A Treatise on Electricity and Magnetism, Vol. 2 (1873) — part 19 of 27

If we make 0=0, the two circles become parallel and on the same axis. To determine the attraction between them we may differentiate M with respect to b. We thus find

' dM

w=*

700.] In calculating the effect of a coil of rectangular section we have to integrate the expressions already found with respect to A, the radius of the coil, and _Z?, the distance of its plane from the origin, and to extend the integration over the breadth and depth of the coil.

In some cases direct integration is the most convenient, but there are others in which the following method of approximation leads to more useful results.

Let P be any function of x and ^, and let it be required to find the value of P where

T+i* r+4y Pxy = / / Pdxdy.

J- J -

In this expression P is the mean value of P within the limits of integration.

Let P0 be the value of P when x = 0 and y = 0, then, expanding P by Taylor's Theorem,

Integrating this expression between the limits, and dividing the result by xy> we obtain as the value of P,

7QI.J COIL OF EECT ANGULAR SECTION. 305

In the case of the coil, let the outer and inner radii be A + \ £, and A — ^ respectively, and let the distance of the planes of the windings from the origin lie between JB + ^rj and B—\TI, then the breadth of the coil is r, and its depth £ these quantities being small compared with A or C.

In order to calculate the magnetic effect of such a coil we may write the successive terms of the series as follows :-^-

&C., &c. ;

ft= ™2

= 277^

&c., &c.

The quantities G0, G1, G2, &c. belong to the large coil. The value of o> at points for which r is less than C is

a, = _27T + 2G0- ^ r Ql (0)- G^r* Q2 ((9)-^-&c. The quantities gl9 g^ &c. belong to the small coil. The value of a/ at points for which r is greater than c is

The potential of the one coil with respect to the other when the total current through the section of each coil is unity is

To find M by Elliptic Integrals. 701.] When the distance of the circumferences of the two circles

VOL. II. *

306 CIRCULAR CURRENTS. [701.

is moderate as compared with the radii of the smaller, the series already given do not converge rapidly. In every case, however, we may find the value of M for two parallel circles by elliptic integrals.

For let b be the length of the line joining the centres of the circles, and let this line be perpendicular to the planes of the two circles, and let A and a be the radii of the circles, then

M ••'«»«

/"/"

= / /

the integration being extended round both curves. In this case,

r2 = A2 + a2 + b2-2Aacos((j>-(l>')

e = — $', ds =

•27T

M

/• ~J

where c ==

and F and E are complete elliptic integrals to modulus c.

From this we get, by differentiating with respect to b and re membering that c is a function of b,

—c

If fj and /2 denote the greatest and least values of r, rf =(A + of + V, r* =(A- a)2 + b2,

4*

and if an angle y be taken such that cos y = — ,

where Fy and Ey denote the complete elliptic integrals of the first and second kind whose modulus is sin y.

If A — a,j cot y = — - , and

i Cb

-^-=

The quantity -^- represents the attraction between two parallel circular currents, the current in each being unity.

703.] LINES OF MAGNETIC FOKCE. 307

Second Expression for M. An expression for M, which is sometimes more convenient, is got

by making ^ = — - - , in which case ri + r2

M = 4

To draw the Lines of Magnetic Force for a Circular Current.

702.] The lines of magnetic force are evidently in planes passing through the axis of the circle, and in each of these lines the value of M is constant.

Calculate the value of KQ — ,-= - = — r^ from Legendre's

(/sine — A3in0)

tables for a sufficient number of values of 0.

Draw rectangular axes of so and z on the paper, and, with centre at the point x = \ a (sin 0 + cosec d), draw a circle with radius \ a (cosec 0— sin 0). For all points of this circle the value of el will be sin 0. Hence, for all points of this circle,

= ^ and A =

Now A is the value of x for which the value of M was found. Hence, if we draw a line for which x = A, it will cut the circle in two points having the given value of M.

Giving M a series of values in arithmetical progression, the values of A will be as a series of squares. Drawing therefore a series of lines parallel to zy for which x has the values found for A, the points where these lines cut the circle will be the points where the corresponding lines of force cut the circle.

If we put m = 4 a a, and M = nm, then

A — x = n2Ke a. We may call n the index of the line of force.

The forms of these lines are given in Fig. XVIII at the end of this volume. They are copied from a drawing given by Sir W. Thomson in his paper on ' Vortex Motion*.'

703.] If the position of a circle having a given axis is regarded as defined by 6, the distance of its centre from a fixed point on the axis, and «, the radius of the circle, then M, the coefficient of induction of the circle with respect to any system whatever

• Trans. R. 8.t Edin., vol. xxv. p. 217 (1869). X 2

308 CIRCULAR CURRENTS. [703.

of magnets or currents, is subject to the following equation

d2M d2M I dM fc x-v

da2 db2 a da

To prove this, let us consider the number of lines of magnetic force cut by the circle when a or b is made to vary.

(1) Let a become a + ba, b remaining constant. During this variation the circle, in expanding, sweeps over an annular surface in its own plane whose breadth is 8 a.

If V is the magnetic potential at any point, and if the axis of y be parallel to that of the circle, then the magnetic force perpen-

dV

dicular to the plane of the ring- is -7- •

dy

To find the magnetic induction through the annular surface we have to integrate

where 6 is the angular position of a point on the ring.

But this quantity represents the variation of M due to the

variation of #, or -= — 8 a. Hence da

dM ^ f2nadT d0 (2]

(2) Let 6 become 6 + 85, a remaining constant. During this variation the circle sweeps over a cylindric surface of radius a and length 8£.

The magnetic force perpendicular to this surface at any point is

-s- where r is the distance from the axis. Hence dr

dM PIT dV JQ ...

— — = — / a-j-dB. (3)

db JQ dr

Differentiating equation (2) with respect to a, and (3) with respect to I, we get

dM P«d7 7 f« dzY . -— - = / -j-dO+l a-—rde, (4)

da2 JQ dy J0 dr dy

dM r' d^v .... ( .

-—— = — a-f-j-dB, (5)

oar J0 dr dy

-j

Hence —^ + -— - = / -j-dO, (6)

da* db2 JQ dy

\dM

= a-da-^y^ Transposing the last term we obtain equation (1).

704.] TWO PARALLEL CIRCLES. 309

Coefficient of Induction of Two Parallel Circles when the Distance betiveen the Arcs is Small compared with the Hadlus of either Circle.

704.] We might deduce the value of M in this case from the expansion of the elliptic integral already given when its modulus is nearly unity. The following method, however, is a more direct application of electrical principles.

First Approximation.

Let A and a be the radii of the circles, and b the distance between their planes, then the shortest distance between the arcs is

We have to find M19 the magnetic induction through the circle A, due to a unit current in a on the supposition that r is small compared with A or a.

We shall begin by calculating the magnetic induction through a circle in the plane of a whose radius is a — c, c being a quantity small com pared with a (Fig. 49).

Consider a small element ds of the circle a. At a point' in the plane of the circle, distant p from the middle of ds, measured in a direction making an angle 6 with the direction of ds, the magnetic force due to ds is perpendicular to the plane, and equal to

—s sin 6 ds. P2

If we now calculate the surface-integral of this force over the space which lies within the circle a, but outside of a circle whose centre is ds and whose radius is c, we find it

*2asin0 j

— g sin 6 ds d0 dp = {log 8 a — log c — 2} ds.

If c is small, the surface-integral for the part of the annular space outside the small circle c may be neglected.

We then find for the induction through the circle whose radius is a— c, by integrating with respect to ds,

Mac = ^ -n a (logStf— logc — 2}, provided c is very small compared with a.

Since the magnetic force at any point, the distance of which from a curved wire is small compared with the radius of curvature,

/JT /

/ /

J J c

310 CIRCULAR CURRENTS. [705.

is nearly the same as if the wire had been straight, we can calculate the difference between the induction through the circle whose radius is a — c, and the circle A by the formula MaA—Mac = 4: 7t a {logo— log r}.

Hence we find the value of the induction between A and a to be

MAa = 4 77 a (log 8 a— log r— 2) approximately, provided r is small compared with a.

705.] Since the mutual induction between two windings of the same coil is a very important quantity in the calculation of ex perimental results, I shall now describe a method by which the approximation to the value of M for this case can be carried to any required degree of accuracy.

We shall assume that the value of M is of the form

1 j . j vu j / ri j **/ A f w rf n

where A = a -f ^i# + A2 — - A2- — -A3-^-}-A3 -~ + &c.,

a a, a* a*

and B = —2a + B,uo+B9 — + B'^- + B^ + B»°^ +&c.,

2 # 2 & 3 «2 3 «2

where « and « + o? are the radii of the circles, and y the distance between their planes.

We ; have to determine the values of the coefficients A and B. It is manifest that only even powers of y can occur in these quan tities, because, if the sign of y is reversed, the value of M must remain the same.

We get another set of conditions from the reciprocal property of the coefficient of induction, which remains the same whichever circle we take as the primary circuit. The value of M must there fore remain the same when we substitute a + % for a, and —a? for a? in the above expression.

We thus find the following conditions of reciprocity by equating the coefficients of similar combinations of x and y,

A . A A 7?__JLJL

^3 - """^2 ~~^3> -°3 — 3 ~ 2 -<

7°6.]

COIL OF MAXIMUM SELF-INDUCTION.

311

From the general equation of M, Art. 703, d2M d*M 1 dM

dx2 dy* a + x dx we obtain another set of conditions,

O // I O J' .. A

2 l" *^ 2 "™ ~^1 3

2 + 2A'

'= 2A

2

&c.;

4 A2+ Al =

Solving these equations and substituting the values of the co efficients, the series for If becomes

M —

log

• &C.J

O ^^ 1. I

— ^5 — 2 -p

• &c

]

To find the form of a coil for which the coefficient of self-in duction is a maximum, the total length and thickness of the wire being given.

706.] Omitting the corrections of Art. 705, we find by Art. 673

where n is the number of windings of the wire, a is the mean radius of the coil, and R is the geometrical mean distance of the transverse section of the coil from itself. See Art. 690. If this section is always similar to itself, R is proportional to its linear dimensions, and n varies as Rz.

Since the total length of the wire is 2 TT an, a varies inversely as n. Hence

dn _ dR , da dR

• = 2-^-, and — = — 2 -^- , n R a R

and we find the condition that L may be a maximum

312 CIRCULAR CURRENTS. [7°6-

If the transverse section of the coil is circular, of radius <?, then,

by Art. 6 9 2, R

Iog7=-i,

and log — = ^,

whence a = 3.22 c ;

or, the mean radius of the coil should be 3.22 times the radius of the transverse section of the coil in order that such a coil may have the greatest coefficient of self-induction. This result was found by Gauss *.

If the channel in which the coil is wound has a square transverse section, the mean diameter of the coil should be 3.7 times the side of the square section.

• Werl-e, Gottingen edition, 1867, vol. v. p. 622.

CHAPTER XV.

ELECTROMAGNETIC INSTRUMENTS.

Galvanometers.

707.] A GALVANOMETER is an instrument by means of which an electric current is indicated or measured by its magnetic action.

When the instrument is intended to indicate the existence of a feeble current, it is called a Sensitive Galvanometer.

When it is intended to measure a current with the greatest accuracy in terms of standard units, it is called a Standard Galva nometer.

All galvanometers are founded on the principle of Schweigger's Multiplier, in which the current is made to pass through a wire, which is coiled so as to pass many times round an open space, within which a magnet is suspended, so as to produce within this space an electromagnetic force, the intensity of which is indicated by the magnet.

In sensitive galvanometers the coil is so arranged that its windings occupy the positions in which their influence on the magnet is greatest. They are therefore packed closely together in order to be near the magnet.

Standard galvanometers are constructed so that the dimensions and relative positions of all their fixed parts may be accurately known, and that any small uncertainty about the position of the moveable parts may introduce the smallest possible error into the calculations.

In constructing a sensitive galvanometer we aim at making the field of electromagnetic force in which the magnet is suspended as intense as possible. In designing a standard galvanometer we wish to make the field of electromagnetic force near the magnet as uniform as possible, and to know its exact intensity in terms of the strength of the current.

314 ELECTROMAGNETIC INSTRUMENTS. [708.

On Standard Galvanometers.

708.] In a standard galvanometer the strength of the current has to be determined from the force which it exerts on the sus pended magnet. Now the distribution of the magnetism within the magnet, and the position of its centre when suspended, are not capable of being determined with any great degree of accuracy. Hence it is necessary that the coil should be arranged so as to produce a field of force which is very nearly uniform throughout the whole space occupied by the magnet during its possible motion. The dimensions of the coil must therefore in general be much larger than those of the magnet.

By a proper arrangement of several coils the field of force within them may be made much more uniform than when one coil only is used, and the dimensions of the instrument may be thus reduced and its sensibility increased. The errors of the linear measurements, however, introduce greater uncertainties into the values of the electrical constants for small instruments than for large ones. It is therefore best to determine the electrical constants of small instruments, not by direct measurement of their dimensions, but by an electrical comparison with a large standard instrument, of which the dimensions are more accurately known ; see Art. 752.

In all standard galvanometers the coils are circular. The channel in which the coil is to be wound is carefully turned. Its breadth

Fig. 50.

is made equal to some multiple, n, of the diameter of the covered wire. A hole is bored in the side of the channel where the wire is

709.] MEASUKEMENT OF THE COIL. 315

to enter, and one end of the covered wire is pushed out through this hole to form the inner connexion of the coil. The channel is placed on a lathe, and a wooden axis is fastened to it; see Fig. 50. The end of a long string is nailed to the wooden axis at the same part of the circumference as the entrance of the wire. The whole is then turned round, and the wire is smoothly and regularly laid on the bottom of the channel till it is completely covered by n windings. During this process the string has been wound n times round the wooden axis, and a nail is driven into the string at the ^th turn. The windings of the string should be kept exposed so that they can easily be counted. The external circumference of the first layer of windings is then measured and a new layer is begun, and so on till the proper number of layers has been wound on. The use of the string is to count the number of windings. If for any reason we have to unwind part of the coil, the string is also unwound, so that we do not lose our reckoning of the actual number of windings of the coil. The nails serve to distinguish the number of windings in each layer.

The measure of the circumference of each layer furnishes a test of the regularity of the winding, and enables us to calculate the electrical constants of the coil. For if we take the arithmetic mean of the circumferences of the channel and of the outer layer, and then add to this the circumferences of all the intermediate layers, and divide the sum by the number of layers, we shall obtain the mean circumference, and from this we can deduce the mean radius of the coil. The circumference of each layer may be measured by means of a steel tape, or better by means of a graduated wheel which rolls on the coil as the coil revolves in the process of winding. The value of the divisions of the tape or wheel must be ascertained by comparison with a straight scale.

709.] The moment of the force with which a unit current in the coil acts upon the suspended apparatus may be expressed in the series ^ gin Q + ^ gin Q ^ ^ + &c ^

where the coefficients G refer to the coil, and the coefficients g to the suspended apparatus, 0 being the angle between the axis of the coil and that of the suspended apparatus ; see Art. 700.

When the suspended apparatus is a thin uniformly and longitud inally magnetized bar magnet of length 2 1 and strength unity, suspended by its middle,

^i = 2^, #2 = 0, #3=2£3, &c.

316 ELECTROMAGNETIC INSTRUMENTS. [7IQ-

The values of the coefficients for a magnet of length 2 1 magnetized in any other way are smaller than when it is magnetized uni formly.

710.] When the apparatus is used as a tangent galvanometer, the coil is fixed with its plane vertical and parallel to the direction of the earth's magnetic force. The equation of equilibrium of the magnet is in this case

m^HcosO = my sin0 {6^+ G2$2 Q/^ + fec.},

where mg^ is the magnetic moment of the magnet, .7? the horizontal component of the terrestrial magnetic force, and y the strength of the current in the coil. When the length of the magnet is small compared with the radius of the coil the terms after the first in G and g may be neglected, and we find

TT

y = -=• cot 0. Gi

The angle usually measured is the deflexion, b, of the magnet which is the complement of 0, so that cot 0 = tan 8.

The current is thus proportional to the tangent of the deviation, and the instrument is therefore called a Tangent Galvanometer.

Another method is to make the whole apparatus moveable about a vertical axis, and to turn it till the magnet is in equilibrium with its axis parallel to the plane of the coil. If the angle between the plane of the coil and the magnetic meridian is 8, the equation of equilibrium is

&c-l >

whence y = -^ - 5 — .sin 8.

(G^-fec.)

Since the current is measured by the sine of the deviation, the instrument when used in this way is called a Sine Galvanometer.

The method of sines can be applied only when the current is so steady that we can regard it as constant during the time of adjusting the instrument and bringing the magnet to equi librium.

711.] We have next to consider the arrangement of the coils of a standard galvanometer.

The simplest form is that in which there is a single coil, and the magnet is suspended at its centre.

Let A be the mean radius of the coil, £ its depth, rj its breadth, and n the number of windings, the values of the coefficients are

712.] TANGENT GALVANOMETEE. 317

£4 = 0, &c. The principal correction is that arising1 from G3. The series

becomes G^ yt ( 1 — | -p ^ (cos2 0 — J sin2 0)) •

V 1

The factor of correction will differ most from unity when the magnet is uniformly magnetized and when 0 = 0. In this case it

I2

becomes 1 — 2 ~^ • It vanishes when tan 0 = 2, or when the de- .4

flexion is tan"1 4, or 26°34'. Some observers, therefore, arrange their experiments so as to make the observed deflexion as near this angle as possible. The best method, however, is to use a magnet so short compared with the radius of the coil that the correction may be altogether neglected.

The suspended magnet is carefully adjusted so that its centre shall coincide as nearly as possible with the centre of the coil. If, however, this adjustment is not perfect, and if the coordinates of the centre of the magnet relative to the centre of the coil are os, y, z, z being measured parallel to the axis of the coil, the factor of

correction is (l 4- 3 °° ) •

When the radius of the coil is large, and the adjustment of the magnet carefully made, we may assume that this correction is insensible.

Gaugavn?* Arrangement.

712.] In order to get rid of the correction depending on G3 Gaugain constructed a galvanometer in which this term was ren dered zero by suspending the magnet, not at the centre of the coil, but at a point on the axis at a distance from the centre equal to half the radius of the coil. The form of G is

and, since in this arrangement B = \ A, G3 = 0.

This arrangement would be an improvement on the first form if we could be sure that the centre of the suspended magnet is

318 ELECTROMAGNETIC INSTRUMENTS. [713.

exactly at the point thus defined. The position of the centre of the magnet, however, is always uncertain, and this uncertainty intro duces a factor of correction of unknown amount depending on G2 and

of the form (l — £ -r) , where z is the unknown excess of distance

^4

of the centre of the magnet from the plane of the coil. This correction depends on the first power of -j . Hence Gaugain's coil

with eccentrically suspended magnet is subject to far greater un certainty than the old form.

Helmholtz's Arrangement,

713.] Helmholtz converted Gaugain's galvanometer into a trust worthy instrument by placing a second coil, equal to the first, at an equal distance on the other side of the magnet.

By placing the coils symmetrically on both sides of the magnet we get rid at once of all terms of even order.

Let A be the mean radius of either coil, the distance between their mean planes is made equal to A^ and the magnet is suspended at the middle point of their common axis. The coefficients are

& =

G3 = 0.0512 — (31 £2 - 36rj2),

GB= -0.73728

where n denotes the number of windings in both coils together.

It appears from these results that if the section of the coils be rectangular, the depth being f and the breadth 17, the value of 6r3, as corrected for the finite size of the section, will be small, and will vanish, if £ is to 77 as 36 to 31.

It is therefore quite unnecessary to attempt to wind the coils upon a conical surface, as has been done by some instrument makers, for the conditions may be satisfied by coils of rectangular section, which can be constructed with far greater accuracy than coils wound upon an obtuse cone.

The arrangement of the coils in Helmholtz's double galvanometer is represented in Fig. 54, Art. 725.

715.] GALVANOMETER OF THREE COILS. 319

The field of force due to the double coil is represented in section in Fig. XIX at the end of this volume.

Galvanometer of Four Coils.

714.] By combining four coils we may get rid of the coefficients G2, G3, G±, G5, and G6. For by any symmetrical combinations we get rid of the coefficients of even orders Let the four coils be parallel circles belonging to the same sphere, corresponding to angles 6, (j>, TT— <£, and TT — 0.

Let the number of windings on the first and fourth coil be ny and the number on the second and third pn. Then the condition that G3 = 0 for the combination gives

ft sin2 0 q; (0) + ^ft sin2 $ Q9' (c/>) = 0, (1)

and the condition that G5= 0 gives

ft sin2 6 <25' (6) + pn sin2 <£ Q/ (<#>) = 0, (2)

Putting sin2 0 = x and sin2 $ = y^ (3)

and expressing Q3' and Q5' (Art. 698) in terms of these quantities, the equations (1) and (2) become

= 0, (4)

= 0. (5)

Taking twice (4) from (5), and dividing by 3, we get

6#2-7#3 + 6j?y2-7j^3 = 0. (6)

Hence, from (4) and (6),

_ x 5x— 4_ x2 7# — 6

P=y I=5j=/6=7^' and we obtain

7 a?— 6 32 7x— 6

= f

— 4

Both x and y are the squares of the sines of angles and must therefore lie between 0 and 1 . Hence, either x is between 0 and -f , in which case y is between -f- and 1, and p between co and ^%, or else x is between f and 1, in which case y is between 0 and f, and p between 0 and |f.

Galvanometer of Three Colls.

715.] The most convenient arrangement is that in which x = 1. Two of the coils then coincide and form a great circle of the sphere whose radius is C. The number of windings in this compound coil is 64. The other two coils form small circles of the sphere. The radius of each of them is / C. The distance of either of

320 ELECTROMAGNETIC INSTRUMENTS. [715.

them from the plane of the first is /'i C. The number of windings on each of these coils is 49.

1 20

The value of G1 is ~-^- < L>

This arrangement of coils is represented in Fig. 51,

Fig. 51.

Since in this three-coiled galvanometer the first term after G1 which has a finite value is (r7, a large portion of the sphere on whose surface the coils lie forms a field of force sensibly uniform.

If we could wind the wire over the whole of a spherical surface, as described in Art. 627, we should obtain a field of perfectly uniform force. It is practically impossible, however, to distribute the windings on a spherical surface with sufficient accuracy, even if such a coil were not liable to the objection that it forms a closed surface, so that its interior is inaccessible.

By putting the middle coil out of the circuit, and making the current flow in opposite directions through the two side coils, we obtain a field of force which exerts a nearly uniform action in the direction of the axis on a magnet or coil suspended within it, with its axis coinciding with that of the coils; see Art. 673. For in this case all the coefficients of odd orders disappear, and since

Hence the expression for the magnetic potential near the centre of the coil becomes

^ QG W + &C.J

7 1 6.] THICKNESS OF THE WIRE. 321

On the Proper Thickness of the Wire of a Galvanometer, the External Resistance being given.

716.] Let the form of the channel in which the galvanometer coil is to be wound be given, and let it be required to determine whether it ought to be filled with a long thin wire or with a shorter thick wire.

Let I be the length of the wire, y its radius, y + b the radius of the wire when covered, p its specific resistance, g the value of G for unit of length of the wire, and r the part of the resistance which is independent of the galvanometer.

The resistance of the galvanometer wire is

„ P i

Jt= -- 5 •

ity*

The volume of the coil is

7= 4l(y + b)2.

The electromagnetic force is y G, where y is the strength of the current and G — gl.

If E is the electromotive force acting in the circuit whose resistance is R + r, E = y (R + r).

The electromagnetic force due to this electromotive force is

G

which we have to make a maximum by the variation of y and I. Inverting the fraction, we find that

_P J r TT<? f gl is to be made a minimum. Hence

pdy rdl & - o H — 75— = 0.

7T^3 I2

If the volume of the coil remains constant

dl dy

-y- + 2 -*- = 0.

1 y + 6

Eliminating dl and dy, we obtain

p y + b _ r

r or

R y

Hence the thickness of the wire of the galvanometer should be such that the external resistance is to the resistance of the gal vanometer coil as the diameter of the covered wire to the diameter of the wire itself.

VOL. IT. Y

322 ELECTROMAGNETIC INSTRUMENTS. [717.

On Sensitive Galvanometers.

717.] In the construction of a sensitive galvanometer the aim of every part of the arrangement is to produce the greatest possible deflexion of the magnet by means of a given small electromotive force acting between the electrodes of the coil.

The current through the wire produces the greatest effect when it is placed as near as possible to the suspended magnet. The magnet, however, must be left free to oscillate, and therefore there is a certain space which must be left empty within the coil. This defines the internal boundary of the coil.

Outside of this space each winding must be placed so as to have the greatest possible effect on the magnet. As the number of windings increases, the most advantageous positions become filled up, so that at last the increased resistance of a new winding diminishes the effect of the current in the former windings more than the new winding itself adds to it. By making the outer windings of thicker wire than the inner ones we obtain the greatest magnetic effect from a given electromotive force.

718.] We shall suppose that the windings of the galvanometer are circles, the axis of the galvanometer passing through the centres of these circles at right angles to their planes.

Let r sin Q be the radius of one of these circles, and r cos 0 the distance of its centre from the centre of the galvanometer, then, if I is the length of a portion of wire coinciding with this circle,

and y the current which flows in it, the magnetic force at the centre of the gal vanometer resolved in the direction of the axis is sin Q

y'-p-

If we write r2 = x2 sin 0, (1)

this expression become^ y —^ •

x

Hence, if a surface be constructed similar to those represented in section in Fig. 52, whose polar equation is

r2 = x* sin 0, (2)

where a?x is any constant, a given length of wire bent into the form of a circular g ' arc will produce a greater magnetic

effect when it lies within this surface than when it lies outside it.

719.] SENSITIVE GALYANOMETEK. 323

It follows from this that the outer surface of any layer of wire ought to have a constant value of x, for if x is greater at one place than another a portion of wire might be transferred from the first place to the second, so as to increase the force at the centre of the galvanometer.

The whole force due to the coil is y G, where

G

•n-

the integration being extended over the whole length of the wire, x being considered as a function of I.

719.] Let y be the radius of the wire, its transverse section will be 7r^2. Let p be the specific resistance of the material of which the wire is made referred to unit of volume, then the resistance of a

length I is — ^ } and the whole resistance of the coil is

*f

/* 77

(4)

where y is considered a function of I.

Let Y2 be the area of the quadrilateral whose angles are the sections of the axes of four neighbouring wires of the coil by a plane through the axis, then Y2l is the volume occupied in the coil by a length I of wire together with its insulating covering, and including any vacant space necessarily left between the windings of the coil. Hence the whole volume of the coil is

r=jY»dl,

where Y is considered a function of /.

But since the coil is a figure of revolution

V — 2 TT jjr2 sin 0 dr dO, (6)

or, expressing r in terms of x, by equation (2),

V = 2 TT I j a? (sin 0)* dan dB. (7)

Now 27T / (sill 0)$ dO is a numerical quantity, call it JV, then •'o

where F"0 is the volume of the interior space left for the magnet.

Let us now consider a layer of the coil contained between the surfaces x and x + das.

Y 2,

324 ELECTROMAGNETIC INSTRUMENTS. [7J9-

The volume of this layer is

x = Y2dl, (9)

where dl is the length of wire in this layer.

This gives us dl in terms of dx. Substituting this in equations (3) and (4), we find

where f/(r and f/.S represent the portions of the values of G and of It due to this layer of the coil.

Now if E be the given electromotive force,

where r is the resistance of the external part of the circuit, in dependent of the galvanometer, and the force at the centre is

G

si We have therefore to make -= — a maximum, by properly ad-

JK -- T

justing the section of the wire in each layer. This also necessarily involves a variation of Y because Y depends on y.

Let G0 and JRQ be the values of G and of R + r when the given layer is excluded from the calculation. We have then

R0+dR

and to make this a maximum by the variation of the value of y for the given layer we must have

£,*«

(13>

.

ay

C1 Since dx is very small and ultimately vanishes, ^- will be sensibly,

**o

and ultimately exactly, the same whichever layer is excluded, and we may therefore regard it as constant. We have therefore, by (10) and (11), X2 Y dy. PR + r

f 0 + 7 3r) = 1-^- = constant- (14)

If the method of covering the wire and of winding it is such that the proportion between the space occupied by the metal of

720.] SENSITIVE GALVANOMETER. 325

the wire bears the same proportion to the space between the wires whether the wire is thick or thin, then

and we must make both y and Y proportional to x, that is to say, the diameter of the wire in any layer must be proportional to the linear dimension of that layer.

If the thickness of the insulating covering is constant and equal to d, and if the wires are arranged in square order,

Y=2(y + b\ (15)

and the condition is

= constant. (16)

In this case the diameter of the wire increases with the diameter of the layer of which it forms part, but not in so high a ratio.