by allowing them to act on such a massive and slowly-moving pendulum that there is no appreciable change of position until the impulses have ceased to act. The impulses thus act on the pendulum in one known position (viz. when it is hanging vertically); and the sum total of these impulses can be deduced from the extent to which the pendulum swings, its mass and dimensions being known.

In the ballistic galvanometer we have a needle of such a mass that it does not move appreciably from zero until the discharge has entirely passed. Thus each portion of the discharge acts on the needle with couples whose arms are all the length of the needle. The extent of swing of the needle then depends solely upon the sum of (each current-strength) × (the_infinitesimal time during which it remains constant). In spite of the fact that the current changes continuously, thus making this 'sum' a matter for the 'infinitesimal calculus,' even a beginner can see that the sum of all the products above given measures the total quantity of electricity that has passed. If Q be this quantity, and be the maximum angle of deflexion of the needle, it can be shown that

where k is a constant depending upon the construction of the instrument and upon the strength H of the earth's horizontal field. This constant can be found or

calculated.'

Note. The student can find the proof of the above formula in more advanced books.

In

M

طح

Use of ballistic galvanometer. -We may use the foregoing instrument to measure, or compare, the capacities of condensers. the accompanying figure xy a b is a condenser, The plates ab are to earth, while the opposite plates xy can be connected, by means of a key M, either with the pole P of a battery, or with the one terminal of a ballistic galvanometer G. The other pole C of the battery, and the other terminal of the galvanometer, are to earth.

E

E

1

2

When the plates

xy are connected with the pole P of the battery, the condenser is charged so that ry are at a potential V, while ab are at zero potential. (It is clear that V measures the unpolarised E. M.F. of the battery, for the other pole C is to earth, and there is no permanent current.)

Then if K be the capacity of the condenser, the quantity Q of the charge will be

where Q, is in coulombs, if V is in volts and K, is in farads (see Chapter XVIII. § 4).

Now let the condenser be discharged through the ballistic galvano

meter.

The total impulse given to the needle is, we have seen, directly proportional to Q; and this impulse is also, we have stated, proportional to sin 1 for mechanical reasons that we have not given; 1

being, as above, the 'angle of throw' of the needle.

k

V

So, for another condenser, we have K2 = . sin

; whence

We can thus compare any condensers with standards, or with each other.

Note.-Let t be the time in seconds of a complete to-and-fro oscillation of the needle when no current passes. Let @ measure the angle of throw in some unit of angle such as degrees. Let r, measure in ohms that resistance through which the E. M. F. measured by V (see above) would drive a steady current, giving a permanent deflexion of one unit of angle—the same unit as is used in measuring 0. And let K be the capacity of the condenser in farads. Then it can be shown that

This formula is given in Fleeming Jenkin's Electricity.' The more massive the needle, and the less friction there is in its movement, the truer is this formula.

·

§ 13. Sir W. Thomson's Graded' Potential Galvanometer -To meet the present requirements of exact electrical measurement, Sir W. Thomson has invented two galvanometers of convenient form and of wide range of sensibility, the one designed especially for measurements of E.M.F.s or of potential differences, the other for measurements of currents. In both there are arrange. ments for altering to a known extent and through a wide range the sensibility of the instrument; and both are so marked by the makers that, with simple corrections for the strength of the earth's horizontal field H at each place, results can be read off in volts and ampères respectively. For these reasons the instruments are called 'graded galvanometers. A very full account of them is given in Gray's Absolute Measurements in Electricity and Magnetism.' They have not, however, for reasons alluded to in § 17, satisfied practical requirements. Sir W. Thomson has since invented other instruments, alluded to at the end of Chapter XXV.

The volt-meter galvanometer.-The general principle of a voltmeter galvanometer has been already explained in Chapter XIV. § 11. In the figure C is the coil. This has a great number of turns of wire, and a large resistance, generally exceeding 5,000 ohms.

M is the magnetometer, or system of needles. This consists of a system of short needles, parallel to one another, disposed symmetri cally about a common pivot. We can consider this system to be equivalent to one short needle, and shall often speak of it as 'the needle.' To this is rigidly fixed a long and light aluminium index, standing at right angles to the direction of the needle. It is this index that is seen in the figure. It moves over a scale properly graduated.

Over the magnetometer and resting on the magnetometer box, but removable at pleasure, is seen a semicircular magnet. By the use of

U

this magnet the ‘restoring-field' (which would otherwise be only the earth s horizontal field H) may be much increased in strength. Further, when the magnet is used, any error in the measurement of H at any particular place will be a smaller percentage of the whole field than it would be if H alone were the whole field, and so will cause a smaller error in the measurements made with the instrument, providing that the field due to the magnet is known. (This latter has, in practice, to be re-determined from time to time.) But it must be remembered that according to the principle of § 9, any strengthening of the restoringfield diminishes the sensibility of the instrument.

The instrument is so constructed that the axis of the coil C, along which runs the central line of force (see (III.) of § 2), passes through the point of suspension of the needle, and thus the current tends to direct the needle along the line of the axis of the coil.

On the other hand, the magnet is so placed that its magnetic axis passes through the point of suspension of the needle, in a direction perpendicular to the last.

When used, the instrument is levelled, and is so placed that the planes of the coil and of the magnet are in the magnetic meridian. When this is the case, the index should stand at zero when no current passes, and equal currents in opposite directions should give equal deflexions from zero.

In this position we have as a restoring-field the earth's horizontal field H, to which may be added at pleasure the stronger horizontal field due to the magnet, coinciding in direction with H. The deflecting field due to the coil is horizontal, and is perpendicular to the last field. Thus the theory of the instrument, since its needle is relatively short and moves in a uniform field, is that of the tangent galvanometer.

The sensitiveness of the instrument can be varied, both abruptly, by means of the magnet, which may be used or removed, and also, step by step, by sliding the magnetometer box nearer to, or further from, the coil. The feet of the box slide in a groove in such a way that the whole moves parallel to itself, while the point of suspension of the needle moves along the axis of the coil. Instructions as to the conversion of the readings into volts under all conditions of sensitiveness are sent with each instrument.

The terminals are of an ingenious construction, designed partly to obviate accidents in measuring very large ▲V.s, and the 'leads' are so arranged that the current passing through them has no action on the needle.

§ 14. Sir W. Thomson's Graded Current Galvanometer.-In the construction of the last instrument it was desired to measure

the AV between two points without appreciably diminishing this V by lessening the resistance between them. Hence we saw that the coil was made of a very high resistance.

In the present case it is desired to measure currents without appreciably diminishing them by the introduction of resistance. The coil, therefore, must be of as low a resistance as is consistent with the requisite sensitiveness.

The current galvanometer, or am-meter, is in general construction almost identical with the last. But the coil is composed of a few turns

1000

of thick copper wire, or copper strip, of a resistance equal to about 10 ohm. There is also a special arrangement of the terminals, to obviate inconveniences or accidents in dealing with large currents, and the 'leads' are so arranged that the current passing through them has no action on the needle.

$ 15. Weber's Electro-Dynamometer.

It is sometimes desired to measure currents by some instrument that does not depend upon the magnitude of the earth's field of force at any particular place-in fact, to construct an instrument that can be used without reference to the variable quantity 'H,' that has appeared in the formulæ of the preceding instruments.

Now we shall see in later Chapters that a coil carrying a current acts as a magnet of the same shape whose poles answer to the two faces of the coil. It is found that the magnetic moment of such a coil depends solely upon its shape and the strength of the current flowing through it; so that, as long as these remain constant, we have an absolutely invariable magnetic needle.

Let the large coil of § 5 (or one with more turns of wire, if suitable for the purpose) stand with its plane vertical and coinciding with the