Z. Comparison of a Coefficient of Mutual Induction with the Product of a Resistance and a Time.

A method similar to that last described will enable us to measure electrically a coefficient of mutual induction, or rather, to be more exact, to compare it with the product of a resistance and a time.

For let us recur to the equation (2), of § Y, giving the relation between the coefficient of induction of two coils and the throw of the galvanometer needle. Let k be the reduction factor of the mirror galvanometer, that of the tangent instrument, and let 0 be the deflexion of the tangent galvanometer; then

If, now, we know k and k', the other quantities on the right-hand side can all be observed, and thus M is found in terms of R, and T the time of swing.

It will be observed that the ratio k/k' only is required. This may be determined directly by another electrical experiment. It will be found that in practice, in order to get convenient deflexions, the mirror galvanometer must be many times more sensitive than the tangent; thus k/k' is small.

Shunt the mirror galvanometer; let K be its resistance and s the resistance of the shunt, and connect it in series with a battery, the tangent galvanometer, and a resistance box. Take out of the resistance box plugs to give a convenient deflexion on the tangent galvanometer, and adjust the shunt until a suitable deflexion 0 is obtained in

the sensitive galvanometer. For this purpose it may be necessary to put a large resistance in series with the mirror galvanometer, and to shunt the whole.

The current through the mirror galvanometer is then equal to

This expression will be simplified if, by means of the resistance box, we make the current in the second part of the experiment equal to the primary current in the induction experiment; we have then 0' 0", and thus

M=

R. T. (I + 1⁄2 ^\). s. sin

π

(K+s) tan 0.

(5)

In this experiment a value is found for м by electrical observations. If the experiment be performed with a pair of coils for which м can be calculated by direct measurement, we have a means of verifying the statement that the coefficient of mutual induction is equal to the number of lines of induction, due to unit current in the one coil, which pass through the other.

In comparing the results with calculation, care must be taken as to the units of measurement employed. T is, of course, in seconds; A, T, S/(K+s), sin / tan 0, are all ratios of quantities of the same kind, and are therefore pure numbers; and we have only left R, the resistance. Now R will usually be measured in ohms; but the absolute unit of resistance based on the C.G.S. system is not 1 ohm, but 10-9 of an ohm. If, then, we wish to get м in absolute C.G.S. units, i.e. in centimetres, we must multiply the value of R in ohms by 10".

The experiment described is really one of the original methods by which the absolute resistance of a wire was determined. For suppose we work with a pair of coils of which the coefficient of mutual induction can be found by calculation based on the value of their radii and their relative position, then м is known in centimetres, and we may re-write the equation thus :

The various ratios on the right-hand side can be observed, and the value of R found in terms of the units of length and time.

It follows from the above equation that the dimensions of R are those of a velocity.

It will readily be understood that to obtain high accuracy in a complicated experiment like the above various precautions, which have only been just alluded to, or even passed over in silence, are necessary. It is inserted here chiefly by way of introduction to the system of absolute

measurement.

Experiment.-Determine in terms of a resistance and a time the coefficient of mutual induction of two coils.

Enter the results thus :

R = Resistance of secondary = 5840 ohms.

K = 90000 + 5600 ohms.

S = 5.62 ohms.

B (in scale divisions) 95 (mean of 4 observations).

=

Distance from mirror to scale, in scale divisions = 1000.

г. Comparison of a Coefficient of Mutual Induction with the Capacity of a Condenser and the Product of Two Resistances.

The method of making this measurement has been already indicated in § Y (1), fig. xlvii. In order to carry it out connect up the apparatus as in fig. 1, placing a key, K2, in the secondary circuit. . Connect one set of plates of the

condenser to one end of the resistance box x, the other set being connected through a key, K3, to the mirror galvanometer M G. Connect the galvanometer to the other end of the resistance box x. Thus, when K, is made and K3 broken, the galvanometer is in the secondary circuit; when K, is made and K, broken, it is in the condenser chcuit. The tangent galvanometer in the primary circuit is not needed for the present experiment. To perform the experiment, make K, and break K3. On making contact at K, we get

an induction throw in the galvanometer, and if 0, be the throw, we have—

Now break the battery circuit; then break K, and make K3. On again making the battery the condenser is charged through the galvanometer. The potential difference between the plates of the condenser will be that between the ends of the box X, or xi. Thus the quantity of electricity which flows through the galvanometer is cxi, c being the capacity of the condenser; and if 0, is the throw,

very approximately, if d1, d2 be the two observed deflexions. Thus

We have seen that in experiments with a condenser a high-resistance galvanometer is required. The quantity of electricity required to charge the condenser is small, and the same quantity will pass round the galvanometer, whatever be its resistance. By having a large number of turns this quantity is made to circulate round the needle a large number of times, and hence to produce a measureable effect. For measuring the induction current the galvanometer resistance need not be very large. In the above experiment we must use a galvanometer which will make the quantities, and d, not very different.

Suppose that the condenser has a capacity of about 1 microfarad; a pair of coils such as those described in § Y may have a coefficient of mutual induction comparable with 108 centimetres, and I microfarad is equal to 10-15 C.G.S. units of capacity.