The Determination of Coefficients of Induction by the Electric Balance.

(43) The electric balance consists of six conductors joining four points, A,C,D,E, two and two. One pair, AC, of these points is connected through the battery B. The opposite pair, DE, is connected through the battery B. The opposite pair, DE, is connected through the galvanometer G. Then if the resistances of the four remaining conductors are represented by P,Q,R,S, and the currents in them by

and

the current through G will be

{\displaystyle z}. Let the potentials at the four points be A,C,D,E. Then the conditions of steady currents may be found from the equations

(21) Solving these equations for

{\displaystyle z}, we find

(22)

In this expression F is the electromotive force of the battery,

{\displaystyle z} the current through the galvanometer when it has become steady. P, Q, R, S the resistances in the four arms. B that of the battery and electrodes, and G that of the galvanometer.

(44) If PS=QR, then

{\displaystyle z=0}, and there will be no steady current, but a transient current through the galvanometer may be produced on making or breaking circuit on account of induction, and the indications of the galvanometer may be used to determine the coefficients of induction, provided we understand the actions which take place.

We shall suppose PS=QR, so that the current

{\displaystyle z} vanishes when sufficient time is allowed, and

Let the induction coefficients between P, Q, R, S be given by the following table, the coefficient of induction of P on itself being

{\displaystyle p}, between P and Q, h {\displaystyle h}, and so on.

Let

{\displaystyle g} be the coefficient of induction of the galvanometer on itself, and let it be out of the reach of the inductive influence of P,Q,R,S (as it must be in order to avoid direct action of P,Q,R,S on the needle). Let X,Y,Z be the integrals of

{\displaystyle x,y,z} with respect to t {\displaystyle t}. At making contact

{\displaystyle x,y,z} are zero. After a time z {\displaystyle z} disappears, and x {\displaystyle x} and y {\displaystyle y} reach constant values. The equations for each conductor will therefore be

(24) Solving these equations for

{\displaystyle Z}, we find

(25)

(45) Now let the deflection of the galvanometer by the instantaneous current whose intensity is Z {\displaystyle Z} be α {\displaystyle \alpha }.

Let the permanent deflection produced by making the ratio of P S {\displaystyle PS} to Q R {\displaystyle QR}, ρ {\displaystyle \rho } instead of unity, be θ {\displaystyle \theta }.

Also let the time of vibration of the galvanometer needle from rest to rest be T {\displaystyle T}. Then calling the quantity

(26) we find

(27)

In determining τ {\displaystyle \tau } by experiment, it is best to make the alteration of resistance in one of the arms by means of the arrangement described by Mr. Jenkin in the Report of the British Association for 1863, by which any value of ρ {\displaystyle \rho } from 1 to 1.01 can be accurately measured.

We observe ( α {\displaystyle \alpha }) the greatest deflection due to the impulse of induction when the galvanometer is in circuit, when the connections are made, and when the resistances are so adjusted as to give no permanent current.

We then observe ( β {\displaystyle \beta }) the greatest deflection produced by the permanent current when the resistance of one of the arms is increased in the ratio of 1 to ρ {\displaystyle \rho }, the galvanometer not being in circuit till a little while after the connection is made with the battery.

In order to eliminate the effects of resistance of the air, it is best to vary ρ {\displaystyle \rho } till β

2 α {\displaystyle \beta =2\alpha } nearly; then

(28)

If all the arms of the balance except P {\displaystyle P} consist of resistance coils of very fine wire of no great length and doubled before being coiled, the induction coefficients belonging to these coils will be insensible, and τ {\displaystyle \tau } will be reduced to p P {\displaystyle {\tfrac {p}{P}}}. The electric balance therefore affords the means of measuring the self-induction of any circuit whose resistance is known.

(46) It may also be used to determine the coefficient of induction between two circuits, as for instance, that between P and S which we have called m {\displaystyle m}; but it would be more convenient to measure this by directly measuring the current, as in (37), without using the balance. We may also ascertain the equality of p P {\displaystyle {\tfrac {p}{P}}} and q Q {\displaystyle {\tfrac {q}{Q}}} by there being no current of induction, and thus, when we know the value of p {\displaystyle p}, we may determine that of q {\displaystyle q} by a more perfect method than the comparison of deflections.

Exploration of the Electromagnetic Field.

(47) Let us now suppose the primary circuit A {\displaystyle A} to be of invariable form, and let us explore the electromagnetic field by means of the secondary circuit B {\displaystyle B}, which we shall suppose to be variable in form and position.

We may begin by supposing B {\displaystyle B} to consist of a short straight conductor with its extremities sliding on two parallel conducting rails, which are put in connection at some distance from the sliding-piece. Then, if sliding the moveable conductor in a given direction increases the value of M {\displaystyle M}, a negative electromotive force will act in the circuit B {\displaystyle B}, tending to produce a negative current in B {\displaystyle B} during the motion of the sliding-piece.

If a current be kept up in the circuit B {\displaystyle B}, then the sliding-piece will itself tend to move in that direction, which causes M to increase. At every point of the field there will always be a certain direction such that a conductor moved in that direction does not experience any electromotive force in whatever direction its extremities are turned. A conductor carrying a current will experience no mechanical force urging it in that direction or the opposite.

This direction is called the direction of the line of magnetic force through that point.

Motion of a conductor across such a line produces electromotive force in a direction perpendicular to the line and to the direction of motion, and a conductor carrying a current is urged in a direction perpendicular to the line and to the direction of the current.

(48) We may next suppose B {\displaystyle B} to consist of a very small plane circuit capable of being placed in any position and of having its plane turned in any direction. The value of M {\displaystyle M} will be greatest when the plane of the circuit is perpendicular to the line of magnetic force. Hence if a current is maintained in B {\displaystyle B} it will tend to set itself in this position, and will of itself indicate, like a magnet, the direction of the magnetic force.

On Lines of Magnetic Force.

(49) Let any surface be drawn, cutting the lines of magnetic force, and on this surface let any system of lines be drawn at small intervals, so as to lie side by side without cutting each other. Next, let any line be drawn on the surface cutting all these lines, and let a second line be drawn near it, its distance from the first being such that the value of M {\displaystyle M} for each of the small spaces enclosed between these two lines and the lines of the first system is equal to unity.

In this way let more lines be drawn so as to form a second system, so that the value of M {\displaystyle M} for every reticulation formed by the intersection of the two systems of lines is unity.

Finally, from every point of intersection of these reticulations let a line be drawn through the field, always coinciding in direction with the direction of magnetic force.

(50) In this way the whole field will be filled with lines of magnetic force at regular intervals, and the properties of the electromagnetic field will be completely expressed by them.

For, 1st, If any closed curve be drawn in the field, the value of M {\displaystyle M} for that curve will be expressed by the number of lines of force which pass through that closed curve.

2ndly. If this curve be a conducting circuit and be moved through the field, an electromotive force will act in it, represented by the rate of decrease of the number of lines passing through the curve.

3rdly. If a current be maintained in the circuit, the conductor will be acted on by forces tending to move it so as to increase the number of lines passing through it, and the amount of work done by these forces is equal to the current in the circuit multiplied by the number of additional lines.

4thly. If a small plane circuit be placed in the field, and be free to turn, it will place its plane perpendicular to the lines of force. A small magnet will place itself with its axis in the direction of the lines of force.

5thly. If a long uniformly magnetized bar is placed in the field, each pole will be acted on by a force in the direction of the lines of force. The number of lines of force passing through unit of area is equal to the force acting on a unit pole multiplied by a coefficient depending on the magnetic nature of the medium, and called the coefficient of magnetic induction.

In fluids and isotropic solids the value of the coefficient μ {\displaystyle \mu } is the same in whatever direction the lines of force pass through the substance, but in crystallized, strained, and organized solids the value of μ {\displaystyle \mu } may depend on the direction of the lines of force with respect to the axes of crystallization, strain, or growth.

In all bodies μ {\displaystyle \mu } is affected by temperature, and in iron it appears to diminish as the intensity of the magnetization increases.

Magnetic Equipotential Surfaces

(51) If we explore the field with a uniformly magnetized bar, so long that one of its poles is in a very weak part of the magnetic field, then the magnetic forces will perform work on the other pole as it moves about the field.

If we start from a given point, and move this pole from it to any other point, the work performed will be independent of the path of the pole between the two points; provided that no electric current passes between the different paths pursued by the pole.

Hence, when there are no electric currents but only magnets in the field, we may draw as series of surfaces such that the work done in passing from one to another shall be constant whatever be the path pursued between them. Such surfaces are called Equipotential Surfaces, and in ordinary cases are perpendicular to the lines of magnetic force.

If these surfaces are so drawn that, when a unit pole passes from any one to the next in order, unity of work is done, then the work done in any motion of a magnetic pole will be measured by the strength of the pole multiplied by the number of surfaces which it has passed through in the positive direction.

(52) If there are circuits carrying electric currents in the field, then there will still be equipotential surfaces in the parts of the field external to the conductors carrying the currents, but the work done on a unit pole in passing from one to another will depend on the number of times which the path of the pole circulates round any of these currents. Hence the potential in each surface will have a series of values in arithmetical progression, differing by the work done in passing completely round one of the currents in the field.

The equipotential surfaces will not be continuous closed surfaces, but some of them will be limited sheets, terminating in the electric circuit as their common edge or boundary. The number of these will be equal to the amount of work done on a unit pole in going round the current, and this by the ordinary measurement

{\displaystyle =4\pi \gamma }, where γ {\displaystyle \gamma } is the value of the current.

These surfaces, therefore, are connected with the electric current as soap-bubbles are connected with a ring in M. Plateau’s experiments. Every current γ {\displaystyle \gamma } has

{\displaystyle 4\pi \gamma } surfaces attached to it. These surfaces have the current for their common edge, and meet it at equal angles. The form of the surfaces in other parts depends on the presence of other currents and magnets, as well on the shape of the circuit to which they belong.