Capacitors, Capacitance, and Dielectrics

David J. Jeffery Department of Physics, University of Idaho, PO Box 440903, Moscow, Idaho 83844-0903,

U.S.A.

ABSTRACT Lecture notes on what the title says. Subject headings: capacitors -- capacitance -- dielectrics

1. INTRODUCTION Capacitors are simple circuit devices that have immense practical utility. e.g., for tuning radio receivers, as power supplies, etc. They're everywhere. An older name for capacitor is condenser which almost never used in modern usage, but turns older works.

2. DEFINITION OF CAPACITOR AND CAPACITANCE A capacitor is a device consisting of two conductors called PLATES (which sometimes are plates or rolled up plates) separated by vacuum or more usually a dielectric--but we discuss dielectrics later.

- 2 - Almost always the plates are metals--and so metal capacitors are really all one thinks of.

In their conventional operation, the plates carry equal and opposite charges: Q and -Q: the charge of a capacitor is just called Q.

In electrostatic equilibrium, the plates are equipotentials. The potential differece V between the plates is the capacitor potential: it's always given as positive and so it is positive plate potential minus negative plate potential.

CAPACITANCE is defined to be

C = QV (1) which leads to

Q = CV and V = QC . (2)

The unit of capaciticance is the farad (F) named for Michael Faraday:

1 F = 1 C/V . (3)

It turns out that 1 farad is a bloody big CAPACITANCE. Fig. 1.-- Generic capacitor and capacitor symbol.

- 3 - So typical small device capacitors are often in microfarads (uF or, confusingly, mF which are not millifarads) or picofarads (pF which are sometimes called puffs (e.g., Wolfson & Pasachoff 1990, p. 621)).

Ideally, CAPACITANCE is actually independent of Q and V --which seems odd given its official definition.

Ideally, CAPACITANCE should depend on the geometry of the plates and the DIELECTRIC (insulating material) in which the plates are embedded.

To understand, imagine a capacitor with charge Q and potential V . Any bit of charge on the capacitor is held in static equilibrium by the forces of all the other charge and the force of the conductor wall.

(The conductor surface force is actually an electrostatic force caused by the nature of the conductor. But we idealize this force as that of an impenetrable wall.)

Now electrostatic electric field and potential just depend linearly on charge:

~E(~r) = X

i

kqi| ~r - ~ri|3 (~r - ~ri) and V (~r) = Xi

kqi| ~r - ~ri| . (4)

So if one just scales up all charges by a common factor, then electric field and potential

Fig. 2.-- Generic capacitor with forces on a single charge indicated

- 4 - at any point just scale up.

The electric force just at any point just scales up too and the conductor surface force scales up to match the force on charges the charges.

So charges in electrostatic equilibrium should stay that way. So scaling up all the charges and thus Q of a capacitor should result in V scaling up and a continued electrostatic configuration.

Thus, the ratio Q/V should be unaffected by the scale-up--but that is just the capacitance C itself.

A simple dielectric will just effectively change the permitivity ffl0, and will not introduce a dependence of C on Q or V .

But the C will depend on the dielectric constant ^.

2.1. Charging a Capacitor Typically, one just connects a capacitor to a battery: a lead from the positive terminal go to one plate which becomes the positive plate and a lead from the negative terminal go to the other plate which becomes the negative plate.

The battery pulls electrons off of the positive plate and pushes them on to the negative plate.

It does work to do this and this work is transformed into the capacitor potential energy as we'll discuss in $ 5.

- 5 - 3. CALCULATING CAPACITANCE In intro physics classes, one typically thinks of parallel plate capacitors for doing examples.

But there are lots of other geometries. Here we look at a few with no dielectrics.

3.1. Spherical Capacitors Imagine two concentric spherical shells of radii R1 and R2: R1 < R2. These are our plates. We put a charge Q on plate 1 and -Q are plate 2. From Gauss's law and integration of the electric field, the potential between the shells is

V (r) = kQr , (5)

where we have chosen the arbitrary constant of the potential to be zero. The potential

Fig. 3.-- Charging a capacitor.

- 6 - difference--which is the capacitor potential between the plates is

V = kQ ` 1R

1 -

1 R2 ' . (6)

The capacitance is

C = QV = 1k (1/R

1 - 1/R2) =

R1R2 k (R2 - R1) . (7)

The result is independent of Q and V as we proved in general it should be in $ 2.

We note that capacitance scales up as the radii are scaled up. This means that it is easier to store charge on the capacitor for a given V .

The essential reason for this is that the charge is less concentrated on each plate, and so takes less force to assemble on each plate.

If R2 ! 1, one has in a sense a one-plate capacitor with capacitance

C = 4ssffl0R1 , (8) which is just the Serway & Jewett (2008, p. 724) result.

We note that capacitance scales up

3.2. Cylindrical Capacitors From Gauss's law, one almost immediately has

~E = 2k*r ^r (9)

for the electric field about an infinite cylindrical charge distribution with charge per unit length * when outside of the distribution.

- 7 - By integration

V (r) = -2k* ln ` rr

0 ' , (10)

where r0 is some fiducial cylindrical radius that we set to zero potential.

Now imagine two con-axial cylindrical conducting shells of radii R1 and R2: R1 < R2. We treat them in the infinite length approximation, but their length is in fact `. These are the plates of a cylindrical capacitor. The potential difference between the plates with linear charge densities * on the inner plate and -* on the outer plate is

V = -2k* ^ln ` R1r

0 ' - ln `

R2

r0 '* = 2k* ln `

R2 R1 ' (11)

which is greater than zero note.

Thus the capacitance is

C = QV = `2k ln (R

2/R1) (12)

and the capacitance per unit length is

C

` =

1 2k ln (R2/R1) . (13)

Fig. 4.-- A cylindrical capacitor.

- 8 - We note that capacitance scales down as the radii are scaled up. This means that it is harder to store charge on the capacitor for a given V .

This is opposite the behavior for the spherical shell capacitor. In the cylindrical case, keeping the plates closer so that the charge on one plate is closer to the other-sign charge on the other plate overrules the concentration of charge of one sign on one plate.

Or that is the interpretation that the math suggests. The essential reason for increased capacitance for smaller radii is the smaller separation between the charge different sign, and so takes less input energy to assemble on each plate.

3.3. Parallel-Plate Capacitors The parallel-plate capacitor is actually the usual example capacitor in intro physics. It's sort of the prototype simplest capacitor. From Gauss's law, one almost immediately has

~E = oeffl

0 ^r (14)

for the electric field between an infinite parallel-plane charge distribution with charge per unit are oe.

Thus, immediately one has

V = oedffl

0

as the potential difference between the plates, where d is the distance between them.

Lets say we have finite plates of area A that we treat in the infinite plate approximation.

- 9 - The capacitance is

C = QV = Qffl0oed = ffl0Ad . (15)

We see that capacitance increases with area A and decreases with separation distance d.

The increase of capacitance as d is made smaller is because the charge is more nearly canceling as the plates are brought closer together.

This means that it takes less energy to assemble a charge distribution on the capacitor which is a charge separation relative to the initial uncharged state of the capacitor.

3.3.1. Example 1: Fiducial Parallel-Plate Capacitance Say we have fiducial values d = 1 m and A = 1 m2, then C = ffl0Ad = 8.854 . . . * 10-12 * ` A1 m2 ' ` 1 md ' F = 8.854 . . . * ` A1 m2 ' ` 1 md ' pF . (16)

Maybe d = 1 mm = 0.001 m would have been a better choice for fiducial separation distance:

C = ffl0Ad = 8.854 . . . * 10-9 * ` A1 m2 ' ` 1 mmd ' F = 8.854 . . . * ` A1 m2 ' ` 1 mmd ' nF .

(17)

3.4. Practical Capacitors There are all kinds of practical capacitors. But a typical design is two long strips of aluminum foil for plates with layer of dielectric plastic between them.

- 10 - The strips are rolled up for compactness and sealed in insulating cover which is probably usually plastic (e.g. Wolfson & Pasachoff 1990, p.621).

The capacitance of the capacitor is marked on the outside usually. The dielectric between the plates increases the capacitance $ 7. Dielectrics with high dielectric constants ^ can greatly increase the capacitance. But it is also probably easier to constract structurally sturdy capacitor if there is no air or vacuum gap between the plates.

However, capacitors with vacuum between their plates have special uses: see

http://en.wikipedia.org/wiki/Capacitor#Capacitor_types

3.5. Zero Capacitance Capacitors This is odd entity is what an ideal open circuit is: typically this is an unconnected wire. For example, say you have two wires connected to battery terminals, but not connected to each other.

The wires have the battery potential difference between them, but ideally they should be neutral. Thus, ideally

C = QV = 0 . (18)

Actually, this can't be quite true, but typically dangling wires have low capacitance and it is usually assumed to be zero.

- 11 - 3.6. Infinite Capacitance Capacitors Now what the devil are these. You can put charge on them without a potential. Actually, current flowing through an ideal conductor is like this. For example, consider a point on a wire. Current of positive charge flows into the point and current out of the point. This can go on endlessly loading one side of the point with positive charge while the other side of the point is getting negatively charged. Or so one can imagine it. And there is no potential difference across the point. Thus,

C = QV = 1 . (19)

Now this way of regarding points in conductors may seem a bit useless. But when one solves circuit problems with capacitors and one wonders what the solution is with capacitor taken out, but circuit connected where the capacitor was, the appropriate mathematical operation is to let C go to infinity.

4. CAPACITORS IN COMBINITIONS Capacitors in combinations occur in circuits. One usually thinks of circuits as having current flowing. They a network of conductors with various electrical devices in which current flows. But here we just consider them in electrostatic mode. Circuit diagrams are standardized schematic diagrams of circuits in which the wires are

- 12 - represented by straight lines that are usually only horizontal and vertical and circuit symbols stand for standard circuit elements or devices.

Note the ideal battery has a fixed potential difference between its terminals no matter what: if current is flowing or not.

Real batteries can't quite do this when current is flowing, but are close ideal in electrostatic cases.

4.1. In Parallel In an electrostatic case, any conductors connected by conductors to a battery terminal will be at the battery potential.

Thus if one has capacitors in parallel and a battery put across them, each positive plate is at the battery positive potential and each negative plate is at the battery negative potential.

For battery potential difference V and capacitor i, one has

V = QiC

i , (20)

Fig. 5.-- Circuit symbols for battery, capacitor, open switch and closed switch.

- 13 - where Ci is the capacitance and Qi is the charge of capacitor i.

Thus

Qi = CiV (21)

and the total charge Q on the capacitors collectively is

Q = X

i

Qi = X

i

CiV . (22)

If one defines the equivalent capacitance of the collection of capacitors in parallel by

C = QV , (23) then it follows that one has

CV = Q = X

i

Qi = X

i

CiV (24)

and, dividing out the common factor V ,

C = X

i

Ci . (25)

This is a nice simple, memorable result. Note that

C >= max(Ci) . (26)

Fig. 6.-- Capacitors in parallel with a battery put across them.

- 14 - 4.2. In Series If capacitors are connected one after the other, they are are in series. In this case, as go across each capacitor i there is a potential drop

Vi = QC

i , (27)

where the Q is common to all the capacitors.

If positive plate of one capacitor has charge Q, then its negative plate has charge -Q. The positive plate of the next capacitor must have charge Q to preserve neutrality. Remember the plates and connecting wires are equipotentials and the sum of potential difference between the batteries must be the same for any path between the batteries.

Thus, the battery potential difference must equal the sum of the potential differences of the capacitors:

V = X

i

Vi = X

i

Q Ci . (28)

If one defines the equivalent capacitance of the collection of capacitors in series by

C = QV , (29)

Fig. 7.-- Capacitors in series with a battery put across them.

- 15 - then it follows that one has

V = QC = X

i

Vi = X

i

Q Ci . (30)

Dividing through by the common factor Q gives

1 C = Xi

1 Ci . (31)

Note that

1 C >= max(

1 Ci ) , (32)

and thus

C <= min(Ci) . (33)

So connecting capacitors in series gives a smaller capacitance than any single capacitor alone.

Why would one use capacitors in series? The only practical reason is to construct a capacitor of the size needed for some purpose from a set of standard sized capacitors.

4.2.1. Example: Capacitors of Equal Capacitance in Series Say that all the capacitors in series have equal capacitance C0 and there are N of them. Then the equivalent capacitor is given by

1 C =

N C0 or C =

C0

N (34)

This case gives some insight into why capacitance goes down by putting capacitors in series.

- 16 - Viewing the series capacitors as hidden in a black box, they have separated charge

Q = CV (35) but inside the box they have separated they have separated charge N CV which is just the charge one alone would have separated since

N CV = C0V . (36)

Of course, that separation inside the black box ideally has no consequence in the outside world.

4.2.2. Example: Two Capacitors in Series In this case

1 C =

1 C1 +

1 C2 (37)

which immediately gives

C = C1C2C

1 + C2 . (38)

5. POTENTIAL ENERGY STORED IN A CAPACITOR Typically, we charge capacitors to a battery. The battery creates an electric field that pulls electrons off the positive plate (which is like pushing positive charge onto it) and pushes electrons onto the negative plate.

What actually goes on in the battery is beyond our scope--it's chemistry. But schematically, chemical forces create electrical forces that transfer the charge between the plates and transform chemical energy into the electrical P E of the separated charge.

- 17 - The details of the process actually do not matter to the state of the plates. The same plate P E is created for the same amount of charge transfer no matter how it is done.

!ul? This is sort of mind-boggling when one thinks about it too hard--just keep saying to yourself, energy is conserved. !/ul?

So we don't have to consider any detail process. We can just imagine transfering the charge abstractly through the potential difference between the plates of a capacitor of capacitance C.

Say the potential difference between the plates was V 0 at some time in the transfer process. The P E change transfering a differential amount of charge dq at that time is

dP E = V 0 dq = qC dq , (39) where q is the capacitor charge at time of transfer and V 0 = q/C follows from the definition of capacitance. We can now just integrate to get total P E for a transfer of total charge Q:

P E = Z

Q

0

q C dq =

Q2 2C =

CV 2

2 , (40)

where again we have used the definition of capacitance C = Q/V .

We see that capacitor can be used to store energy as well as charge. There are actually practical limits to storing charge and energy in a capacitor. If the potential difference gets too large (which imples are large electric field), charge will start to flow between the plates. It can be pulled off the surface of the plates and if there is a dielectric between the plates (which is usual) then the dielectric can break down (i.e., start to conduct).

- 18 - 5.1. A Bit of History In the 18th century people thought of electricity as a sort of fluid--which is actually not such a bad idea. Perhaps that idea led to the idea that electricity could be contained in a jar of water. This idea led to the Leyden jar which is a capacitor consisting of metal foils on the inside and outside of a glass jar--the water turned out to be unnecessary. The foils are the plates and the glass is the dielectric that separates the plates.

Large charges could be built up in Leyden jars--sometimes leading to large shocks. Benjamin Franklin realized that there was no need for a jar and used foils on plates of glass. For one experiment he built up a large charge and tried to use it to kill a turkey--he probably would have had it for dinner--but ended up shocking himself: "I tried to kill a turkey, but nearly succeeded in killing a goose" (e.g., Tipler &Mosca 2008, p. 803).

5.2. Energy of the Electric Field We have mentioned before that P E is stored in the electric field. We can now elucidate this idea. Consider an ideal parallel-plate capacitor with no dielectric:

C = QV = ffl0Ad , (41)

where A is the plate area and d is the plate separation. For, an ideal capacitor, the E-field exists only between the plates and satisfies V = Ed. Thus

P E = CV

2

2 =

1 2

ffl0A

d (Ed)

2 = 1

2 ffl0E

2Ad . (42)

Now Ad is the volume of the region between the plates. If we assume that energy is actually in the E-field, then the energy density u is plausibly

u = 12 ffl0E2 . (43)

- 19 - This result is indeed correct for vacuum E-fields and is general. Proving this is beyond our scope though. Note the E-field energy density is proportional to the square of E and is always a postive quantity.

5.3. Energy in the Electric Field Puzzles It is probably most fundamental (i.e., most true) to view electrical P E as being the energy of the electric field which has vacuum energy density

u = 12 ffl0E2 . (44) This perspective is useful in dealing with electromagnetic radiation and essential in general relativity (e.g., Griffiths 1999, 96). But it does present some puzzles.

To elucidate consider a thin spherical shell of charge: charge Q and radius R. The energy of assembling this shell follows from the formula

P E = Z

Q

0 V (q) dq . (45) In this case, the integral is simple since the potential of the shell relative to infinity is

V = kqR (46) as we obtain from Gauss's law, and thus

P E = Z

Q

0

kq

R dq =

kQ2

2R . (47)

This is the P E, but where is the P E? Maybe the question doesn't need an answer. But since we claim that the energy is the electric field, we better be able to calculate it from the electric field. The E-field for the shell is

~E = 8!: kQr2 ^r r >= R;

0 r < R.

(48)

- 20 - from Gauss's law. If we now integrate up the electric field density over all space, we obtain

P E = Z 1

R

1 2 ffl0E

24ssr2 dr = Z 1

R

1 2 ffl0

k2Q2

r4 4ssr

2 dr = 1

2 (4ssffl0)

k2Q2

R =

kQ2

2R , (49)

where we have used

k = 14ssffl

0 . (50)

So we have agreement between the potential approach and the E-field energy density approach. That's nice.

But the E-field energy density approach always gives us a positive energy and we know that there are negative potential energy cases: e.g., the potential energy of a negative and positive point charge. And the resolution is that the potential energy approach did not count up the energy of assembling the point charges. In calculating the energy from the E-field energy density approach we get the total energy of the ensemble of charge including that of the point charges. The potential energy approach only gives the difference in energy from having the charges infinitely far apart and finitely far apart. If you add that in energy of assembling the point charges, then one would guess that one would get agreement again.

But what the devil is the energy of a point charge? If we take our shell charge result and let R ! 0, the energy goes to infinity. One resolution may be that there are no real point charges. The proton for example is not really point-like: it has an electromagnetic RMS radius of , 0.8 * 10-15 (e.g. Enge 1966, p. 35) and quark structure. But on the other hand, the electron maybe be truly point-like or maybe not: perhaps modern string theory will eventually solve this point. In any case, the electron does not have infinite energy. From the Einstein equation E = mc2, we know its energy is 0.511 . . . MeV and we can create electrons using this amount of energy. The paradoxes of true point charges have not been resolved in modern physics

- 21 - 6. CONCLUSION 7. CAPACITORS WITH DIELECTRICS

8. CONCLUSION 9. CONCLUSION 10. CONCLUSION

Support for this work has been provided by the Department of Physics of the University of Idaho.

REFERENCES Arfken, G. 1970, Mathematical Methods for Physicists (New York: Academic Press) Enge, H. A. 1966, Introduction to Nuclear Physics (Reading, Massachusetts: Addison-Wesley

Publishing Company)

Griffiths, D. J. 1999, Introduction to Electrodynamics (Upper Saddle River, New Jersey:

Prentice Hall)

Serway, R. A. & Jewett, J. W., Jr. 2008, Physics for Scientists and Engineers, 7th Edition

(Belmont, California: Thomson)

Tipler, P. A., & Mosca, G. 2008, Physics for Scientists and Engineers, 6th Edition (New

York: W.H. Freeman and Company)

Wolfson, R. & Pasachoff, J. M. 1990, Physics: Extended with Modern Physics (London:

Scott, Foresman/Little, Brown Higher Education)

- 22 - This preprint was prepared with the AAS LATEX macros v5.2.