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» . »

Assistant Professor ok Physics


Leland Stanford, J r^, University

■» •-

Jfcb Work


'903 V u4// rights reserved

I'M I.' *■ P'.' \' I '" "

I) [ :



B 1941! L


TO CfHTtM «^^VftVl

Copyright, 1903 By the MACMILI-AN COMPANY

Set up, dectrotyped and pnnted Sq^tember, 1903


* k


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P«iss or

iNt Nn E^A P«l*fl«0 COHM««.





Page 4, line i6, /or unelectrified (six//i word) read electrified

P^igG 33. line \o,for intensity rr^^/ tension

Page 127, line \,for Ad^ read d^A

Page 1 28, line 7, iftsert 6. ^/ beginning of line.

Page 130, line 8,/^ 6 read 6

Page 203, lines 5, 6-7, 24, cancel isotropic

Page 204, line \,for F, V^ read V^^

Page 212, lines 6—7, cancel the expression />/ brackets.

Page 216, lines 27 and 30, cancel (39) ^7//rf (40)

Page 221. line 28, for V,- V^ = ^,^ read V,^ = "iT^^

Page 244, line 4, ^7/*/^r to /Af^^rV rdne forjn of)

Page 200, line lA./or'S'^rcadS: : .- \

Page 315, line 22, ./or ^i^read iS' ' ^ / '.-V---

■» * " ^"^ ^'' '

Page 426, line g, after and /V/^^t/ even f6r -sinaH values of r

Page 426, line io^./pf comparable with ^rvad greater than a small fraction of ' ..V- -

Page 441, line 29V/(?>^(9) read {a)

Page 450, lines \^\j:^,' substitute small Jbo^ie5 with equal and opposite charges are-fnadeta vibrate symmetrically with (ap- proximately) simple harnionic motioft -ik a straight line about a fixed point, a wave system ^ , -- \ V



* . *



'• m.

*-> C

* *

k * to * * to* fc

to to to to to


to to «,

- to

to ^ to









«• ' (

IM i




B 1942 L

,•«•.« TO C.f«lT«M WfSf"^*

Copyright, 1903 By the MACMILI^N COMPANY

Set up, electrotyped and printed September, Z903

* ^ •■


••• '

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lat Htw EiA PtiRnnc Commmi. LAacA«ru,nL


In this treatise I have tried to present in systematic and defi- nite form a simple, rigorous, and thoroughly modern introduc- tion to the fundamental principles of electromagnetic theory, together with some of the simpler of their more interesting and important non -technical applications. The work makes no pre- tense to completeness, but is written for the serious student of physics, who will make liberal use of more detailed treatises, of hand-books, and of journals, as occasion demands.

I am of course indebted to many books and memoirs. My obligations are especially great, as the most cursory examination of the book will show, to the works of Maxwell, Heaviside, and Poynting. I am also much indebted to Professor A. G. Webster for the use of a number of excellent diagrams from his treatise on electrical theory.

S. J. Barnett.

Stanford University, California June, 1903.


Cbaptsx. Pack.

I. General Electrostatic Theory i

II. Ideal Electric Fields and Condensers with Homo- geneous Dielectrics 57

III. Standard Condensers. Condenser Systems. Electrom-

eters 122

IV. Electric Fields with Two or More Dielectrics 139

V. Reversible Thermal Effect. Electrostriction 168

VI. Electric Absorption. Electrets 176

VII. Comparison of Dielectric Constants. Specific Ind^c-

TivE Capacity...... 192

VIII. The Electric Conduction Current. Intrinsic Elec- tromotive Force 199

IX. Electrolytic and Metallic Conduction 228

X. Thermal and Voltaic Electromotive Forcms 246

XI. Magnets. Magnetostatic Fields 265

XII. The Magnetic Field of the Conduction Current 286

XIII. Electromagnetic Induction 332

XIV. Units and Dimensions 415

XV. Convection and Displacement Currents. The CiENkkal

Electric Current 424

XVI. The Flux of Electromagnetic Energy. Klixtkic

Waves 433


* *

. ••

. •. v.; ..•..• :••••

•• ' .

- » - m "


••• •••••

•. ••••El^EMENTSV.<3F •" .• ••• *••




1. Electrification by Contact. Positive and Negative Charges.

Let one end of an ebonite rod and a dry woolen cloth be rubbed or strongly pressed together and then separated ; and let a second rod and cloth be treated in the same way : The rubbed part of each cloth will be found, on trial, to be attracted toward the rubbed part of each rod, while the rubbed part of each cloth will be repelled from the rubbed part of the other cloth, and the rubbed part of each rod from the rubbed part of the other rod. These are examples of electric phenomena. The region in which they are manifested is called an electric field 1 1), and the medium which permeates this region air and aether in the above case and through which electric influences are trans- mitted is called a dielectric. The parts of the ebonite and wool rubbed together are said to be electrified^ or to possess electric cliarges. The two pieces of woolen cloth are said to have like charges, since they were similarly treated and since what is repelled from one is repelled from the other, and what is attracted toward one is attracted toward the other. Similarly, the two ebonite rods are said to have like charges. But the wool and the ebonite are said to have tnilike or opposite charges, since what is repelled from one is attracted toward the other.


Like ebonite and y^o6i;'eny- two diffete^t i?ubstances, or por- tions of the same . substance in diflereitt physical conditions, exhibit electric propeulies after intimate coht^t and separation. One of the bodies .t)eihaves like ebonite rubbed with wool, the

« « *

other like the wool.l....

An electric char£e*lik^-that of wool after , con tact with ebonite is called 2^ positive cha*^-g!e;-jmtd a charge iil>e that of the ebonite, a 7i€gative charge. Tl>e^erms positive a"nd luegative are justified

by the opposite properties -of the twa kifids of electrification, but

* «• » ^ . ^ *

there is no reason except convention and resulting convenience why the two terms should not be interchanged.

In addition to the forces between electrified bodies, forces are found to exist, in general, between an electrified body and an insulator 2) not electrified (Chapters IV. and VI.).

2. Conductors and Insulators. Electrification by Conduction.

A rod of ebonite electrified at one end exhibits electric properties only at that end ; while a rod of metal, held by an ebonite handle and electrified at one end, becomes electrified at once (apparently) all over its surface. Substances like the metals, by which an electric charge is distributed with extreme rapidity, so as to come into a state of equilibrium within (usually) a small fraction of a second, are called electric conductors, A body charged by connection with an electrified body through a con- ductor, like the far end of the metal rod mentioned above, is said to be electrified by conduction. Substances like ebonite, over or through which an electric charge is transferred only with extreme slowness, are called electric insidators or non-conductors.

Among ordinary molecular substances perfect insulators and perfect conductors do not exist, no such substance completely and for an indefinite time preventing all transfer of electrification, and all offering more or less obstruction to such transfer. There is every reason to believe, however, that free aether (a ** vacuum **) and clean dry gases containing no (elcctrolytically) dissociated molecules have the properties of a perfect insulator (Chapter IX.).



Among substao'qcVT^Ssessing ftilrhVconduCtivity are the metals,

0 tt 0 # A

graphite, and saftoi'^^cid solutions; among t"hose with high in- sulating propefitie§*«fe (undissociated) gas4§;'fused quartz (cold and in the soljd state), ebonite, cold gla^-^silk, and wool. A substance which is^^ excellent insulator 'm;6^e condition, how- ever, may in ancftfie^'^condition have the-p^ojjerties of a conductor. Thus cold glass fe atl^jexq^Uont iosn^^t^Jvi^ul ^s the temperature is raised its insulatir\g*^rc^lties disappear. Also, under very great electric stress thfc •fo^ftjatvig -{Jroperties of all molecular substances break down.

A body completely surrounded with insulators is said to be ifisulated,

A conductor can be completely discliarged by bringing it into contact at any one point with the inner surface of a hollow closed conductor 4), such as the walls of the room within which the experiments are performed, provided there are no (insulated) electrified bodies within. When connected to the walls of the room, or the earth, the conductor is said to be earthed. From an insulator the electrification can be entirely removed only by applying a conductor at every electrified point, e. g,, by immers- ing it in a conducting gas or liquid.

3. Electrification by Induction. An insulated conductor, when brought near an electrified body, i. r., into an electric field, itself becomes electrified. Examined by the methods of § i, the charges of the more remote and nearer ends of the conductor are found to be similar and opposite, respectively, to that of the original electrified body. A conductor electrified in this manner is said to be electrified by inductioji.

If the conductor, while still insulated, is removed from the electric field, all signs of electrification disappear. But if, while still in the field, it is connected with the walls, or earthed, the electrification similar to that of the original charged body disap- pears, while the opposite electrification of the near end remains. If the conductor is now insulated and removed from the original


electric field, this charge Becomes mofe-'eVfenly distributed over

its surikce 42). In this manner any nhmKer of conductors

*••••* *.**'■

may be given charges .opposite to that of a gi>/en electrified body


without, as may be proved by the method of § 5, diminishing or

increasing the latterV'elcctrification.

••• -•


4-8. Experiments . >|vQl* ^Uqw .Cld^e^:^ Oohdnotors. Electric Screens. Let A denote ,an-msulate"dfaoIl6\V 'conductor having a closely fitting conducting lid, ^^ yv{f6 an Insulating handle. Let A be connected with an electroscope or electrometer (Chapter III.), C, by means of which any change in the state of electri- fication of its exterior (or interior) surface may be detected ; and let A be kept closed except when another body is being intro- duced into its cavity, or removed therefrom, or its position in- side (or outside) altered.

4. (i) Let the electrometer be placed outside of A. If A is initially unelectrified, and an insulated unelectrified conductor, D, is now introduced into A without touching it, the inner and outer surfaces of A will become electrified by induction 3) with charges opposite and similar, respectively, to that of D, And the electrification of the external surface, as indicated by the electrometer, >\ill be found to remain absolutely unaltered how- soever D is moved about within, even when it is brought into contact with A ; but D, on being insulated after contact, and then removed from A's interior, will be found completely discharged. This process may be repeated indefinitely, D always becoming completely discharged on coming into contact with the inner surface of A. If A is initially electrified in any manner, the phe- nomena will be precisely the same, except that the external electrification and the corresponding indication of the electrom- eter will be different.

(2) Let the electrometer be placed within A, either connected with A metallically, or insulated therefrom. In this case it will be found that if there are insulated charged bodies within A, the electrometer will give a certain deflection ; that if there are no


insulated electrified bodies within A, the electrometer will give no deflection ; and that its indication in either case will remain absolutely unaltered howsoever the electrification of the exterior of A or of external bodies is changed, even if -^4 is connected to the walls of the room.

These experiments are due to Faraday, who constructed for the purpose of performing (2) a closed conductor large enough to en- able him to make the observations while himself inside the cavity.

An experiment similar in principle to those of Faraday, but less general, performed earlier by Cavendish and repeated later by Maxwell with all the precision of modem investigation, gave identical results.

From the experiments just described it follows that, when there is electrical equilibrium,

1. Ah electric charge cajinot exist in the substance of a con- ductor^ or on the inner sufface of a hollow closed conductor (unless there are insulated electrified bodies within). For D, on being removed from A, of whose substance it formed a part, electrically, while in contact, was always unelectrified.

2. An electric field 11) does not exist within tlie hollow of a closed conductor (unless there are charges inside). For in (2) the electrometer was unaffected (by induction or otherwise) no matter what the external electrification, except when there were insulated charges within.

3. Hie electric charges and electric fields within and without a hollow closed conductor are absolutely independent of one anot/icr. The conducting shell thus completely screens each of these re- gions from all static effects in the other.

4. A71 electric field does not exist within the substance of a con- ductor. See § 15.

6. Equal Charges. Two electric charges of the same sign are, by definition, of the same magnitude if they produce the same effect on the electrification of the vessel A when intro- duced in succession separately.


Similarly, two charges of opposite signs are, by definition, equal in magnitude if they produce no effect on the electrification of A when introduced simultaneously.

These definitions are independent of the particular closed conductor A used, as two charges defined as equal by means of one such vessel are found to remain equal when tested in the same way with any other hollow closed conductor.

6. Positive and Negative Charges are Always Developed Simultaneously in Equal Amounts. If two bodies electrified by contact are introduced into the vessel A simultaneously, the in- dication of the electrometer remains unaltered.

If an electrified body is insulated within A, and if an insulated uncharged conductor is then introduced in addition, the latter becomes electrified by induction, in conformity with § 3, but the indication of the electrometer remains unaltered.

In these cases, therefore, positive and negative charges are developed in equal amounts (§5); and in the same way it may be shown that this is always the case, howsoever the electrifica- tion is produced.

7. The Total Quantity of Electrification is Unaltered by Con- duction. If the two insulated bodies of the last experiment are brought into contact with one another while inside the vessel A, or if they are brought into contact with the inner surface of A itself, conduction occurs, but no effect on the external electrification is produced. From this it follows that when conduction occurs, the total (algebraic) amount of electrification is unaltered.

Corollary, The charges induced on the inner and outer sur- faces of A when an electrified body is introduced ajid instdatcd zvithin, as in § 4, are each of the same magnitude as that of the instdatcd body. For when D touches A, the charges of D and the inner surface of A completely disappear by conduction, since D on removal is unelectrified ; thus their algebraic sum is zero. And the (opposite) charges on the inner and outer surfaces, being induced, must, by § 6, be equal in magnitude.


8. Electric Charges of Both Kinds Measured in Terms of a Single Arbitrary Unit. In addition to the hollow conductor A of §§ 4-7, let there be provided another similar insulated vessel B^ sufficiently large to admit A through its opening ; and let the conductor D be given a certain charge (suppose positive for the sake of definiteness), which will be adopted as a provisional unit.

If now D is brought within A and kept insulated, the outer surface of A will have unit positive charge. \{ A is brought in- side B and then into contact with it, this charge will disappear, as will also the charge induced on ^*s inner surface, leaving the outside of B with unit positive charge. If A is now removed from B'% interior and then D from A^ the negative charge in- duced on As inner surface will pass to the outer surface and will disappear when A is discharged. This complete process may be repeated any number of times. Each time B will acquire an additional unit positive charge, and thus may be given a measured positive charge which is any integral multiple of the original unit.

To give B a negative charge measured in terms of the same unit, the outer surface of A must be brought into contact with the inner surface of a hollow closed conductor after the introduction of Z>, when the positive charge will disappear from the outside, leaving unit negative charge upon the inner surface. When D is removed, this charge will pass to the outer surface of A, and will be given up wholly to B when A is brought into contact with -ff's interior. B will now have unit negative charge, and by removing A and repeating the process may be given any number of units negative charge desired.

To obtain any submultiple, i/«, of the original charge, it is only necessary to arrange symmetrically in contact the original conductor S> and n i precisely similar and equal conductors, all other bodies, except the surrounding dielectric, supposed homogeneous and isotropic, being so remote as to have no appre- ciable effect. Then, by the principle of symmetry, each con- ductor will take i of the original charge.


9. The Law of Coulomb. Let two small spherical insulated conductors which can be given any charge desired, measured in terms of some provisional unit by the methods of §§5 and 8, be so connected with a dynamometer, such as a gravity balance, that the force F between them can be measured as their charges, q^ and q^, the distance L between their centers, and the surround- ing dielectric are varied. Then it is found by experiment that,

(i) However the distance L and the charges q^ and q^ are varied, provided all the experiments are performed in the same dielectric, and provided that this dielectric is homogeneous and isotropic and extends to a great distance on all sides of the elec- trified bodies, F is in the straight line joining the centers of the conductors ; is directly proportional to the product of their charges, being repulsive (considered positive) when the charges are like and attractive (considered negative) when the charges are unlike, as already known from § i ; and the greater L in com- parison with the linear dimensions of the charged bodies, the more nearly inversely proportional to Lr.

(2) In different dielectrics, with all other conditions the same, the force is different, and always less than in vacuo (free aether).

The general expression for F, when the linear dimensions of the (not necessarily spherical) charged bodies are negligible in comparison with their distance apart, is therefore

F= Aq,qJcD (l')

where r is a constant depending on the medium in which the ex- periments are performed, called its permittivity or dielectric con- stant, and y4 is a positive constant depending on the units in which ^j, ^2, L, Fy and c are expressed, (i') expresses Coulomb's law.

The Eational Electrostatic XTnit Charge. XTnit Permittivity. In

what follows, unless the contrary is stated, the centimeter will be used as unit length, the dyne as unit force, the permittivity of free aether, which will be denoted by c^, as unit permittivity,


and as unit charge the charge which each of two indefinitely small bodies must have in order that when at a distance of i cm. apart in a vacuum the force between them may be i/47r dyne. This unit charge is called by its originator, Oliver Heaviside, the rational electrostatic unit cliarge, and c^ is called the electrostatic unit pertnittiinty.

Methods of measuring permittivity are discussed in Chapter


The conventions just made give, by the above equation, A =s 1/4^, and the equation reduces to

which, in addition to being a particular case of (i'), is a particular case of (2).

The direct experimental investigation of the law of force is due to Coulomb, but is not capable of great precision. The law, as stated by Coulomb, is most satisfactorily established by the consideration that all experimental knowledge is in perfect accord with an electrical theory based largely upon the assumption that the laws expressed in (i) are exact.* A reason for the law of inverse squares and a justification of the term rational unit will be given in §§5, IL, and 24.

The dimensions of electric charge and the other electric quan- tities, as well as other systems of units, will be considered in Chapter XIV.

For rational electrostatic the abbreviation RES will hereafter be employed.

10. If any one of the experiments described above is repeated in different dielectrics, the results in all cases will be identical, except that, in conformity with § 9, the force between two charged bodies will always depend on the surrounding dielec- tric.

*The common deduction of the law of inverse squares from the results of the Cavendish experiment cannot oe accepted as valid. Sec The Physical Rnnnu^ Sep- tember, 1902, p. 175.


11. Electric Field. Electric Intensity. Any region in which an electrified body is acted upon by a mechanical force in virtue of its charge, or in which an uncharged conductor is charged by induction, is called an electric field. Such a field exists, for ex- ample, around an electrified body i), but may also exist with- out the presence of electrification (Chapters VI. and XIII.).

As a result of experiment, it may be stated that the force F acting upon a small charged body, or small portion of a charged body, at any point of an electric field is proportional to its charge q provided that the distribution of electric charge (real and apparent, Chapter IV.) originally accompanying the electric field remains undisturbed by the introduction of q. Expressed in the form of an equation, this relation is

F=Eq (2)

where is a constant for the given point of the field called the electric intensity^ electric farce ^ or voltivity at the point.

The conditions for the rigorous proof of this relation by direct experiment would be impossible to realise, and the remark at the close of §9 with reference to the establishment of Coulomb's law applies without alteration to (2).

As (2) shows, E is not a mere number, but a physical quan- tity specifying the state of the field and such that its product by an electric charge is a mechanical force. E is clearly a vector quantity, its direction being that of the force on a positively charged body, and its magnitude the number of dynes per unit charge. When q is expressed in the RES unit charge and F in dynes, E is said to be expressed in the RES unit electric inten- sity. •

The term electric field is often used to denote the collective in- tensity in a region, instead of the region itself. The direction of the field at any point is the direction of the intensity, and the strength of the field is the magnitude of the intensity.

12. The Superposition of Electric Fields. Experiment also shows that any number of electric fields (up to a certain limit,


when the dielectric breaks down and conduction occurs) may bo superposed upon one another, the effect of each beinj^ intlciHjn- dent of all the rest. Electric intensities, bcin^ vectors, may therefore be compounded like all other vectors for which the principle of superposition holds, the resultant intensity at any point being the geometric or vector sum of the component in- tensities.

An electric field is uniform if its intensity is the same at every point. Since E is a vector, this condition necessitates a constant direction as well as a constant magnitude. In most cases E varies from point to point. Examples of uniform and other elec- tric fields, as well as of the superposition of electric fields, will be given below.

IS. Electric Displacement or Induction. Electrisation. The physical nature of every electric qu«intity is at present unknown. Many phenomena, however, support the hypothesis that c is an elastic permittivity (/. r., the reciprocal of an elastic modulus) and that E is an elastic .stress. For the sake of constructing a mechanical conception of the electric fiehl we shall provisionally assume c and ^ to be a permittivity and a stress, respectivt^ly. The so-called permittivity c will then be the actual j>ermittivity of the aither or a:ther entangled in matter for the Cunknown) kind of strain concerned.

Now, in the case of ordinary elastic substances subjected to slight mechanical strains we have, very approximately, the rela- tion (Hooke's law): strain' itras ^ i modulus ^^ pirmittivity, or strain permittrinty x stress. If then / is a jx-rmittivity of a orr- tain type and E a stress of the corresj^indinj,^ *yi>*-» thrir product r£must measure the corrcsi>^>ndin'^^ strain or displa'-<'in'nt of the dielectric.

Whether this conc<,-]>tioii is ^orr<';t or not, tin produtt f l\ x-^ ealle J tht electric di:>piruim* nt ^a)v> th<: ehftrir indu/tiou), and is denoted by IJ. That is

/>-//; (3;


E being a vector, and c being, the same for every direction of the intensity, since isotropic substances only are to be considered here, Z? is a vector with the same direction as that of E, When c and E are expressed in RES units, D is said to be expressed in the RES unit displacement (or induction),

A substance in which there is electric displacement is also said to be in a state of electrisation, or to be electrised. If the dis- placement and permittivity are uniform throughout, the electrisa- tion is said to be uniform,

14. Mechanical Conception of the Electric Field. A definite conception of the electric field based on the assumptions made above will now be given. According to this conception (which leads to results by no means wholly consistent, however) the aether is the simplest possible kind of dielectric and is composed of two kinds of minute, incompressible, elastic cells, called

«. No electric displacement b. Electric dispiaccmfcnt

directed to left

Fig. 1.

positive and negative cells, respectively, so arranged (in rows). Fig. I, ^, that only unlike kinds are in contact, and that no slip between adjacent cells is possible.

When the aether supports an electric field, the cells remain un- changed in volume, but their shapes are distorted and their centers of volume displaced. Fig. i, b, the centers of the positive cells in the direction of the electric intensity, and the centers of the negative cells in the opposite direction. The electric displace- ment is measured by the relative linear displacement of the centers of volume of the cells of a positive row with reference to the centers of volume of the adjacent negative rows divided


by the distance between t\vo adjacent rows. The electric inten- sity is the force per unit area in the direction of J) actin^^f wynm the positive cells^ or the force per unit area in tlie opposite tlirec- tion to that of D acting upon the mgathc (r//s, in any plane passing through the direction of D, For small displaceniontH, the displacement and intensity so measured will be pro|H>rtional, as required by (3) in ^// cases. The total mechanical force acting upon the wAo/e substance within any element of volume is zero.

From what precedes and from the nature of the distortion as shown in the figure, it is clear that there is a ttHsion in the aether parallel to the intensity, and a pressure in all tlirectloni normal to the intensity. That this deduction from our mechanical conception is consistent with fact is demonstrated in §S 40-41.

When the dielectric, instead of free athcr, is a molecular .sub- stance permeated by aether, the .same general conception is umc- ful. Like the aether which permeates the matter, its moletnilcM may be thought of as composed each of two constituents, positive and negative atoms, or atomic groups, or corpuscles (CJhapter IX.), which suffer a displacement .similar and in addition to that of the aether cells entangled among them. However this may be, the permittivity of all molecular substances yet invcsUYjatitd is greater than that of free aether. Thus, in ordinary maiUtr t greater displacement than in free ;ether accompanies a given intensity.

In perfect insulators, according if) our concqHimi, the otlU cannot slip over one another, and thus elasiU i\\n\AmMnutui ^mly can accompany electric intensity. In an impcrUuX jn^ulat/zr 0»e cells can slip only with extreme hh^wnenn, and fiM/re hUfwly tim more highly insulating the substanMr, In a i^nuUuiUff t:U'jAru, stress can exist only temp^-^rarily (iitiU:*^H ati jnipM j>v:d *i\fi*Aro' motive force, ChsiyiU^r VIIL, i^ <x/ntiniiouj>)y a^iitiyj, and U always accompanied by rof^d >Jjp. 7 hat th*r ^^i\/>AAWA: *A a '^m- ductor cannot bup^x/Tt ekctri/, di>,j/)i^>;in^rnt in a ^Wm, f»' Jd wilJ be shown in | 15, TiM: uu:*}iAStu^\ 'y/fi//rj/li'/n ^4 'l^^Afif. v/f* duction will reodve furtl^rr ^ymi>vi'if4U'jtt Wut on (<A$4{A^'j IX^i-


15. Electric Displacement and Intensity Zero within a Condnc- tor in a Static Field. We may now restate (4), § 4, as a corol- lary of (3), § 4 : A static field cannot exist within the substance of a conductor. For the fields within and without a hollow closed conductor are absolutely independent of one another, however thin the conducting shell. Hence they cannot be con- nectcd by an electric field or electrically strained medium, and the whole substance of a conductor, except an extremely thin surface layer, is without electrical significance (in a static field). Thus the electric intensity and displacement in the outer region terminate at the outer surface of the conductor, and the electric intensity and displacement of the inner region (if the conductor is hollow and encloses insulated electrified bodies) terminate at the inner surface.

16. Lines and Tnbes of Intensity, Displacement, etc. A line so drawn in an electric field as to have at every point along its length the direction of the electric intensity (electric force), elec- trisation, or displacement (induction) is called a line of intcfisity {force), electrisation, or displacement (inductioji),

A tubular suriace the elements of which consist wholly .of lines of intensity or induction (etc.) is called a tube of intensity or induction (etc.).

The strength of a tube of induction or displacement is defined in § 23.

17. Voltage, Electromotive Force, and Difference of Potential

The work done by the electric field in carrying an indefinitely small body with electric charge q along an element dL CFig. 2) of a path L between two points P^ and P^ of an electric field, if dL makes an angle 6 with the electric intensity E, is

dW^ qE cos e dL (4)

and the total work done in carrying q along L from /\ to P^ is

W^qJEcosOdL (5)



the integral being taken from /\ to P^. To carry q from P^ to P, along the same path would of course require the expenditure of the same amount of work against the field by an outside agent.

In the same way the work ^E

done by the field in carrying the body with charge q from /\ to P^ along another path U is

W ^ qjE'cosO'dU.

If the electric field is a static Pi field, IV = IV\ and therefore

cos 0dL = f£' cos 0'dL\ Fig. 2.

For if the work done along any path L were greater than that done along any other path Z', a positive amount of work, IV ^ IV, would be done on the charged body by the field dur- ing each completion of a circuit from P^ to P^ along L and back along Z', and yet the energy of the field would remain unaltered. Since this is inconsistent with the principle of the conservation of energy, cos 0 dL is the same for every path between two given points in an electrostatic field.

The line integral of the electric intensity, fji cos 0dL^ If7?» along a path L from /\ to P^ is called the electromotive force (e.m.f.) or voltage along the path L from P^ to P^, When, as in the case just considered, this quantity is the same for every path from Pj to Pj, it is called also the differ e7ice of potential between P^ and P^ or thufall of potential from P^ to P^.

Since a voltage is a quantity of work divided by a charge, it is evidently not a vector.

When ^ris expressed in ergs and q in the RIIS unit char^;^, or when E\s expressed in the RliS unit intensity, and L in cm., the voltage ( = IV'q = jE cos 0 dL) is said to be expressed in the RES unit voltage. In magnitude, the voltage between two points is equal to the work done in carrying unit charge from


one point to the other along the given path, or any path if the voltage is a potential difference.

18. Potential. Equipotential Surfaces. The fall of potential from a given point P to any point at an infinite distance from all electrified bodies is called the electric potential at P,

This term is also commonly applied to the fall of potential from P to any point of the earth. That the two definitions are not identical will be shown in § 6, Chapter II.

The symbol V will be used to denote the potential at a point P, In conformity with this notation, the fall of potential from a point /\ to a point P^ will be written FJ V^, V^^, or, where there is no danger of confusion, simply V,

A surface which is everywhere normal to the electric intensity, and between any two points of which there is therefore no voltage, is called an equipotential surface, or simply an equipotential. It is clear that an equipotential surface is always a closed surface or else (in certain ideal fields) an infinite plane.

19. Electric Intensity in a Static Field the Space Rate of Dimi- nution of Potential. For the voltage from P^ to P^ we have

by writing Ej^ for E cos 0, the component of electric intensity in the direction oidL. That is, the potential of P, exceeds that at P^ byfEi^dL from P, to P^ ; or the diminution of potential from P^

to P^ is jEi^ dL from P^ to P^, If the two points are taken an

infinitesimal distance dL apart, the diminution of potential along dL becomes ^dV, and the integral becomes simply E^^ dL, Thus

we have

- dV^ Ej, dL and therefore

^^ = _ dVldL (6)

That is, the component of electric intensity in any direction is the space rate of diminution of the electric potential in that direction.



V obviously diminishes most rapidly along a line of intensity, and not at all along a line in an equipotential surface.

20. Electric Field Mapped out by a System of Eqnipotentials. If a line of intensity is denoted. by N^ the last equation gives


From this it follows that an electric field can be completely mapped out by a system of equipotential surfaces so drawn that the voltage between successive surfaces is constant. For the direction of the intensity at any point is that of the normal to the equipotential passing through the point ; and its magnitude is, by the above equation, proportional to the number of successive equipotential surfaces crossed at the point per unit length by this normal or line of intensity. Maxwell's method of drawing such an equipotential system is described in §§ 7, 11, 13, 14, II.

21. Electric Flux. Let dS, Fig. 3, denote an element of area at any point of an electric field where the displacement is D, and let the angle between D and the normal N to dS be denoted by d. The product of dS into

the component of D normal to dS, that is, D cos 0 dS, is called the electric flux across dS,

To obtain the electric flux, n, across an extended sur- face 5, over which D may vary in any manner, the in- tegral of D cos 6 dS must be taken over the whole sur- face. Thus ^

U = fD cos 0 dS

Fig. 3.


22. Oaas8*s Theorem: The electric flux outward across any closed surface 5 so drawn as to enclose a total charge q is equal to q.


The theorem will first be established for the case in which all space is filled up with a single homogeneous isotropic dielectric with permitti\dty c (or with any number of isotropic dielectrics all of which have the same permittivity c\

Fig. 4.

Consider first the field about a charge q concentrated at P^ Fig. 4, any point within 5, a closed surface of any shape. For the magnitude of the displacement, D, at any element of area dS, distant L from P, we have from (i), (2), and (3)

D = cE=^ c(g'4 TT c/J) = ql^irD

In direction, D and E are evidently radial from P (or to P if q is negative).

For the flux across dS we have therefore

dH = D cos 6 dS = q dS cos Oi^irL? = q dS' l^irD = qiA'rr dta

where dS^ = dS cos 0 is the projection of