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increase. In general we might expect a temperature to be soon reached at which the latter action is greater than the former.
I should expect the curves which give the relation between temperature and resistance, in the case of a steady current, to be somewhat (to express myself loosely) asymptotic, the asymptote being the line which gives the relation between the temperature and resistance when the bismuth is in zero field.
In addition, taking my theory--which may be applied to currents of the frequencies under consideration or to those of 10,000 per second as imagined, perhaps correctly, by Lenard— as a working hypothesis, I have formed the expectation that the difference between the resistances as measured by telephone and galvanometer would diminish with increase of temperature; this is perhaps worthy of investigation.
There is another action which may exist,-perhaps also worthy of research; I have not had the advantage of studying the original paper, but Geronza and Finzi * find that an alternating current influences the susceptibility of iron, nickel, and steel; possibly the susceptibility of bismuth may be influenced in like manner, and indirectly affect the resistance.
I hope at some future time to carry on some experiments with alternating currents perfectly harmonic in character, a telephone responding only to the frequency under consideration, and to endeavour to determine what is the lowest frequency at which a change in resistance is perceptible; also to examine whether the shape of the section of the bismuth wire is of any importance.
The paper contains a few new experiments dealing with the action of alternating currents when sent through the coils of a galvanometer.
A convenient and satisfactory arrangement is described for diminishing the spark on breaking a galvanic circuit.
A description is given of experiments, made with the bismuth spiral, all of which (and others of which no description is given) gave negative results.
A theory is elaborated which explains, to a limited extent, the fact that in a strong field the resistance of bismuth is greater when measured with an alternating than with a constant current.
* Beiblätter, vol. xviii. No. 3, p. 375. Phil. Mag. S. 5. Vol. 39. No. 238. March 1895.
There are some surmises with regard to the action of a rise of temperature.
Addendum I. contains a thorough discussion of the behaviour of the method of differential winding for diminishing the spark on breaking a galvanic circuit.
Addendum II. gives a theoretical treatment of the working of the interrupter which may be useful when it is necessary to design an interrupter for high frequencies.
Addendum III. describes the method adopted to compare self-inductions.
In conclusion I have to express my thanks to Prof. A. Schuster for initiating my research and for much friendly criticism throughout.
In the method of differential winding two wires of equal length and diameter are wound into coils side by side, and their ends so connected that the wires are in parallel arc and that the equal currents which circulate the coils, when a steady current is flowing, go in opposite directions. When a steady current goes through the arrangement no lines of force are produced within the combination coil; one naturally associates the production of a spark at a break in a coil with lines of force, and there is a danger of erroneously assuming that no spark due to induction will be formed by causing a break in one of the coils. What is, of course, necessary, is that the rate of change of the number of lines of force enclosed within the coil should equal zero.
Let us imagine that one coil can be suddenly broken at any point, and that the current in the other coil can flow on undisturbed if this happened the lines of force enclosed within the combination coil would instantaneously vary from zero to a finite number, i. e., the rate of change of the number of lines of force would equal infinity and the E.M.F. tending to produce a spark across the gap would also equal infinity.
What actually happens is, probably, somewhat as follows: since the self-induction of each coil is equal to the mutual induction of the two, on breaking one coil the current in the other is suddenly stopped, it then begins immediately to increase, and whilst increasing causes, by induction, a spark across the gap in the broken coil.
The spark produced by the break is therefore analogous to the "make" spark of an ordinary induction-coil.
The stoppage of both currents by the break of only one
coil is capable of explanation by means of a mechanical analogy.
In the figure, A represents a disk capable of rotation about a vertical axis; M a heavy body which is capable of rotation about a vertical axis XY; C a disk capable of rotation about an axis (variable in position) joining M to XY; B a disk similar to A. The disks A, B, and C are supposed to be of inappreciable mass. If A and B (corresponding to currents) rotate with equal velocities in opposite directions, the disk C will rotate, but M (corresponding to induction) will remain stationary. If, now,
B be suddenly stopped, as M, owing to its inertia, must instantaneously remain at rest, C will be instantaneously stopped also, and likewise A. If the motive power of A be kept continuously applied, it will immediately begin to rotate, and gradually increase in velocity from zero to its maximum speed.
The cause of the production of the spark is also capable of mathematical treatment.
For simplicity, let it be assumed that the E.M.F. between the ends of each coil is kept constant and equal to E.
Let i, be the current in coil (1).
L coefficient of self-induction of each coil.
Since LM, equations (1) and (2) may be written
The problem is to find how i, varies if a break is made in coil (1), i. e. if r1 varies from a finite to a great or infinite
Adding corresponding sides of (3) and (4),
Substituting these values in (6),
0=L(2E—¿‚r',) d'! +r' ̧2rgi,+Ldig —r2E + Lrır, dig
The last equation would probably, under any assumption, produce an intractable differential equation. However, the difficulty can be avoided by not troubling about r1 for the present, and assuming that i, varies from a finite value to zero; later it will be proved that the assumption is consistent with the conditions of the problem, e. g. that the expression deduced for r, is quite a legitimate one.
When t=0, iio, substituting and manipulating,
It will now be necessary to prove that r is a possible function of t. It is needless to give the details, but it can
readily be shown that
Let k>, then an examination of (9) shows that r1 is
initially equal to and continually increases with t.
We may now, with safety, study the nature of the solution
the value of this when x is very great is approximately
ior2 -kio+ L
This result shows that if x is very great, i, initially diminishes very rapidly.
It is useless to give the details, but it can be shown that the minimum value of i2 is
If x=∞, the above becomes an indeterminate expression the limiting value of which is zero, showing that if the coil (1) is completely and rapidly broken, the current in coil (2) is, under the given conditions, instantaneously diminished to