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The probable relation between ig and t when k is great, but not infinite, is indicated in the figure below.
ADDENDUM II. The apparatus used as an interrupter is diagrammatically sketched below.
A C represents the vibrating wire, B the battery, D and E electromagnets, F a vessel containing mercury, I the inductioncoil. Some cells, consisting of lead plates in dilute sulphuric acid, were used as a shunt across the spark-gap; they are not shown in the figure, nor are ihey taken account of in the theoretical considerations which are given below.
It was found experimentally, as one would expect from telephony, that initial magnetization by means of subsidiary coils increased the amplitude of vibration, at any rate for low freqnencies.
It was also found that a piece of iron placed below one of the electromagnets increased the amplitude ; this is a corresponding device to that used in Auer's telephone. The probable explanation is that the iron produces a strong and divergent field in the neighbourhood of the iron wire,
If there is a strong initial magnetization, we may assume that the efficient pull on the wire will vary as the current through the coils. Let F=efficient pull,
i =current, then Foci.
Some rough calculations were made, and the conclusion was arrived at that the self-induction of the electromagnets was of the dimensions of 109 in C.G.S. units.
The resistance of the primary circuit was about 3 x 109 C.G.S. units. Let T=time of vibration,
T let is=maximum current in the primary circuit, i. e. the
current just before the break. Let it be assumed that the platinum point is in contact with
T the mercury for the time
where E equals the E.M.F. and R the resist
R' ance of the primary circuit.
The equation which represents the rise in the current after contact is made is
(1) When t is small,
1 Calculations showed that if T=2, i;=io *.94667; T 1
1 i;=i, X-2541 ; T=50; i;=i,* •03; T=
1 T= 1000
i;=i* .0015; and that if n>50, formula (2) may, with very little error, be taken as correct. It can be shown that the average current in the primary circuit varies inversely as the frequency.
The energy given to the vibrating wire in one complete oscillation will now be considered.
Since the current is stronger whilst the wire is rising than when descending, the work done on the string by magnetic forces when rising must be greater than the work done by the string against magnetic forces when descending; the spark will also prolong the pull upwards,
In the following the effect of the spark will be entirely omitted.
As a matter of fact, the oscillation of the wire cannot be perfectly harmonic; nevertheless, let us assume that the motion of the wire is represented by the equation
x=d sin pt, where d= maximum distance traversed from the middle point,
T' x= distance of platinum point below its middle position. The work done by the electromagnets on the wire can be put in the form [Fdx, taken between the proper limits where
iR F varies as i, i. e. F=kt, where k varies as
L' It is needless to go into details, but the result is that the resultant work done on the string in one oscillation
I which shows that the energy given to the wire in one oscillation varies inversely as the frequency if the amplitude is unaltered.
When the vibrating wire has settled down, the energy given to the wire in each oscillation by magnetic causes must equal the energy lost through other causes.
I. Air resistance, which may be taken to vary as the square of the velocity of the string.
II. Resistance due to induced currents or damping, which may be taken to vary as the velocity.
III. Other frictional losses, which will be neglected.
By equating the energy given to the energy lost, and dealing only with the losses under (I.), I obtain the result that
E varies as no; so that, as 4 cells were used to obtain a frequency of 500, to obtain a frequency of 1000 of the same amplitude 32 cells would be required.
If the chief losses come under (II.), then E varies as në.
If n increases, the amplitude diminishes, the relation being that d varies as n-.
The wire ceases to vibrate if io is not sufficiently great ; it is hard to say what decides the cessation of vibration ; if the platinum point acted perfectly, i. e, if the least rise broke the circuit completely, and the least depression made the circuit, there would, I think, be no limit to the frequency obtainable,
If we assume that it is necessary for the platinum point to move a certain niinimum distance before a break is made in the circuit, then, for the wire to continue vibrating,
E must vary as ns. Theory indicates that the frequency obtainable by the apparatus discussed could be raised by increasing E and at the same time adding resistance free from induction.
The preceding discussion applies to the apparatus under consideration ; but the subject can be treated more generally.
Let it still be assumed that F varies as i; so that we may put F=Ki.
Let r=a sin pt, where x=distance of platinum point below its middle position,
a=maximum deflexion of vibrating wire.
The work done by the wire against magnetic forces whilst going downwards equals sfdx or sKidx, taken between the proper limits; the work done on the wire whilst ascending also equals (Kidx, taken, of course, between other limits.
I omit the details, which are tedious, and give the final result, which is that if W equals the resultant work done on the wire in one vibration, then
R? +L'p? If T is small, it can be shown that this expression agrees with the previously obtained result.
In order to obtain the greatest amplitude for a particular frequency, W must be a maximum ; its maximum value cannot easily be found—at any rate, by the ordinary treatment; without finding the particular value, however, it can be indicated that there is a certain value of L which produces the most efficient result.
If L=0, W equals 0; if L=, W equals zero ; so that there is some value of L between 0 and co which it is best to employ.
The actual problem, which I have not gone into, is more complicated still; for both K and L depend on the winding: for precise treatment, a certain type of winding might be adopted, and K and L made to depend on some variable depending on the character of the winding ; perhaps experiments would give results more readily than mathematics.
ADDENDUM III. Each resistance of variable self-induction was made of two bobbins, one of which rotated within the other, after the manner of those described by Lord Rayleigh in Phil. Mag. 1886, xxii. p. 473.
To compare and graduate the self-inductions I used a method which was very convenient and sensitive, a modification of that described in Clerk-Maxwell, vol. ii. art. 757.
The adjoining figure represents a Wheatstone's bridge; the letters have
B their usual significance. Let self-induction of AD=L, of CD=L'.
P, Q, R, S were arranged so that with a steady current and galvanometer no current went through the galvanometer.
The galvanometer was then replaced 1} by a telephone, the steady by an alternating current, and the rotating bobbin of one of the resistances was rotated until no noise was heard in the telephone. In this case
L R P
XXIV. On Electromagnetic Stress. By E. TAYLOR JONES,
B.Sc., Science Scholar of Royal Commission for Exhibition of 1851, nominated by the University College of N. Wales *. THE problem of finding experimentally the true relation
between electromagnetic stress, or "lifting-power” per unit area of magnets, and magnetization has been attacked by many experimenters during the last sixty years; but no one
Communicated by Lord Playfair, F.R.S,