) How is the bottle constructed? How can it be electrified?

• b) How is the electrification distributed on, or in, the bottle? Is there a maximum amount of electrification that it can hold? What happens if one attempts to add more?

• c) In what sense are positive and negative electricity balanced when the bottle is electrified? Why does Franklin find this distribution of electrification wonderful—and a bit puzzling?

• d) When a person standing on a block of wax holds the bottle in his hand, can he experience a shock from the bottle? How?

Ques. 4.2. How does Franklin demonstrate that the outside and inside of a Leyden jar have opposite charges?

• a) Is a small neutral cork-ball attracted to the wire projecting from an electrified Leyden jar? What happens when the cork-ball touches the wire? What does this suggest?

• b) What happens to a cork suspended between the ring-ends of two wires attached to the outside and inside, respectively, of an electrified Leyden jar (as in Fig. 4.1b of Franklin’s third experiment)? How is this similar to Franklin’s fourth experiment?

• c) Can a fouled or wet Leyden jar be electrified? Why do you suppose this is the case, in light of Franklin’s fifth experiment?

• d) Why does the gold foil coating of an old book momentarily catch fire when, as in Franklin’s eleventh experiment, it connects the outside and inside of a Leyden jar?

Ques. 4.3. Where, exactly, is the electrification of a Leyden jar stored? How do you know?And can a flat plate of glass (rather than a glass bottle) store electricity?

Ques. 4.4. Describe Franklin’s electrical battery. What are the two different configurations he considers? Does Franklin’s battery generate electrification, or merely store electrification?

Ques. 4.5. Can all electrical forces be somehow understood as purely attractive?

• a) What happens to two balls suspended from the prime conductor inside a Leyden jar when a charged rod approaches the end of the conductor protruding through the cork? What happens when the charged rod is withdrawn? What does this imply?

• b) What happens to the balls if a finger is briefly touched to the prime conductor while the charged rod is held near it? What does this imply?

• c) Are there examples of non-electrical repulsive forces in nature? Why does Franklin mention this?

• d) Which is simpler, a purely repulsive theory of electricity, or one that involves both attraction and repulsion? Which is more convenient? By what standard should scientific theories be judged?

Ques. 4.6. What is Mr. Kinnersley’s established, “orthodox” law of the electric fluid? What problem(s) does Franklin find with this law? And how does Franklin qualify, or restate the law?

4.4 Exercises

Ex. 4.1 (Müschenbroek’s bottle ). Suppose that an uncharged Müschenbroek bottle in placed on a wax block. Describe the state of electrification of both the interior and the exterior of the bottle (a) initially, (b) after a plastic rod is rubbed with rabbit fur and brought near the wire protruding through the cork of the jar, (c) after the rod has briefly touched the wire, and (d) after an external copper wire has briefly connected the protruding wire to the exterior of the bottle.

Ex. 4.2 (Capacitance, Charge and Electric Potential Laboratory). A Leyden jar is a scientific apparatus specifically designed to store electricity. But as a matter of fact, all bodies—whether Müschenbroek bottles, metallic spheres, or fleshy persons—have the ability to store an excess of positive or negative electrical charge. Nonetheless some have the capacity to store more charge than others. The so-called electrical capacitance of a body depends on its size and shape as well as on the type of substance surrounding the body. As a simple example, the capacitance, $C$, of an isolated metallic (conducting) sphere of radius $R$ is given by

$$ \begin{aligned} C = 4 \pi \varepsilon R. \end{aligned} $$

(4.1)

Here, $\varepsilon$ is the electrical permittivity of the medium surrounding the sphere. In the international system of units (SI) the permittivity of a region of space devoid of all matter (a vacuum) is $\varepsilon_0 = 8.854 \times 10^{-12}$ F/m; the farad is then the SI unit of capacitance. As another example, the capacitance of a parallel-plate capacitor, which consists of two flat parallel conducting plates, is (approximately)

$$ \begin{aligned} C = \frac{\varepsilon A}{d}, \end{aligned} $$

(4.2)

where $A$ is the area of the plates, $d$ is their separation, and $\varepsilon$ is the permittivity of the insulator sandwiched between them.¹³ More generally, the capacitance of a body expresses the relationship between the electric potential, $V$, which is used to electrify the body and the quantity of unbalanced electrical charge, $Q$ on the body. This relation is expressed mathematically as

$$ \begin{aligned} Q = C V. \end{aligned} $$

(4.3)

To better understand Eq. 4.3, consider an analogous situation from hydrostatics involving two upright cylindrical vessels having different cross-sectional areas. When both vessels are filled to the same height with fluid, the one with the larger cross sectional area will store a larger volume of fluid. Similarly, when two bodies are charged to the same electric potential, the one with the larger capacitance will store a larger quantity of charge.

Fig. 4.4

Electrostatics equipment consisting of two conductive spheres (top), a volt power supply (middle), parallel-plate capacitor (right), charge producers and proof plane (bottom), and an electrometer wired to a faraday ice pail (left)

In the following laboratory experiments, we will explore the relationship between electric potential (measured in volts), electric charge (measured in coulombs) and capacitance (measured in farads). We will make use of an electrometer, an adjustable parallel-plate capacitor, a Faraday ice pail, charge producers and a proof-plane, conductive spheres, and a volt electrostatics voltage source (see Fig. 4.4).¹⁴

Fig. 4.5

a A schematic diagram of the interior of an electrometer. b An external capacitor has been attached across the leads

Electrometer and Faraday Ice Pail

The electrometer is a device used to measure the presence of electrical charge; it consists of a voltmeter hooked up in parallel with an internal capacitor (see Fig. 4.5). When a charge $Q$ is placed on the plates of the interior capacitor, $C_{int}$, an electric potential, $V$, is registered by the voltmeter.¹⁵ An external capacitor, $C_{ext}$, may be attached to the leads of an electrometer so as to allow charge measurements by induction. This is depicted on the right side of Fig. 4.5. When charging by induction, a charged object is brought near the upper plate of the external capacitor, attracting negative charges to that plate. These negative charges must come from somewhere. Since the upper plate of the external capacitor is connected to the upper plate of the internal capacitor by a conducting wire, negative charge will flow from the upper plate of the internal to the upper plate of the external capacitor. Hence, the upper plate of the internal capacitor is left with a positive charge. In addition, the negative charge on the upper plate of the external capacitor will attract positive charge to the lower plate of the external capacitor. This leaves the lower plate of the internal capacitor with a negative charge. The net result of all this is that the voltmeter will register a positive voltage if a positively charged object is brought near the upper plate of the external capacitor and a negative voltage if a negatively charged object is brought near the upper plate of the external capacitor. In our experiments using the electrometer, a Faraday ice pail will act as the external capacitor. It consists of two concentric conducting cylinders: an “outer shield” and an “inner pail”. The outer shield and inner pails then serve as the lower and upper capacitor plates respectively. The region between the cylinders is the region between the capacitor plates.

Measuring Charge

Let us begin by becoming acquainted with the measurement of charge using the electrometer and ice pail. Connect the electrometer to the ice pail using low-capacitance test-leads. The black (ground) clip should be attached to the outer shield; the red clip should be attached to the inner pail. Briefly touch a finger to the inner and outer pails simultaneously so as to discharge the ice pail; the electrometer should read zero. The sensitivity knob may be adjusted to keep the electrometer readings on scale. Now rub the charge producers together, insert one of them deep into the ice pail (without touching the sides!), and read the electrometer. Remove the object, and again note the reading. Next, note the reading when you insert and remove the other charged object from the ice pail. Then touch one of the objects to the ice pail, remove it and note the reading. Discharge the ice pail and reinsert the same object. Did the object retain any charge after once touching the ice pail? You might also try inserting both (initially uncharged) charge producers into the ice pail and rubbing them together, then noting the electrometer reading when neither, one, or both charge producers are removed from the ice pail. Can you make sense of all these results?

Charge Distribution

Is any excess charge on an electrified object distributed uniformly over its surface? Let us explore the charge distribution on a conducting sphere by sampling the charge at various locations with a small hand-held proof-plane. Connect one terminal of the volt power supply to ground and the other terminal to an aluminized sphere perched atop an acrylic (insulating) rod. The other, identical, sphere should be placed at least half a meter from the first (electrified) sphere and then momentarily grounded. Now touch the face of the proof plane to various points on the second (un-electrified) sphere, insert it into the ice pail (without touching the pail), and read the voltage on the electrometer. Are your readings what you’d expect? Now move the first sphere so that it is just 1 cm from the second sphere. Again, sample the charge at various locations on the second sphere using the proof plane, ice pail and electrometer. Is the charge distribution symmetric?¹⁶ Next, with the first and second spheres nearby, momentarily ground the second sphere and then repeat the process of sampling the charge on the second sphere. Is the charge distribution (and polarity) the same as before? Finally, without touching the second sphere, move the first sphere at least half a meter away. What is the charge distribution on the second sphere now? Does any charge remain? Is it symmetric?

Parallel Plates, Constant Spacing

Next, we will explore the relationship between the electric potential and charge contained on a variable-width parallel plate capacitor. First, we will keep the plate separation—and hence the capacitance—constant, and measure how the voltage across the capacitor plates (as measured by the electrometer) varies as charge is gradually transferred to the capacitor plates. Begin by grounding the stationary capacitor plate and attaching the the movable plate to the (ungrounded) electrode of the electrometer. Also, electrify one of the aluminum spheres using a volt power supply. Briefly touch your finger simultaneously to both capacitor plates to remove any excess charge from the plates. Now, keeping the plate separation at about 2 mm, use the proof plane to “scoop” charge from the electrified sphere and onto the ungrounded capacitor plate. Note the electrometer reading each time charge is added to the capacitor. Consider: when scooping charge onto the ungrounded plate, does the grounded plate become electrified? If so, where does its charge come from? Next, double the plate separation and repeat the previous experiments. Are your results the same?

Parallel Plates, Constant Potential

In the previous experiment, we kept the plate spacing constant and measured the potential difference between the plates as we gradually charged the capacitor. Now we will keep the potential difference between the plates constant and measure how the charge on the plates varies with plate separation. Using Eq. 4.3, try to predict what might happen to the amount of charge held by the plates as the plate separation is varied. Now begin your experiments with a plate separation of about 5 cm. Establish a potential difference across the plates using the volt power supply. Then use a proof-plane to sample the charge at various points on the inner and outer surfaces of the capacitor plates: touch the proof plane to various points and insert it into an ice pail attached to an electrometer.¹⁷ Is the charge distribution uniform? Now investigate how the charge at the center point of of one of the plates varies as the plate separation is varied. Do your results match your expectations? What do you think would happen if a pane of glass were inserted between the plates?

Ex. 4.3 (Potential of $N$ charged spheres). Suppose you have a 1 kV power supply. Its hot (high-voltage) terminal is touched briefly to a 1 cm diameter aluminized sphere which is suspended from an insulating silk thread. This first sphere is then touched briefly to a second identical suspended sphere. Likewise, this second sphere is then touched briefly to a third identical sphere, the third to a fourth, and so on until $N$ spheres are charged. After this series of operations, how much charge (in coulombs) resides on each of the $N$ spheres, and what is the electric potential (in volts) of each sphere? (Hint: See Eq. 4.3 and the surrounding discussion.)

Ex. 4.4 (Electricity essay). Is Franklin’s theory of electricity different than the modern theory of electricity? If so, how? And which is correct? In answering these questions, you might consider the following: How does Franklin define positive and negative electrification? Does he assume the existence of two distinct electrical fluids? What role does the concept of equilibrium play in Franklin’s theory? Can his theory explain both electrical attraction and electrical repulsion? If not, then how might his theory be modified in order to do so?

4.5 Vocabulary

• 1. Equilibrium

• 2. Vacuum

• 3. Plenum

• 4. Phial

• 5. Fillet

• 6. Obstinate

• 7. Restitution

• 8. Decant

• 9. Dexterous

• 10. Armature

• 11. Contrivance

• 12. Unintelligible

• 13. Flax

• 14. Denomination

• 15. Analogous

• 16. Ascribe

• 17. Asunder

• 18. Orthodoxy