The Mathematical Theory of Electricity and Magnetism (5th ed, 1927) — part 26 of 39

Law of Signs. If an observer is imagined to stand on that side of the "equivalent magnetic shell" which contains the negative poles, the current flows round him in the same direction as that in which the sun moves round an observer standing on the earth's surface in the northern hemisphere.

We can also state the law by saying that to drive an ordinary right- handed screw {e.g. a cork-screw) in the direction of magnetisation of the shell, the screw would have to be turned in the direction of the

current. c < <_^.

Current The law of signs expresses a fact of nature, not a -J- + -f- +

mathematical convention. At the same time, it must be

noticed that the law does not express that nature shews

any preference in this respect for right-handed over left- . '

i-ij m i-iiii i Direction of Magnetisation

nancled screws. 1 wo conventions have already been made f . , , 77

in deciding which are to be called the positive directions

of current and of magnetisation, and if either of these

conventions had been different, the word " right-handed " in the law of signs would have

had to be replaced by " left-handed. n

483. Since, by § 346, any system of currents can be regarded as the superposition of a number of simple closed currents, it follows that the magnetic field produced by any system of currents can always be regarded as that produced by a number of magnetic shells, each of uniform strength.

Electromagnetic Unit of Current.

484. If i is the strength of the current flowing in a circuit, and <£ the strength of the equivalent magnetic shell, then

<f> = ki,

where k is a constant, which is positive if the law of signs just stated has been obeyed in determining the signs of c/> and i.

In the system of units known as Electromagnetic, we take k=l, and define a unit current as one such that the equivalent magnetic shell is of unit strength. The strength of a current, in these units, is therefore measured by its magnetic effects. Obviously the strength measured in this way will be entirely different from the strength measured by the number of electrostatic units of electricity which pass a given point. This latter method of measurement is the electrostatic method. A full discussion of systems of units will be given later (§ 585); at present it may be stated that a current which is of unit strength when measured electromagnetically in c.G.s. units is of strength 3 x 1010 (very approximately) when measured electrostatically. The practical unit of current, the ampere, is, as already stated, equal to 3 x 109 electrostatic units of current, so that the electromagnetic unit of current is equal to 10 amperes.

428 The Magnetic Field produced by Elecfric Currents [en. xm

A unit charge of electricity in electromagnetic units will be the amount of electricity that passes a fixed point per unit time in a circuit in which an electromagnetic unit of current is flowing. It is therefore equal to 3 x 10 electrostatic units.

10

Fig. 121.

-S^.

Work done in threading a Circuit.

485. In  fig.  121  let  the  thick  line  represent  a  circuit  in  which  a  current 
     

is flowing, and let the thin line through ^ ._

the point P represent the outline of /' 'x

any equivalent magnetic shell, P

being any point in the shell. Let us

imagine that we thread the circuit by (

any closed path beginning and ending \

at P, this path being represented by

the dotted line in the figure. At every ^

point of this path except P, we have a "***--.

full knowledge of the magnetic forces.

It will be convenient to regard the shell as having a definite, although infinitesimal, thickness at P. Let P+, P. denote the points in which the path intersects the positive and negative faces of the shell. Then we may say that the forces are known at all points of the path, except over the small range i+P.

The original current can, however, be represented by any number of equivalent magnetic shells, for any shell is capable of representing the current, provided only it has as boundary the circuit in which the current is flowing.

Let any other equivalent shell cut the path in the points Q+Q-. From our knowledge of the forces exerted by this shell, we can determine the forces exerted by the current at all points of the path except those within the range of Q+Q_. In particular we can determine the forces over the range P+FL, and it is at once obvious that on passing to the limit and making the range P+P- infinitesimal, the forces at the points P+, P., and at all points on the infinitesimal range P^R. must be equal. Obviously the forces are also finite.

The work done on a unit pole in taking it round the complete circuit from P. back to P., is accordingly the same as that done in taking it from P. round the path to P+. This can be calculated by supposing the forces to be exerted by the first equivalent shell, for the path is entirely outside this shell. If the potential due to the shell is ClP at P+ and is HP_ at P, the work done is flP — flP_ .

Now n, the potential of the shell at any point, is, as we know (§ 419), equal to ia>, Avhere <o is the solid angle subtended by the shell and i is the

Fig. 122.

484-486] Magnetic Potential of Field 429

current, measured in electromagnetic units. The change in the solid angle as we pass from P_ to P+ is, as a matter of geometry, equal to 4>ir. Thus

ftp -fiP_ = 47Ti (410).

The work done in taking a unit pole round the path described is accord- ingly 4nri.

Magnetic Potential of a Field due to Currents.

486. Let  us  fix  upon  a  definite  equivalent  shell  to  represent  a  current  of 
     

strength i. Let us bring a unit pole from in- finity to any point A% by a path which cuts the equivalent shell in points P, Q, ... Z. For simplicity, let us at first suppose that at each

of these points the path passes from the ^V— — ^//^*^^:>^>^L

positive to the negative side of the shell, and let the points on the two sides of the shell be denoted, as before, by P+> P_; Q+, Q_; and FlG- 123-

so on.

Then, if il denotes the magnetic potential due to the equivalent shell,

the work done in bringing the unit pole from infinity to P+ will be ftp . In

the limit i+ and R. are coincident, so that the work in taking the unit pole on from i+ to P_ is infinitesimal. In taking it from P_ to Q+ work is done of amount £lQ — ftP_, from Q+ to Q_, the work is infinitesimal, and so on, until

ultimately we arrive at A. Thus the total work done in bringing the unit pole to A is

aP +(nQ -Op ) + (nR -nQ )+... + (ciA-nz ),

"T" + ~" +" —

or, rearranging, is

&A + (^+ ~ ^P_) + (fy>+ - &Q_) + • • • •

Now each of the terms 0,P+— £lP_> £lQ —£lQ_, etc. is equal by equation

(410) to 4>iri, so that if n is the number of these terms, the whole expression is equal to

£lA + 4<7rni.

Replacing QA by iw, where w is the solid angle subtended by the shell at A, we find for the potential at A due to the electric current

(co + 4>7rn)i (411).

If the path cuts the equivalent shell n times in the direction from + to — , and m times in the opposite direction, the quantity n must be replaced by n — m.

Expression (411) shews that the potential at a point is not a single- valued function of the coordinates of the point. The forces, which are obtained by differentiation of this potential, are, however, single-valued.

430 The Magnetic Field produced by Electric Currents [on. xin

Fig. 124.

Current in infinite straight wire.

487. As an illustration of the results obtained, let us consider the magnetic field produced by a current flowing in a straight wire which is of such great length that it may be regarded as infinite, the return current being entirely at infinity.

Let us take the line itself for axis of z. Any semi-infinite plane termi- nated by this line may be regarded as an equivalent magnetic shell. Let us fix on any plane and take it as the plane of xz.

Consider any point P such that OP, the shortest distance from P to the axis of z, makes an angle 6 with Ox. The cone through P which is subtended by the semi-infinite plane Ox, is bounded by two planes — one a plane through P and the axis of z ; the other a plane through P parallel to the plane zOx. These contain an angle 7r — 6, so that the solid angle subtended by the plane zOx at P is 2 (-7T — 9). Giving this value to co in formula (411), we obtain as the magnetic potential at P

a = {2 (tt - 6) + 4wtt} i.

rsry

Since -^— = 0 it is clear that there is no radial magnetic force, and the dr &

force at any point in the direction of 6 increasing

d£l _2i

rdd r '

This result is otherwise obvious. If the work done in taking a unit pole round a circle of circumference 27rr is to be 4nri, the tangential force at

every point must be — .

488- This result admits of a simple experimental confirmation.

Let PQR be a disc suspended in such a way that the only motion of which it is capable is one of pure rotation about a long straight wire in which a current is flowing. On this disc let us suppose that an imaginary unit pole is placed at a distance r from the wire. There will be a couple tending to turn the disc, the

2i

moment of this couple being — xr ov 2i. Similarly

if we place a unit negative pole on the disc there is a couple — 2i.

On placing a magnetised body on the disc, there will be a system of couples consisting of one of moment 2i for every positive pole and one of moment — 2i for every negative pole. Since the total charge

Fig. 125.

487-489]

Magnetic Potential of Field

431

in any magnet is nil, it appears that the resultant couple must vanish, so that the disc will shew no tendency to rotate. This can easily be verified.

Circular Current.

489. Let us find the potential due to a current of strength i flowing in a circle of radius a. The equivalent magnetic shell may be supposed to be a hemisphere of radius a bounded by this circle.

The potential at any point on the axis of the circle can readily be found. For at a point on the axis distant r from the centre of the circle, the solid angle w subtended by the circle is given by

<y

= 27r(l-cosa) = 27r(l-

r

Va2 +

so that the potential at this point is Q = 2ni ( 1 -

sfaF+r'-

This expression can be expanded in powers of r by the binomial theorem. We obtain the following expansions :

if r < a.

( a 2 a3 v 2.4... 2n \aj

if r > a, _ ft . fl a2 . . w+1 1 . 3 ... 2n- 1 fa\2n

Fig. 126.

2r

2.4... 2n

.(412),

.(413).

From this it is possible to deduce the potential at any point in space. Let us take spherical polar coordinates, taking the centre of the circle as origin, and the axis of the circle as the initial line 6 = 0. Inside the sphere r = a, the potential is a solution of V2f2 = 0 which is symmetrical about the axis 6 = 0, and remains finite at the origin. It is therefore capable of expansion in the form

O = XAnrnPn (cos 6). o

Along the axis we have 6 = 0, so that this assumed value of H becomes

n = ZAnrn,

o

and the coefficients may be determined by comparison with equation (412).

432 The Magnetic Field produced by Electric Currents [ch. xiii

Thus we obtain for the potentials,

H = 2m \ 1 - - P, (cos 6) + \ - P3 (cos 0) - . . .

• (- l)n+1

-) Pm+1(cos0) + .. .\ ...(414),

when r < a, and

[1 a2

2.4...2w Va

3 a4

^^i?(cos6>)-g-^(cos^)-...

■f (- 1)^ 1 ;3 •;• 2n~ 1 (ff Pm (cos 0) + ...} ...(415),

2.4

when r >a.

ml

At points so near to the origin that — may be neglected, the potential is

a*

Q = 2iri (1 cos

a

»)-«(!-£).

where 2; = r cos ^, and the magnetic force is a uniform force — -~- =■

dz a

parallel to the axis.

Fig. 127.

Solenoids.

490 A cylinder, wound uniformly with wire through which a current can be sent, is called a "solenoid."

Consider first a circular cylinder ot radius a and height h, having a wire coiled round it at the uniform rate of n turns per unit length, the wire carrying a current i Let z be a coordinate measuring the distance of any cross-section from the base of the

solenoid. Then the small layer between z and z + dz,

being of thickness dz, will contain ndz turns of wire. The currents flowing in all these turns may be re- garded as a single current nidz flowing in a circle, this circle being of radius a and at distance z from the base of the solenoid. The magnetic potential of this current may be written down from the formula of the last section, and the potential of the whole solenoid follows by integration.

491. Endless Solenoid. In the limiting case in which the solenoid is of infinite length (or in which the ends are so far away that the solenoid may be treated as though it were of infinite length), the field can be determined in a simpler manner.

Consider first the field outside the solenoid. In taking a unit pole round any path outside the solenoid which completely surrounds the solenoid, the work done is, by § 485, 4nri. The current flowing per unit length of the

Faf

Q

R'

F3

R

489-492] Galvanometers 433

solenoid is ni. In general we are concerned with cases in which this is finite n being very large and i being very small. The quantity im may accordingly be neglected, and we can suppose that the work done in taking unit pole round the solenoid is zero.

It follows that the force outside the solenoid can have no component at right angles to planes through the axis, and clearly, by a similar argument, the same must be true inside the solenoid. Hence the lines of induction must lie entirely in the planes through the axis of the solenoid. From symmetry, there is no reason why the lines of induction at any point should converge towards, rather than diverge from, the axis, or vice versa. Hence the lines of induction will be parallel to the axis, and the force at every point will be entirely p parallel to the axis.

Let the lines PQR, P'Q'R' in fig. 128 be radii meeting the axis, the lines PF, QQ', RR' being parallel to the axis and each of length e. Let the FlG ^3

magnetic forces along these lines be Flt F2 and F3 respectively.

In taking unit pole round the closed path PP'Q'QP the work done is

i?e - F26,

and since this must vanish, we must have F1 = F2. Hence the force at all points outside the solenoid must be the same ; it must be the same as the force at infinity and must consequently vanish. Thus there is no force at all outside the solenoid.

In taking unit pole round the closed path PP'R'RP, the work done is F3e, and this must be equal to 4nrnie, so that we must have F3e = 4tirni. Thus the force at any point inside the solenoid is a force 4nrni parallel to the axis.

Thus the field of force arising from an infinite solenoid consists of a uniform field of strength 4nrni inside the solenoid, there being no field at all outside. The construction of a solenoid accordingly supplies a simple way of obtaining a uniform magnetic field of any required strength.

Galvanometers,

492. A galvanometer is an instrument for measuring the strength of an electric current, the method of measurement usually being to observe the strength of the magnetic field produced by the current by noting its action on a small movable magnet.

There are naturally various classes and types of galvanometers designed to fulfil various special purposes.

j. 23

434 The Magnetic Field produced by Electric Currents [oh. xiii

The Tangent Galvanometer,

493. In the tangent galvanometer the current flows in a vertical circular coil, at the centre of which a small magnetic needle is pivoted so as to be free to turn in a horizontal plane.

Before use, the instrument is placed so that the plane of the coil contains the lines of magnetic force of the earth's field. The needle accordingly rests in the plane of the coil. When the current is allowed to flow in the coil a new field is originated, the lines of force being at right angles to the plane of the coil and the needle will now place itself so as to be in equi- librium under the field produced by the superposition of the two fields — the earth's field and the field produced by the current.

As the needle can only move in a horizontal plane, we need consider only the horizontal components of the two fields. Let H, as usual, denote the horizontal component of the earth's field. Let i be the current flowing in the coil, measured in electromagnetic units, let a be the radius and let n be the number of turns of wire. Near the centre of the coil the field produced by the current is, by § 489, a uniform field at right angles to

the plane of the coil, of intensity . The total

horizontal field is therefore compounded of a field of strength H in the plane of the coil, and a field of

strength at right angles to it.

The resultant will make an angle 6 with the plane of the coil, where

/2-rrin\

tanfl= % ' (416),

xz

and the needle will set itself along the lines of force of the field. Thus the needle will, Avhen in equilibrium, make an angle 6 with the plane of the coil, where 6 is given by equation (416). If we observe 6 we can determine i from equation (416). We have

i = ?tan0 (417),

where G is a constant, known as the galvanometer constant, its value being

The instrument is called the tangent galvanometer from the circum- stance that the current is proportional to the tangent of the angle 6.

493, 494] Galvanometers 435

The tangent galvanometer has the advantage that all currents, no matter how small or how great, can be measured without altering the adjustment of the instrument. A disadvantage is that the readings are not very sensi- tive when the currents to be measured are large — only a very small change in the reading is produced by a considerable change in the current. Let the current be increased by an amount di, and let the corresponding change in 6 be dd, then from equation (417),

d0 so that if i is large, -p is small. Thus, although the instrument may be

used for the measurement of large currents, the measurements cannot be effected with much accuracy.

A second defect of the instrument is caused by the circumstance that the field produced by the current is not absolutely uniform near the centre of the coil. If a is the radius of the coil, and b the distance of either pole of the magnet from its centre, the poles will be in a part of the field in which the intensity differs from that at the centre of the coil by terms of

b3

the order of — . For instance, if the magnet is one inch long, while the

Cb

coil has a diameter of 10 inches, the intensity of the field will be different from that assumed, by terms of the order of (yV)3> so that the reading will be subject to an error of about one part in a thousand.

By replacing the single coil of the tangent galvanometer by two or more parallel coils, it is possible to make the field in the region in which the magnet moves, as uniform as we please. It is therefore possible, although at the expense of great complication, to make a tangent galvanometer which shall read to any required degree of accuracy

The Sine Galvanometer.

494. The sine galvanometer differs from the tangent galvanometer in having its coil adjusted so that it can be turned about a vertical axis. Before the current is sent through the coil, the instrument is turned until the needle is at rest in the plane of the coil. The coil is then in the direc- tion of the earth's field at the point.

As soon as a current is sent through the coil, the needle is deflected, as in the tangent galvanometer. The coil is now slowly turned in the direction in which the needle has moved, until it overtakes the needle, and as soon as the needle is again at rest in the plane of the coil, a reading is taken, giving the angle through which the coil has been turned. Let 6 be this angle, then the earth's field may be resolved into components, H cos 6 in

28—2

436 The Magnetic Field produced by Electric Currents [ch. xiii

the plane of the coil and H sin 6 at right angles to this plane. Since the needle rests in the plane of the coil, the latter component must be just neutralized by the field set up by the current, this being, as we have seen, entirely at right angles to the plane of the coil. We accordingly have

. 2irin

H sin Q = ,

a

so that we must have

FT" i=g-sm0 (418),

where G, the galvanometer constant, has the same meaning as before.

This instrument has the disadvantage that it cannot be used to measure

TT

currents greater than -~ • ^ is> however, sensitive over the whole range

through which it can be used : if d6 is the increase in 6 caused by a change di in i, we have

d6 = -^ sec 6 di, 11

so that the greater the current the more sensitive the instrument.

The great advantage of this form of galvanometer, however, is that when the reading is taken the magnet is always in the same position relative to the field set up by the current in the coil. Thus the deviations from uniformity of intensity at the centre of the field do not produce any error in the readings obtained : they result only in the galvanometer constant having a value different from that which it has so far been supposed to have. But when once the right value has been assigned to the constant G, equation (418) will be true absolutely, no matter how large the movable needle may be in comparison with the coil.

Other galvanometers.

495. There are various other types of galvanometers in use to serve various purposes other than the exact measurement of a current. For full descriptions of these the reader may be referred to books treating the theory of electricity and magnetism from the more experimental side. The following may be briefly mentioned here •

I. The D'Arsonval Galvanometer. This instrument is typical of a class of galvanometer in which there is no moving needle, the moving part being the coil itself, which is free to turn in a strong magnetic field. The coil is suspended by a torsion fibre between the poles of a powerful horseshoe magnet. When a current is sent through the coil, the coil itself produces the same field as a magnetic shell, and so tends to set itself across the

494, 495] Galvanometers 437

lines of force of the permanent magnet, this motion being resisted by no forces except the torsion of the fibre.

II. The Mirror Galvanometer. This is a galvanometer originally designed by Lord Kelvin for the measurement of the small currents used in the trans- mission of signals by submarine cables. The design is, in its main outlines, identical with that of the tangent galvanometer, but, to make the instrument as sensitive as possible, the coil is made of a great number of turns of fine wire, wound as closely as possible round the space in which the needle moves, and the needle is suspended as delicately as possible by a fine torsion-thread. To make the instrument still more sensitive, permanent magnets can be arranged so as to neutralize part of the intensity of the earth's field. The instrument is read by observing the motion of a ray of light reflected from a small mirror which moves with the needle : it is from this that the instrument takes its name. In the most sensitive form of this instrument a visible motion of the spot of light can be produced by a current of 10~10 amperes.

III. The Ballistic Galvanometer. This instrument does not measure the current passing at a given instant, but the total flow of electricity which passes during an infinitesimal interval. If the needle is at rest in the plane of the coil, a current sent through the coil will establish a magnetic field tending to turn the needle out of this plane. So long as the needle is approximately in the plane of the coil, the couple acting on the needle will be proportional to the current in the coil : let it be denoted by ci, where i is the current.

Then if a> is the angular velocity of the needle at any instant, we shall have an equation of the form

ink- -j- = ci, at

where mk* is the moment of inertia of the needle. Integrating through the small interval of time during which the current may be supposed to flow, we obtain

mk2Q =c I idt.

Here O is the angular velocity with which the needle starts into motion, and J idt is the total current which passes through the coil. Thus the total

flow I idt can be obtained by measuring O, and this again can be obtained by

observing the angle through which the needle swings before coming to rest at the end of its oscillation.

438 The Magnetic Field produced by Electric Currents [ch. xiii

Vector-potential of a Field due to Currents.

496. From the formulae obtained in § 446 for the vector-potential of a uniform magnetic shell, we can at once write down expressions for the vector- potential of a field due to currents.

For, by § 483, the field due to any system of currents may be regarded as the field due to a number of shells of uniform strength, so that the vector- potential at any point will be the sum of the vector-potentials due to these different shells. Hence if <£, cf>', ... are the strengths of the various shells, the vector-potential at any point P has components (cf. § 446)

where the summation is over all the shells, and dx, ds' refer to an element of the edge of a shell of strength <p, this element being at a distance r from the point P.

The equations just found may clearly be replaced by

F=(i-pds J r as

.(419),

J r ds

J 7' ds )

where ds is now an element of any wire or linear conductor in which a current of strength i is flowing, and the integration is now along all the conductors in the field.

By the use of equations (376), we may at once obtain the components of magnetic force or induction at any point x, y, z' in the forms

_d_H_dG

a " dij dz'

/'£'©=-£©$** (420)-

= i

Mechanical Action in the Field.

Ampere's rule for the force from a circuit.

497. Let 0 (x, y, z) be the position of any element ds of a circuit, and let P be any point {x, y', z') in free space.

From equations (420) it follows that the magnetic force at P may be regarded as made up of contributions from each element of the circuit such that the contribution from the element ds at 0 has components

l'{a7®£-aT'0)l}^etc-'etc-

496-498]

Mechanical Action

439

On putting r2 = (x - a/)2 + (y — y'f + (z- z'f, and differentiating, these components become

ids \y — y'dz z — z' dy] ids [z — z' dx x — x dz) r ds

dy) ids (z — z' dx x — x dz)

Let us denote

x — x y — y z — z

by h> whi fht these being the direction-

cosines of the line OP, and let ~ , -~ , -r-

ds ds ds

be denoted by l2, m2, n2, these being the direction-cosines of ds. Then the com- ponents of force (421) become

ids

ids . .

(nJt-nJJ,

-^(^m2-4mi) ...(422).

Clearly the resultant is a force at right angles both to OP and to ds, and of amount

^ ■ (423),

where 6 is the angle between OP and ds.

Thus the total force at P may be regarded as made up of contributions such as (423) from each element of the circuit. This is known as Ampere's law.

Mechanical action on a circuit.

498. We are at present assuming the currents to be steady, so that action and reaction may be supposed to be equal and opposite. It follows that the force exerted at a unit pole at P upon the circuit of which the element ds is part, may be regarded as made up of forces of amount

i sin 0

per unit length, acting at right angles to OP and to ds. If we have poles of strength m at P, m' at P', etc., the resultant force on the circuit may be regarded as made up of contributions

im sin 6 im'sinO'

per unit length. The resultant of these forces may be put in the form

iH sin x where H is the resultant magnetic intensity at 0 of all the poles m, m , etc., and x is the angle between the direction of this intensity and ds. This resultant force acts at right angles to the directions of H and of ds.

440 The Magnetic Field produced by Electric Currents [ch. xm

499. We have found that the force from a whole circuit is the same as if each element ids contributed a force ids sin d/r2, and the force on a whole circuit is the same as if each element were acted on by a force iH sin %. But so long as we are dealing only with complete closed currents, it is impossible to discover what the actual force from or on a single element of the current will be. In a later chapter we shall regard a current as a stream of electrons in motion. The element ids will then be treated as a small number of moving electrons, and we shall be able to shew that the actual forces associated with the single element ids are exactly identical with those just found.

Energy of a System of Circuits carrying Currents.

500. The energy of a magnetic field, as we have seen (§ 470), is

g^jjf fi(cc + /32 + ry*)dxdydz (424).

If the energy resides in the medium, this expression may be regarded as the energy of the field, no matter how this field is produced. If the field is produced wholly by currents, expression (424) may be regarded as the energy of the system of currents. As we shall now see, it can be transformed in a simple way, so as to express the energy of the field in terms of the currents by which the field is produced.

The integral through all space, as given by expression (424), may be regarded as the sum of the integrals taken over all the tubes of induction by which space is filled. The lines of induction, as we have seen, will be closed curves, so that the tubes are closed tubular spaces.

If ds is an element of length, and dS the cross-section at any point, of a tube of unit strength, we may replace dxdydz by dSds, and instead of inte- grating with respect to dS we may sum over all tubes. Thus expression (424) becomes

^XJV(a! + /32-f-72)d£}tfs,

where the summation is over all unit tubes of induction. If H2 = a.- + /32 + 72, we have, by the definition of a unit tube, fiHdS = 1, so that

li (a2 + /3°- + r) dS = fiH'dS = H, and the integral becomes

Now I Hds is the work performed on a unit pole in taking it once round

the tube of induction, and this we know is equal to 4nr1'i, where Ifi is the sum of all the currents threaded by the tube, taken each with its proper sign. Thus the energy becomes £2(2't).

499-501] Energy 441

This indicates that for every time that a unit tube threads a current i, a contribution ^i is added to the energy Thus the whole energy is

h^iN (425),

where the summation is over all the currents in the field, and N is the number of unit tubes which thread the current i.

501. We have seen that a shell of strength <j) is equivalent, as regards the field produced at all external points, to a current i, if <f> = i. The energy of a system of currents has however been found to be ^ZiN, whereas the energy of a system of shells was found (§ 450) to be

-it<j)N (426).

The difference of sign can readily be accounted for. Let us consider a single shell of strength </>, and let dS be an element of area, and dn an element of length inside the shell measured normally to the shell. At any point just outside the shell, let the three components of magnetic force be a, j3, y, the first being a component normal to the shell, and the others being components in directions which lie in the shell. On passing to the inside of the shell, the normal induction is discontinuous owing to the permanent magnetism which must be supposed to reside on the surface of the shell. Thus inside the shell,

we may suppose the components of force to be S + -, ft, y, where fx, is the

permeability of the matter of which the shell is composed, and S is the force originating from the permanent magnetism of the shell.

The contribution to the energy of the field which is made by the space inside the shell is

where the integral is taken throughout the interior of the shell ; or

This can be regarded as the sum of three integrals,

(i) ^rl\'lStdndS \

1 f[[faU-^ + ^dnaK- (427).

(fi) 8^111 (T + ^ + A^JdndS

<iii> ~ \

442 The Magnetic Field produced by Electric Currents [ch. xiii

On reducing the thickness of the shell indefinitely, S becomes infinite, for at any point of the shell,

Sdn = — (difference of potential between the two forces of shell)

= — 4>TT(j),

so that S becomes infinite when the thickness vanishes. Thus on passing to the limit, the first integral

becomes infinite. This quantity is, however, a constant, for it represents the energy required to separate the shell into infinitesimal poles scattered at infinity.

The second integral vanishes on passing to the limit, and so need not be further considered.

The third integral can be simplified. We have

M!SttdndS=UNSdn)dS-

Now I Sdn= — 47r<£, while 1 1 adS is the integral of normal induction over

the shell, and may therefore be replaced by N, the number of unit tubes of induction from the external field, which pass through the shell. Thus the third integral is seen to be equal to

In calculating expression (424) when the energy is that of a system of currents, the contribution from the space occupied by the equivalent mag- netic shells is infinitesimal. Thus all the terms which we have discussed represent differences between the energies of shells and of circuits.

Terms such as the first integrals of scheme (427) represent merely that the energies are measured from different standard positions. In the case of the shells, we suppose the shells to have a permanent existence, and merely to be brought into position. The currents, on the other hand, have to be created, as well as placed in position. Beyond this difference, there is an outstanding difference of amount <£iV for each circuit, and this exactly accounts for the difference between expressions (425) and (426).