and further investigation has shown the exact relation that the amount of heat evolved bears to the electrical energy lost in this portion of the circuit.

The nature of his experiments is indicated by the accompanying figure.

A wire of known resistance is immersed in a known mass of water, a current measured by a galvanometer is passed through the wire for a measured time, and the rise in temperature of the water is noted. We thus measure C, R, and the heat evolved. From a comparison of many experiments in which C and R were varied, the law above given was deduced.

Further experiment showed that the same law held with respect to the battery-cell itself. It is then true that in the whole circuit

or in any part of it, the .

(Heat evolved per second) is proportional to (the product C2 R).

If we express our heat, current, and resistance, in suitable units, then this proportionality becomes equality, and we have.

(H per second) = C2 R.

We shall now for convenience introduce a new symbol E having a very simple meaning. Let be the algebraic sum of the E.M.F.s in any portion of the circuit considered; so that, by Ohm's JE; law or by Kirchhoff's extension of it, we have the relation C = R

A

or CR = E. (Thus if we consider the portion of the circuit A B, and if no source of E.M.F. lie between A and B, we have the same as E1 or (V1— V1); but if there be a direct E. M.F. e, or an opposed E. M.F. - e, between A and B, then = E±e). We can then write C instead of C2 R; and we have, when suitable units are employed.

H (per second) = C2 R = C L.

If C, R, and E be measured in ampères, ohms, and volts, then the product C2 R or CE gives us the heat-activity in watts.

If watts and calories per second be compared, by referring both to ergs per second, it can be shown by the data given earlier that . 1 watt = *24 călories per second (approx.).

This relation may be verified directly by experiment.

any circuit

We may then express Joule's law as follows. If in or portion of a circuit C be the current in ampères, R the resistance in ohms, and be the algebraic sum of the E.M.F.s (in such a sense that by Ohm's law CRE) measured in volts, then there will be evolved in that circuit or portion of circuit heat to the amount of CRx24, or CE× 24, calories per second.

We have thus two formulæ for the heat evolved in calories per second; the formula C2 Rx24 is always true; and CX24 is true if be such that C

38

R

§ 5. The Heating of Uniform Wires.-The most important practical application of Joule's law is the application to the case of conducting wires.

We will use formula (i.) of the last section.

(I.) If a wire have resistance R, and there be flowing through it a current C, then there is evolved

C2Rx24 calories per second.

With a given current we can thus calculate the calories per second evolved if we know the length, section, and specific resistance p, of the wire. For, if be given in ohms, or be reduced to ohms, we have that

If be the radius of a wire of circular section (the most usual make of wire) then.

(II.) The temperature to which a wire is raised.-If the heat were not dissipated by convection, conduction, and radiation, no limit (saving that imposed by the fusion of the wire) could be fixed to the temperature to which a wire would rise. As a matter of fact, however, there soon obtains a state of equilibrium between the heat evolved per second and the heat dissipated per second. Hence the temperature will be, with more or less exactness, proportional to C2 and to R; or is higher as the current is larger, and as the wire offers a greater resistance.

Experiment.--Put in the same circuit lengths of Pt and of Ag wire of the same radius. The platinum is the worse conductor of the two; and therefore

It there

for it the resistance is greater, while the current is the same for both. fore is seen to attain a white-heat, while the silver remains much cooler ; in the final condition, the greater dissipation due to the higher temperature compensates for the greater amount of heat evolved per second.

Note. If in the same circuit we have (were it possible to find such) two wires of same radius, same conductivity, same surface power of dissipating heat, but of different masses and specific heats, then it is sometimes stated that the temperatures will depend upon the masses and on the specific heats, varying inversely as both. This, however, is not the case. That wire whose waterequivalent (or mass × specific-heat) is the least, will first attain the final temperature. But, when this is once attained, only the resistance and the dissipating. power come into the question; and, since by hypothesis this is the same for both, it follows that the final temperatures will be the same.

III. Temperature as dependent on radius.-Let us now consider wires of different radii, but of the same material, included in the same circuit, so that the same current necessarily passes through each. And let ✪ represent the number of degrees by which the temperature of the wire exceeds the so-called 'temperature of the air,' when a steady condition of temperature has been arrived at.

We have then, for each unit length of the wire

per]

[Heat Heat produced by current per] = [Heat lost by radiation, &c., per

second

second

If we assume- -[but it is not at all accurate to do so]-that the resistance of the wire does not depend upon its temperature, we may under the conditions assumed say that

(Heat produced per second) ∞

where r is the radius of the wire.

I

2

Again, if we assume-[but at such high temperatures this is not accurate]-that the rate of loss of heat is simply proportional to and to the area of surface exposed, we have

(Heat lost per second) ∞ 2πr. 0.

With these assumptions, then, we have that for wires of same material in the same circuit

§ 6. Distribution of Heat in the Circuit.-Let the current in a circuit be C; the resistances of battery and of various parts of the

circuit be B, R1, R2, R3, &c., respectively; the total resistance be R ; the total heat evolved per second be H calories; the several portions evolved in the above different portions of the circuit be HB, H1, H2, &c., respectively; then we evidently have the following relations holding.

§ 7. Heat Evolved with various Arrangements of n Cells.Let there ben cells, each of resistance B and E.M.F. E. Let R be the total resistance of the whole circuit including the battery. Let H be the total heat per second evolved; and let H; and He be the portions evolved internally to the battery, and in the external circuit, respectively.

(") Let the cells be arranged m end-on and 7 in parallel; and let the external resistance be r. Then we have H= C2 R x 24

calories.

We have also CR = m E by Ohm's law; and R=r+

m B

(see Chapter XIII. § 13, &c.). We can thus substitute for C and R in terms of known quantities; and we have .

m B

(B) If the cells be so arranged that the internal resistance =r, then we have.

§ 8. Case of no Back-E.M.F. in the Circuit. - Where there is in the circuit only the E.M.F. E of the battery, and no other E.M.F., then by Joule's law.

H=C2R = CE x 24 calories-per-second.

or HCE watts.

And by § 3 we know that the total electric activity is

WCE watts.

In this case, therefore, all the electric activity runs down into the form of heat-per-second.

Experiment and theory, moreover, concur in establishing it to be a fact that in a cell where there is no local action, i.e. in which no chemical action occurs until the circuit is closed, all the chemicalpotential-energy lost per second appears as electrical activity. Hence in this case the total chemical action gives first the equivalent electrical energy, and then, finally, the same amount of heat that would have been given had the chemical action taken place without the intermediency of a current.

§ 9. Case of a Back-E.M.F. e in the Circuit-Now let us have in the circuit an electrolytic cell, or some other arrangement giving a reverse E.M.F. e; the total resistance being still R.

As argued in § 3, the battery is expending energy at the rate of WC XE watts.

By Joule's law we have heat evolved at the rate of H = C2 R

watts.

And since by Ohm's law C

be written as C (E − e) or C E – Ce.

Hence of the activity CE expended by the battery, we have accounted for part in heat; but we have not yet accounted for the remainder Ce.