# 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 Galvanometer.

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.

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 suspended 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, ${\displaystyle n}$, of the diameter of the covered wire. A hole is bored in the side of the channel where the wire is 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 ${\displaystyle n}$ windings. During this process the string has been wound ${\displaystyle n}$ times round the wooden axis, and a nail is driven into the string at the ${\displaystyle n}$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

${\displaystyle G_{1}g_{1}\sin \theta +G_{2}g_{2}\sin \theta Q_{2}^{\prime }(\theta )+\mathrm {\&c.} }$,

where the coefficients ${\displaystyle G}$ refer to the coil, and the coefficients ${\displaystyle g}$ to the suspended apparatus, ${\displaystyle \theta }$ 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 longitudinally magnetized bar magnet of length ${\displaystyle 2l}$ and strength unity, suspended by its middle,

${\displaystyle g_{1}=2l}$,${\displaystyle g_{2}=0}$, ${\displaystyle g_{3}=2l^{3}}$, &c.

The values of the coefficients for a magnet of length ${\displaystyle 2l}$ magnetized in any other way are smaller than when it is magnetized uniformly.

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

${\displaystyle mg_{1}H\cos \theta =m\gamma \sin \theta \{G_{1}g_{1}+G_{2}g_{2}Q_{1}^{\prime }(\theta )+\mathrm {\&c.} \}}$,

where ${\displaystyle mg_{1}}$ is the magnetic moment of the magnet, ${\displaystyle H}$ the horizontal component of the terrestrial magnetic force, and ${\displaystyle \gamma }$ 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 ${\displaystyle G}$ and ${\displaystyle \gamma }$ may be neglected, and we find

${\displaystyle \gamma ={\frac {H}{G_{1}}}\cot \theta }$.

The angle usually measured is the deflexion, ${\displaystyle \delta }$, of the magnet which is the complement of ${\displaystyle \theta }$, so that ${\displaystyle \cot \theta =\tan \delta }$.

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 ${\displaystyle \delta }$, the equation of equilibrium is

${\displaystyle mg_{1}H\sin \delta =M\gamma {\frac {1}{g}}\{G_{1}g_{1}-{\frac {3}{2}}G_{3}g_{3}+\mathrm {\&c.} \}}$,

whence

${\displaystyle \gamma ={\frac {H}{(G_{1}-\mathrm {\&c.} )}}\sin \delta }$.

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 equilibrium.

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 ${\displaystyle A}$ be the mean radius of the coil, ${\displaystyle \xi }$ its depth, ${\displaystyle \eta }$ its breadth, and ${\displaystyle n}$ the number of windings, the values of the coefficients are

 ${\displaystyle G_{1}={\frac {2\pi n}{A}}\left\{1+{\frac {1}{12}}{\frac {\xi ^{2}}{A^{2}}}={\frac {1}{}}{\frac {\eta ^{2}}{A^{2}}}\right\}}$, ${\displaystyle G_{2}=0}$, ${\displaystyle G_{3}=-{\frac {\pi n}{A^{3}}}\left\{1+{\frac {1}{2}}{\frac {\xi ^{2}}{A^{2}}}-{\frac {5}{8}}{\frac {\eta ^{2}}{A^{2}}}\right\}}$, ${\displaystyle G_{4}=0}$, &c.
The principal correction is that arising from ${\displaystyle G_{3}}$. The series

${\displaystyle G_{1}g_{1}+G_{3}g_{3}Q_{3}^{\prime }(\theta )}$

becomes

${\displaystyle G_{1}g_{1}\left(1-{\frac {3}{2}}{\frac {1}{A^{2}}}{\frac {g_{3}}{g_{1}}}(\cos ^{2}\theta -{\frac {1}{4}}\sin ^{2}\theta )\right)}$.

The factor of correction will differ most from unity when the magnet is uniformly magnetized and when ${\displaystyle \theta =0}$. In this case it becomes ${\displaystyle 1-{\frac {1}{2}}{\frac {l^{2}}{A^{2}}}}$. It vanishes when ${\displaystyle \tan \theta =2}$, or when the deflexion is ${\displaystyle \tan ^{-1}{\frac {1}{2}}}$, 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 ${\displaystyle x}$, ${\displaystyle y}$, ${\displaystyle z}$, ${\displaystyle z}$ being measured parallel to the axis of the coil, the factor of correction is

${\displaystyle \left(1+{\frac {3}{2}}{\frac {x^{2}+y^{2}-2z^{2}}{A^{2}}}\right)}$.

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

Gaugain's Arrangement

712.] In order to get rid of the correction depending on ${\displaystyle G_{3}}$ Gaugain constructed a galvanometer in which this term was rendered 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 ${\displaystyle G_{3}}$ is

${\displaystyle G_{3}=4\pi {\frac {A^{2}(B^{2}-{\frac {1}{4}}A^{2})}{C^{7}}}}$

and, since in this arrangement ${\displaystyle B={\frac {1}{2}}A}$, ${\displaystyle G_{3}=0}$.

This arrangement would be an improvement on the first form if we could be sure that the centre of the suspended magnet is exactly at the point thus defined. The position of the centre of the magnet, however, is always uncertain, and this uncertainty introduces a factor of correction of unknown amount depending on ${\displaystyle G_{2}}$ and of the form ${\displaystyle \left(1-{\frac {6}{5}}{\frac {z}{A}}\right)}$, where ${\displaystyle z}$ is the unknown excess of distance of the centre of the magnet from the plane of the coil. This correction depends on the first power of ${\displaystyle {\frac {z}{A}}}$. Hence Gaugain's coil with eccentrically suspended magnet is subject to far greater uncertainty than the old form.

Helmholtz's Arrangement.

713.] Helmholtz converted Gaugain's galvanometer into a trustworthy 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 ${\displaystyle A}$ be the mean radius of either coil, the distance between their mean planes is made equal to ${\displaystyle A}$, and the magnet is suspended at the middle point of their common axis. The coefficients are

 ${\displaystyle G_{1}={\frac {16\pi n}{5{\sqrt {5}}}}{\frac {1}{A}}\left(1-{\frac {1}{60}}{\frac {\xi ^{2}}{A^{2}}}\right)}$, ${\displaystyle G_{2}=0}$, ${\displaystyle G_{3}=0.0512{\frac {\pi n}{3{\sqrt {5}}A^{5}}}(31\xi ^{2}-36\eta ^{2})}$, ${\displaystyle G_{4}=0}$, ${\displaystyle G_{5}=-0.73728{\frac {\pi n}{{\sqrt {5}}A^{5}}}}$,
where ${\displaystyle n}$ 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 ${\displaystyle \xi }$ and the breadth ${\displaystyle \eta }$, the value of ${\displaystyle G_{3}}$, as corrected for the finite size of the section, will be small, and will vanish, if ${\displaystyle \xi }$ is to ${\displaystyle \eta }$ 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.

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 ${\displaystyle G_{2}}$, ${\displaystyle G_{3}}$, ${\displaystyle G_{4}}$, ${\displaystyle G_{5}}$, and ${\displaystyle G_{6}}$. 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 ${\displaystyle \theta }$, ${\displaystyle \phi }$, ${\displaystyle \pi -\phi }$, and ${\displaystyle \pi -\theta }$.

Let the number of windings on the first and fourth coil be ${\displaystyle n}$, and the number on the second and third ${\displaystyle pn}$. Then the condition that ${\displaystyle G_{3}=-}$ for the combination gives

${\displaystyle n\sin ^{2}\theta Q_{3}^{\prime }(\theta )+pn\sin ^{2}\phi Q_{3}^{\prime }(\phi )=0}$,

(1)

and the condition that ${\displaystyle G_{5}=0}$ gives

${\displaystyle n\sin ^{2}\theta Q_{5}^{\prime }(\theta )+pn\sin ^{2}\phi Q_{5}^{\prime }(\phi )=0}$,

(2)

Putting

${\displaystyle \sin ^{2}\theta =x}$ and ${\displaystyle \sin ^{2}\phi =y}$,

(3)

and expressing ${\displaystyle Q_{3}^{\prime }}$ and ${\displaystyle Q_{5}^{\prime }}$ (Art. 698) in terms of these quantities, the equations (1) and (2) become

${\displaystyle 4x-5x^{2}+4py-5py^{2}=0}$

(4)

${\displaystyle 8x=28x^{2}+21x^{3}+8py-28py^{2}+21py^{3}=0}$.

(5)

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

${\displaystyle 6x^{2}-7x^{3}+6py^{2}-7py^{3}=0}$.

(6)

Hence, from (4) and (6),

${\displaystyle p={\frac {x}{y}}{\frac {5x-4}{4-5y}}={\frac {x^{2}}{y^{2}}}{\frac {7x-6}{6-7y}}}$,

and we obtain

${\displaystyle y={\frac {4}{7}}{\frac {7x-6}{5x-4}}}$, ${\displaystyle p={\frac {32}{49x}}{\frac {7x-6}{(5x-4)^{3}}}}$.

Both ${\displaystyle x}$ and ${\displaystyle y}$ are the squares of the sines of angles and must therefore lie between 0 and 1. Hence, either ${\displaystyle x}$ is between 0 and ${\displaystyle {\frac {4}{7}}}$, in which case ${\displaystyle y}$ is between ${\displaystyle {\frac {6}{7}}}$ and 1, and ${\displaystyle p}$ between ${\displaystyle \infty }$ and ${\displaystyle {\frac {49}{32}}}$, or else ${\displaystyle x}$ is between ${\displaystyle {\frac {6}{7}}}$ and 1, in which case ${\displaystyle y}$ is between 0 and ${\displaystyle {\frac {4}{7}}}$, and ${\displaystyle p}$ between 0 and ${\displaystyle {\frac {32}{49}}}$.

Galvanometer of Three Coils.

715.] The most convenient arrangement is that in which ${\displaystyle x=1}$. Two of the coils then coincide and form a great circle of the sphere whose radius is ${\displaystyle C}$. 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 ${\displaystyle {\sqrt {\frac {4}{7}}}C}$. The distance of either of them from the plane of the first is ${\displaystyle {\sqrt {\frac {3}{7}}}C}$. The number of windings on each of these coils is 49.

The value of ${\displaystyle G_{1}}$ is ${\displaystyle {\frac {120}{C}}}$.

This arrangement of coils is represented in Fig. 51.

Fig. 51.

Since in this three-coiled galvanometer the first term after ${\displaystyle G_{1}}$ which has a finite value is ${\displaystyle G_{7}}$, 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

${\displaystyle \mu ={\sqrt {\frac {3}{7}}}}$,${\displaystyle Q_{4}^{\prime }={\frac {5}{2}}\mu (7\mu ^{2}-3)=0}$.

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

${\displaystyle \omega ={\sqrt {\frac {3}{7}}}\pi n\gamma \left\{3{\frac {r^{2}}{C^{2}}}Q_{2}(\theta )+{\frac {11}{7}}{\frac {r^{6}}{C^{6}}}Q_{6}(\theta )+\mathrm {\&c.} \right\}}$.

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 ${\displaystyle l}$ be the length of the wire, ${\displaystyle y}$ its radius, ${\displaystyle y+b}$ the radius of the wire when covered, ${\displaystyle \rho }$ its specific resistance, ${\displaystyle g}$ the value of ${\displaystyle G}$ for unit of length of the wire, and ${\displaystyle r}$ the part of the resistance which is independent of the galvanometer.

The resistance of the galvanometer wire is

${\displaystyle R={\frac {\rho }{\pi }}{\frac {l}{y^{2}}}}$.

The volume of the coil is

${\displaystyle V=4l(y+b)^{2}}$

The electromagnetic force is ${\displaystyle \gamma G}$, where ${\displaystyle \gamma }$ is the strength of the current and

${\displaystyle G=gl}$.

If ${\displaystyle E}$ is the electromotive force acting in the circuit whose resistance is ${\displaystyle R+r}$,

${\displaystyle E=y(R+r)}$.

The electromagnetic force due to this electromotive force is

${\displaystyle E{\frac {G}{R+r}}}$,

which we have to make a maximum by the variation of ${\displaystyle y}$ and ${\displaystyle l}$. Inverting the fraction, we find that

${\displaystyle {\frac {\rho }{\pi g}}{\frac {1}{y^{2}}}+{\frac {r}{gl}}}$

is to be made a minimum. Hence

${\displaystyle 2{\frac {\rho }{\pi }}{\frac {dy}{y^{3}}}+{\frac {r\,dl}{l^{2}}}=0}$.

If the volume of the coil remains constant

${\displaystyle {\frac {dl}{l}}+2{\frac {dy}{y+b}}=0}$.

Eliminating ${\displaystyle dl}$ and ${\displaystyle dy}$, we obtain

${\displaystyle {\frac {\rho }{\pi }}{\frac {y+b}{y^{3}}}={\frac {r}{l}}}$,

or

${\displaystyle {\frac {r}{R}}={\frac {y+b}{y}}}$.

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

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.

Fig. 52.
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 ${\displaystyle r\sin \theta }$ be the radius of one of these circles, and ${\displaystyle r\cos \theta }$ the distance of its centre from the centre of the galvanometer, then, if ${\displaystyle l}$ is the length of a portion of wire coinciding with this circle, and ${\displaystyle \gamma }$ the current which flows in it, the magnetic force at the centre of the galvanometer resolved in the direction of the axis is

${\displaystyle \gamma l{\frac {\sin \theta }{r^{2}}}}$.

If we write

${\displaystyle r^{2}=x^{2}\sin \theta }$,

(1)

this expression becomes ${\displaystyle \gamma {\frac {l}{x^{2}}}}$. Hence, if a surface be constructed similar to those represented in section in Fig. 52, whose polar equation is

${\displaystyle r^{2}=x_{1}^{2}\sin \theta }$,

(2)

where ${\displaystyle x_{1}}$ is any constant, a given length of wire bent into the form of a circular arc will produce a greater magnetic effect when it lies within this surface than when it lies outside it. It follows from this that the outer surface of any layer of wire ought to have a constant value of ${\displaystyle x}$, for if ${\displaystyle x}$ 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 ${\displaystyle \gamma G}$, where

${\displaystyle G=\int {\frac {dl}{x^{2}}}}$,

(3)

the integration being extended over the whole length of the wire, ${\displaystyle x}$ being considered as a function of ${\displaystyle l}$.

719.] Let ${\displaystyle y}$ be the radius of the wire, its transverse section will be ${\displaystyle \pi y^{2}}$. Let ${\displaystyle \rho }$ be the specific resistance of the material of which the wire is made referred to unit of volume, then the resistance of a length ${\displaystyle l}$ is ${\displaystyle {\frac {l\rho }{\pi y^{2}}}}$, and the whole resistance of the coil is

${\displaystyle R={\frac {\rho }{\pi }}\int {\frac {dl}{y^{2}}}}$,

(4)

where ${\displaystyle y}$ is considered a function of ${\displaystyle l}$. Let ${\displaystyle Y^{2}}$ 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 ${\displaystyle Y^{2}l}$ is the volume occupied in the coil by a length ${\displaystyle l}$ 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

${\displaystyle V=\int Y^{2}dl}$,

(5)

where ${\displaystyle Y}$ is considered a function of ${\displaystyle l}$. But since the coil is a figure of revolution

${\displaystyle V=2\pi \iint r^{2}\sin \theta \,dr\,d\theta }$,

(6)

or, expressing ${\displaystyle r}$ in terms of ${\displaystyle x}$, by equation (2),

${\displaystyle V=2\pi \iint x^{2}(\sin \theta )^{\frac {5}{2}}\,dx\,d\theta }$.

(7)

Now ${\displaystyle 2\pi \int _{0}^{\pi }(\sin \theta )^{\frac {5}{2}}\,d\theta }$ is a numerical quantity, call it ${\displaystyle N}$, then

${\displaystyle V={\frac {1}{3}}Nx^{3}-V_{0}}$,

(8)

where ${\displaystyle V_{0}}$ is the volume of the interior space left for the magnet.

Let us now consider a layer of the coil contained between the surfaces ${\displaystyle x}$ and ${\displaystyle x+dx}$.

he volume of this layer is

${\displaystyle dV=Nx^{3}\,dx=Y^{2}\,dl}$,

(9)

where ${\displaystyle dl}$ is the length of wire in this layer.

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

 ⁠ ${\displaystyle dG}$ ${\displaystyle {}=N{\frac {dx}{Y^{2}}}}$, ⁠ (10) ${\displaystyle dR}$ ${\displaystyle {}=N{\frac {\rho }{\pi }}{\frac {x^{2}dx}{Y^{2}y^{2}}}}$, (11)

where ${\displaystyle dG}$ and ${\displaystyle dR}$ represent the portions of the values of ${\displaystyle G}$ and of ${\displaystyle R}$ due to this layer of the coil.

Now if ${\displaystyle E}$ be the given electromotive force,

${\displaystyle E=\gamma (R+r)}$,

where ${\displaystyle r}$ is the resistance of the external part of the circuit, independent of the galvanometer, and the force at the centre is

${\displaystyle \gamma G=E{\frac {G}{R+r}}}$.

We have therefore to make ${\displaystyle {\frac {G}{R+r}}}$ a maximum, by properly adjusting the section of the wire in each layer. This also necessarily involves a variation of ${\displaystyle Y}$ because ${\displaystyle Y}$ depends on ${\displaystyle y}$.

Let ${\displaystyle G_{0}}$ and ${\displaystyle R_{0}}$ be the values of ${\displaystyle G}$ and of ${\displaystyle R+r}$ when the given layer is excluded from the calculation. We have then

${\displaystyle {\frac {G}{R+r}}={\frac {G_{0}+dG}{R_{0}+dR}}}$,

(12)

and to make this a maximum by the variation of the value of ${\displaystyle y}$ for he given layer we must have

${\displaystyle {\frac {{\frac {d}{dy}}.dG}{{\frac {d}{dy}}.dR}}={\frac {G}{R+r}}}$.

(13)

Since ${\displaystyle dx}$ is very small and ultimately vanishes, ${\displaystyle {\frac {G_{0}}{R_{0}}}}$ will be sensibly, and ultimately exactly, the same whichever layer is excluded, and we may therefore regard it as constant. We have therefore, by (10) and (11),

${\displaystyle {\frac {x^{2}}{y^{2}}}\left(1+{\frac {Y}{y}}{\frac {dy}{dY}}\right)={\frac {\rho }{\pi }}{\frac {R+r}{G}}=\mathrm {constant} }$.

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

${\displaystyle {\frac {Y}{y}}{\frac {dy}{dY}}=1}$,

and we must make both ${\displaystyle y}$ and ${\displaystyle Y}$ proportional to ${\displaystyle x}$, 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 ${\displaystyle b}$, and if the wires are arranged in square order,

${\displaystyle Y=2(y+b)}$,

(15)

and the condition is

${\displaystyle {\frac {x^{2}(2y+b)}{y^{3}}}=\mathrm {constant} }$.

(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.

If we adopt the first of these two hypotheses, which will be nearly true if the wire itself nearly fills up the whole space, then we may put

${\displaystyle y=\alpha x}$, ${\displaystyle Y=\beta y}$,

where ${\displaystyle \alpha }$ and ${\displaystyle \beta }$ are constant numerical quantities, and
 ${\displaystyle G=N{\frac {1}{\alpha ^{2}\beta ^{2}}}\left({\frac {1}{a}}-{\frac {1}{x}}\right)}$, ${\displaystyle R=N{\frac {\rho }{\pi }}{\frac {1}{\alpha ^{4}\beta ^{2}}}\left({\frac {1}{a}}-{\frac {1}{x}}\right)}$,
where ${\displaystyle a}$ is a constant depending upon the size and form of the free space left inside the coil.

Hence, if we make the thickness of the wire vary in the same ratio as ${\displaystyle x}$, we obtain very little advantage by increasing the external size of the coil after the external dimensions have become a large multiple of the internal dimensions.

720.] If increase of resistance is not regarded as a defect, as when the external resistance is far greater than that of the galvanometer, or when our only object is to produce a field of intense force, we may make ${\displaystyle y}$ and ${\displaystyle Y}$ constant. We have then
 ${\displaystyle G={\frac {N}{Y^{2}}}(x-a)}$, ${\displaystyle R={\frac {1}{3}}{\frac {N}{Yy^{2}}}{\frac {\rho }{\pi }}(x^{3}-a_{1}^{3})}$,
where ${\displaystyle a}$ is a constant depending on the vacant space inside the coil. In this case the value of ${\displaystyle G}$ increases uniformly as the dimensions of the coil are increased, so that there is no limit to the value of ${\displaystyle G}$ except the labour and expense of making the coil.

On Suspended Coils.

721.] In the ordinary galvanometer a suspended magnet is acted on by a fixed coil. But if the coil can be suspended with sufficient delicacy, we may determine the action of the magnet, or of another coil on the suspended coil, by its deflexion from the position of equilibrium.

We cannot, however, introduce the electric current into the coil unless there is metallic connexion between the electrodes of the battery and those of the wire of the coil. This connexion may be made in two different ways, by the Bifilar Suspension, and by wires in opposite directions.

The bifilar suspension has already been described in Art. 459 as applied to magnets. The arrangement of the upper part of the suspension is shewn in Fig. 55. When applied to coils, the two fibres are no longer of silk but of metal, and since the torsion of a metal wire capable of supporting the coil and transmitting the current is much greater than that of a silk fibre, it must be taken specially into account. This suspension has been brought to great perfection in the instruments constructed by M. Weber.

Fig. 53.
The other method of suspension is by means of a single wire which is connected to one extremity of the coil. The other extremity of the coil is connected to another wire which is made to hang down, in the same vertical straight line with the first wire, into a cup of mercury, as is shewn in Fig. 57, Art. 729. In certain cases it is convenient to fasten the extremities of the two wires to pieces by which they may be tightly stretched, care being taken that the line of these wires passes through the centre of gravity of the coil. The apparatus in this form may be used when the axis is not vertical; see Fig. 53.

722.] The suspended coil may be used as an exceedingly sensitive galvanometer, for, by increasing the intensity of the magnetic force in the field in which it hangs, the force due to a feeble current in the coil may be greatly increased without adding to the mass of the coil. The magnetic force for this purpose may be produced by means of permanent magnets, or by electromagnets excited by an auxiliary current, and it may be powerfully concentrated on the suspended coil by means of soft iron armatures. Thus, in Sir W. Thomson's recording apparatus, Fig. 53, the coil is suspended between the opposite poles of the electromagnets ${\displaystyle N}$ and ${\displaystyle S}$, and in order to concentrate the lines of magnetic force on the vertical sides of the coil, a piece of soft iron, ${\displaystyle D}$, is fixed between the poles of the magnets. This iron becoming magnetized by induction, produces a very powerful field of force, in the intervals between it and the two magnets, through which the vertical sides of the coil are free to move, so that the coil, even when the current through it is very feeble, is acted on by a considerable force tending to turn it about its vertical axis.

723.] Another application of the suspended coil is to determine, by comparison with a tangent galvanometer, the horizontal component of terrestrial magnetism.

The coil is suspended so that it is in stable equilibrium when its plane is parallel to the magnetic meridian. A current ${\displaystyle \gamma }$ is passed through the coil and causes it to be deflected into a new position of equilibrium, making an angle ${\displaystyle \theta }$ with the magnetic meridian. If the suspension is bifilar, the moment of the couple which produces this deflexion is ${\displaystyle F\sin \theta }$, and this must be equal to ${\displaystyle H\gamma g\cos \theta }$, where ${\displaystyle H}$ is the horizontal component of terrestrial magnetism, ${\displaystyle \gamma }$ is the current in the coil, and ${\displaystyle g}$ is the sum of the areas of all the windings of the coil. Hence

${\displaystyle H\gamma ={\frac {F}{g}}\tan \theta }$.

If ${\displaystyle A}$ is the moment of inertia of the coil about its axis of suspension, and ${\displaystyle T}$ the time of a single vibration,

${\displaystyle FT^{2}=\pi ^{2}A}$,

and we obtain

${\displaystyle H\gamma ={\frac {\pi ^{2}A}{T^{2}g}}\tan \theta }$.

If the same current passes through the coil of a tangent galvanometer, and deflects the magnet through an angle ${\displaystyle \phi }$,

${\displaystyle {\frac {\gamma }{H}}={\frac {1}{G}}\tan \phi }$,

where ${\displaystyle G}$ is the principal constant of the tangent galvanometer, Art. 710. From these two equations we obtain

${\displaystyle H={\frac {\pi }{T}}{\sqrt {\frac {AG\tan \theta }{g\tan \phi }}}}$,${\displaystyle \gamma ={\frac {\pi }{T}}{\sqrt {\frac {A\tan \theta \tan \phi }{Gg}}}}$.

This method was given by F. Kohlrausch[1].

724.] Sir William Thomson has constructed a single instrument by means of which the observations required to determine ${\displaystyle H}$ and ${\displaystyle \gamma }$ may be made simultaneously by the same observer.

The coil is suspended so as to be in equilibrium with its plane in the magnetic meridian, and is deflected from this position when the current flows through it. A very small magnet is suspended at the centre of the coil, and is deflected by the current in the direction opposite to that of the deflexion of the coil. Let the deflexion of the coil be ${\displaystyle \theta }$, and that of the magnet ${\displaystyle \phi }$, then the energy of the system is

${\displaystyle H\gamma g\sin \theta +m\gamma G\sin(\theta -\phi )-Hm\cos \phi -F\cos \theta }$.

Differentiating with respect to ${\displaystyle \theta }$ and ${\displaystyle \phi }$, we obtain the equations of equilibrium of the coil and of the magnet respectively,

${\displaystyle H\gamma g\cos \phi +m\gamma G\cos(\theta -\phi )+F\sin \theta =0}$,

${\displaystyle -m\gamma G\cos(\theta -\phi )+Hm\sin \phi =0}$.

From these equations we find, by eliminating ${\displaystyle H}$ or ${\displaystyle \gamma }$, a quadratic equation from which ${\displaystyle \gamma }$ or ${\displaystyle H}$ may be found. If ${\displaystyle m}$, the magnetic moment of the suspended magnet, is very small, we obtain the following approximate values
 ${\displaystyle H={\frac {\pi }{T}}{\sqrt {\frac {-AG\sin \theta \cos(\theta -\phi )}{g\cos \theta \sin \phi }}}-{\frac {1}{2}}{\frac {mG}{g}}{\frac {\cos(\theta -\phi )}{\cos \theta }}}$, ${\displaystyle \gamma ={\frac {\pi }{T}}{\sqrt {\frac {-A\sin \theta \sin \phi }{Gg\cos \theta \cos(\theta -\phi )}}}-{\frac {1}{2}}{\frac {m}{g}}{\frac {\sin \phi }{\cos \theta }}}$.

In these expressions ${\displaystyle G}$ and ${\displaystyle g}$ are the principal electric constants of the coil, ${\displaystyle A}$ its moment of inertia, ${\displaystyle T}$ its time of vibration, ${\displaystyle m}$ the magnetic moment of the magnet, ${\displaystyle H}$ the intensity of the horizontal magnetic force, ${\displaystyle \gamma }$ the strength of the current, ${\displaystyle \theta }$ the deflexion of the coil, and ${\displaystyle \phi }$ that of the magnet.

Since the deflexion of the coil is in the opposite direction to the deflexion of the magnet, these values of ${\displaystyle H}$ and ${\displaystyle \gamma }$ will always be real.

Weber's Electrodynamometer.

725.] In this instrument a small coil is suspended by two wires within a larger coil which is fixed. When a current is made to flow through both coils, the suspended coil tends to place itself parallel to the fixed coil. This tendency is counteracted by the moment of the forces arising from the bifilar suspension, and it is also affected by the action of terrestrial magnetism on the suspended coil.

In the ordinary use of the instrument the planes of the two coils are nearly at right angles to each other, so that the mutual action of the currents in the coils may be as great as possible, and the plane of the suspended coil is nearly at right angles to the magnetic meridian, so that the action of terrestrial magnetism may be as small as possible.

Let the magnetic azimuth of the plane of the fixed coil be ${\displaystyle \alpha }$, and let the angle which the axis of the suspended coil makes with the plane of the fixed coil be ${\displaystyle \theta +\beta }$, where ${\displaystyle \beta }$ is the value of this angle when the coil is in equilibrium and no current is flowing, and ${\displaystyle \theta }$ is the deflexion due to the current. The equation of equilibrium is

${\displaystyle Gg\gamma _{1}\gamma _{2}\cos(\theta +\beta )-Hg\gamma _{2}\sin(\theta +\beta +\alpha )-F\sin \theta =0}$.

Let us suppose that the instrument is adjusted so that ${\displaystyle \alpha }$ and ${\displaystyle \beta }$ are both very small, and that ${\displaystyle Hg\gamma _{2}}$ is small compared with ${\displaystyle F}$. We have in this case, approximately,

${\displaystyle \tan \theta ={\frac {Gg\gamma _{1}\gamma _{2}\cos \beta }{F}}-{\frac {Hg\gamma _{2}\sin(\alpha +\beta )}{F}}-{\frac {HGg^{2}\gamma _{1}\gamma _{2}^{2}}{F^{2}}}-{\frac {G^{2}g^{2}\gamma _{1}^{2}\gamma _{2}^{2}\sin \beta }{F^{2}}}}$.

If the deflexions when the signs of ${\displaystyle \gamma _{1}}$ and ${\displaystyle \gamma _{2}}$ are changed are as follows:
 ${\displaystyle \theta _{1}}$ when ${\displaystyle \gamma _{1}}$ is ${\displaystyle {}+{}}$ and ${\displaystyle \gamma _{2}}$ ${\displaystyle {}+{}}$, ${\displaystyle \theta _{2}}$ .mw-parser-output .__ditto{display:inline-block;position:relative;text-indent:0}.mw-parser-output .__ditto_hidden{visibility:hidden;color:transparent;white-space:nowrap}.mw-parser-output .__ditto_text{display:inline-block;position:absolute;left:0;width:100%;white-space:nowrap}{{{1}}}„ ${\displaystyle {}-{}}$ {{{1}}}„ ${\displaystyle {}-{}}$, ${\displaystyle \theta _{3}}$ {{{1}}}„ ${\displaystyle {}+{}}$ {{{1}}}„ ${\displaystyle {}-{}}$, ${\displaystyle \theta _{4}}$ {{{1}}}„ ${\displaystyle {}-{}}$ {{{1}}}„ ${\displaystyle {}+{}}$;
then we find

${\displaystyle \gamma _{1}\gamma _{2}={\frac {1}{4}}{\frac {F}{Gg\cos \beta }}(\tan \theta _{1}+\tan \theta _{2}-\tan \theta _{3}-\tan \theta _{4})}$.

If it is the same current which flows through both coils we may put ${\displaystyle \gamma _{1}\gamma _{2}=\gamma ^{2}}$, and thus obtain the value of ${\displaystyle \gamma }$.

When the currents are not very constant it is best to adopt this method, which is called the Method of Tangents.

If the currents are so constant that we can adjust ${\displaystyle \beta }$, the angle of the torsion-head of the instrument, we may get rid of the correction for terrestrial magnetism at once by the method of sines. In this method ${\displaystyle \beta }$ is adjusted till the deflexion is zero, so that

${\displaystyle \theta =-\beta }$.

If the signs of ${\displaystyle \gamma _{1}}$ and ${\displaystyle \gamma _{2}}$ are indicated by the suffixes of ${\displaystyle \beta }$ as before,
 ${\displaystyle F\sin \beta _{1}=-F\sin \beta _{3}=-Gg\gamma _{1}\gamma _{2}+Hg\gamma _{2}\sin \alpha }$, ${\displaystyle F\sin \beta _{2}=-F\sin \beta _{4}=-Gg\gamma _{1}\gamma _{2}-Hg\gamma _{2}\sin \alpha }$.

and

${\displaystyle \gamma _{1}\gamma _{2}=-{\frac {F}{4Gg}}(\sin \beta _{1}+\sin \beta _{2}-\sin \beta _{3}-\sin \beta _{4})}$.

Fig. 54.

This is the method adopted by Mr. Latimer Clark in his use of the instrument constructed by the Electrical Committee of the British Association. We are indebted to Mr. Clark for the drawing of the electrodynamometer in Figure 54, in which Helmholtz's arrangement of two coils is adopted both for the fixed and for the suspended coil[2]. The torsion-head of the instrument, by which the bifilar suspension is adjusted, is represented in Fig. 55. The

Fig. 55.

equality of the tension of the suspension wires is ensured by their being attached to the extremities of a silk thread which passes over a wheel, and their distance is regulated by two guide-wheels, which can be set at the proper distance. The suspended coil can be moved vertically by means of a screw acting on the suspension-wheel, and horizontally in two directions by the sliding pieces shewn at the bottom of Fig. 55. It is adjusted in azimuth by means of the torsion-screw, which turns the torsion-head round a vertical axis (see Art. 459). The azimuth of the suspended coil is ascertained by observing the reflexion of a scale in the mirror, shewn just beneath the axis of the suspended coil.

The instrument originally constructed by Weber is described in his Elektrodynamische Maasbestimmungen. It was intended for the measurement of small currents, and therefore both the fixed and the suspended coils consisted of many windings, and the suspended coil occupied a larger part of the space within the fixed coil than in the instrument of the British Association, which was primarily intended as a standard instrument, with which more sensitive instruments might be compared. The experiments which he made with it furnish the most complete experimental proof of the accuracy of Ampère's formula as applied to closed currents, and form an important part of the researches by which Weber has raised the numerical determination of electrical quantities to a very high rank as regards precision.

Weber's form of the electrodynamometer, in which one coil is suspended within another, and is acted on by a couple tending to turn it about a vertical axis, is probably the best fitted for absolute measurements. A method of calculating the constants of such an arrangement is given in Art. 697.

726.] If, however, we wish, by means of a feeble current, to produce a considerable electromagnetic force, it is better to place the suspended coil parallel to the fixed coil, and to make it capable of motion to or from it.

Fig. 56.
The suspended coil in Dr. Joule's current-weigher, Fig. 56, is horizontal, and capable of vertical motion, and the force between it and the fixed coil is estimated by the weight which must be added to or removed from the coil in order to bring it to the same relative position with respect to the fixed coil that it has when no current passes.

The suspended coil may also be fastened to the extremity of the horizontal arm of a torsion-balance, and may be placed between two fixed coils, one of which attracts it, while the other repels it, as in Fig. 57.

By arranging the coils as described in Art. 729, the force acting on the suspended coil may be made nearly uniform within a small distance of the position of equilibrium.

Another coil may be fixed to the other extremity of the arm of the torsion-balance and placed between two fixed coils. If the two suspended coils are similar, but with the current flowing in opposite directions, the effect of terrestrial magnetism on the position of the arm of the torsion-balance will be completely eliminated.

Fig. 57.

727.] If the suspended coil is in the shape of a long solenoid, and is capable of moving parallel to its axis, so as to pass into the interior of a larger fixed solenoid having the same axis, then, if the current is in the same direction in both solenoids, the suspended solenoid will be sucked into the fixed one by a force which will be nearly uniform as long as none of the extremities of the solenoids are near one another.

728.] To produce a uniform longitudinal force on a small coil placed between two equal coils of much larger dimensions, we should make the ratio of the diameter of the large coils to the distance between their planes that of ${\displaystyle 2}$ to ${\displaystyle {\sqrt {3}}}$. If we send the same current through these coils in opposite directions, then, in the expression for ${\displaystyle \omega }$, the terms involving odd powers of ${\displaystyle r}$ disappear, and since ${\displaystyle \sin ^{2}\alpha ={\frac {4}{7}}}$ and ${\displaystyle \cos ^{2}\alpha ={\frac {3}{7}}}$, the term involving ${\displaystyle r^{4}}$ disappears also, and we have

${\displaystyle \omega ={\frac {8}{7}}{\sqrt {\frac {3}{7}}}\pi n\gamma \left\{3{\frac {r^{2}}{c^{2}}}Q_{2}(\theta )+{\frac {11}{7}}{\frac {r^{6}}{c^{6}}}Q_{6}(\theta )+\mathrm {\&c.} \right\}}$,

which indicates a nearly uniform force on a small suspended coil. The arrangement of the coils in this case is that of the two outer coils in the galvanometer with three coils, described at Art. 715. See Fig. 51.

729.] If we wish to suspend a coil between two coils placed so near it that the distance between the mutually acting wires is small compared with the radius of the coils, the most uniform force is obtained by making the radius of either of the outer coils exceed that of the middle one by ${\displaystyle {\frac {1}{\sqrt {3}}}}$ of the distance between the planes of the middle and outer coils.

1. Pogg., Ann. cxxxviii, Feb. 1869.
2. In the actual instrument, the wires conveying the current to and from the coils are not spread out as displayed in the figure, but are kept as close together as possible, so as to neutralize each other's electromagnetic action.