683.] If the current is a function of r, the distance from the axis of 2, and if we write

x = r cos 0, and y = r sin 0,

(4)

and B for the magnetic force, in the direction in which 0 is measured perpendicular to the plane through the axis of z, we have

If C is the whole current flowing through a section bounded by a circle in the plane ay, whose centre is the origin and whose radius is r,

= [ ' 2 = r w dr =

C=

0

rw Br.

(6)

It appears, therefore, that the magnetic force at a given point. due to a current arranged in cylindrical strata, whose common axis is the axis of z, depends only on the total strength of the current flowing through the strata which lie between the given point and the axis, and not on the distribution of the current among the different cylindrical strata.

For instance, let the conductor be a uniform wire of radius a, and let the total current through it be C, then, if the current is uniformly distributed through all parts of the section, w will be constant, and

(7) The current flowing through a circular section of radius r, being less than a, is C'wr. Hence at any point within the wire,

=

In the substance of the wire there is no magnetic potential, for within a conductor carrying an electric current the magnetic force does not fulfil the condition of having a potential.

Outside the wire the magnetic potential is

Let us suppose that instead of a wire the conductor is a metal tube whose external and internal radii are a, and a2, then, if C is the current through the tubular conductor,

The magnetic force within the tube is zero. In the metal of the tube, where is between a, and a,

1

B = 20

a2

(12)

and outside the tube,

C

β

= 2

(13)

the same as when the current flows through a solid wire.

684.] The magnetic induction at any point is bμ ẞ, and since,

Мо

A-2 C log r,

(14)

(15)

(16)

where is the value of μ in the space outside the tube, and A is a constant, the value of which depends on the position of the return current.

In the substance of the tube,

In the space within the tube II is constant, and

H = A-2 μ。 C log a1+

но C

(a ̧2 — r2+2a ̧2 log —). (17)

685.] Let the circuit be completed by a return current, flowing in a tube or wire parallel to the first, the axes of the two currents being at a distance b. To determine the kinetic energy of the system we have to calculate the integral

If we confine our attention to that part of the system which lies between two planes perpendicular to the axes of the conductors, and distant from each other, the expression becomes

T=

41 ff Hw dx dy.

(20)

If we distinguish by an accent the quantities belonging to the return current, we may write this

T

27 = [[IIw' da2 dy + [[ I'w dr dy + ff Hw dx dy+ [[ I'w'da' dy'. (21)

Since the action of the current on any point outside the tube is the same as if the same current had been concentrated at the axis of the tube, the mean value of H for the section of the return current is 4-2 μ。 C log b, and the mean value of H' for the section of the positive current is 4-2 Ho C′ log b.

687.]

LONGITUDINAL TENSION.

289

Hence, in the expression for T, the first two terms may be written. AC-2 μo CC' log b, and A'C-2 μo CC' log b.

Мо

=

μο

Integrating the two latter terms in the ordinary way, and adding the results, remembering that C+C' 0, we obtain the value of the kinetic energy 7. Writing this LC2, where I is the coefficient of self-induction of the system of two conductors, we find as the value of L for unit of length of the system

It is only in the case of iron wires that we need take account of the magnetic induction in calculating their self-induction. In other cases we may make Po, p, and μ all equal to unity. The smaller the radii of the wires, and the greater the distance between them, the greater is the self-induction.

To find the Repulsion, X, between the Two Portions of Wire.

686.] By Art. 580 we obtain for the force tending to increase b,

which agrees with Ampère's formula, when po= 1, as in air. 687.] If the length of the wires is great compared with the distance between them, we may use the coefficient of self-induction. to determine the tension of the wires arising from the action of the current.

In one of Ampère's experiments the parallel conductors consist of two troughs of mercury connected with each other by a floating bridge of wire. When a current is made to enter at the extremity of one of the troughs, to flow along it till it reaches one extremity

of the floating wire, to pass into the other trough through the floating bridge, and so to return along the second trough, the floating bridge moves along the troughs so as to lengthen the part of the mercury traversed by the current.

Professor Tait has simplified the electrical conditions of this experiment by substituting for the wire a floating siphon of glass filled with mercury, so that the current flows in mercury throughout its course.

Fig. 40.

This experiment is sometimes adduced to prove that two elements of a current in the same straight line repel one another, and thus to shew that Ampère's formula, which indicates such a repulsion of collinear elements, is more correct than that of Grassmann, which gives no action between two elements in the same straight line ; Art. 526.

But it is manifest that since the formulae both of Ampère and of Grassmann give the same results for closed circuits, and since we have in the experiment only a closed circuit, no result of the experiment can favour one more than the other of these theories.

In fact, both formulae lead to the very same value of the repulsion as that already given, in which it appears that 6, the distance between the parallel conductors is an important element.

When the length of the conductors is not very great compared with their distance apart, the form of the value of L becomes somewhat more complicated.

688.] As the distance between the conductors is diminished, the value of Z diminishes. The limit to this diminution is when the wires are in contact, or when b = a1+a2. In this case

L = 2 l log ((1+a,)2 + }).

(26)

This is the smallest value of the self-induction of a round wire doubled on itself, the whole length of the wire being 27.

Since the two parts of the wire must be insulated from each other, the self-induction can never actually reach this limiting value. By using broad flat strips of metal instead of round wires the self-induction may be diminished indefinitely.

On the Electromotive Force required to produce a Current of Varying Intensity along a Cylindrical Conductor.

689.] When the current in a wire is of varying intensity, the electromotive force arising from the induction of the current on itself is different in different parts of the section of the wire, being in general a function of the distance from the axis of the wire as well as of the time. If we suppose the cylindrical conductor to consist of a bundle of wires all forming part of the same circuit, so that the current is compelled to be of uniform strength in every part of the section of the bundle, the method of calculation which we have hitherto used would be strictly applicable. If, however, we consider the cylindrical conductor as a solid mass in which electric currents are free to flow in obedience to electromotive force, the intensity of the current will not be the same at different distances from the axis of the cylinder, and the electromotive forces themselves will depend on the distribution of the current in the different cylindric strata of the wire.

The vector-potential H, the density of the current w, and the electromotive force at any point, must be considered as functions of the time and of the distance from the axis of the wire.

The total current, C, through the section of the wire, and the total electromotive force, E, acting round the circuit, are to be regarded as the variables, the relation between which we have to find.

Let us assume as the value of H,

H = S+T2+T1 r2 + &c. + T, r2",

where S, To, T, &c. are functions of the time.

Then, from the equation

(1)