A Treatise on Electricity and Magnetism/Part II/Chapter III

by James Clerk Maxwell

CHAPTER III.

ELECTROMOTIVE FORCE BETWEEN BODIES IN CONTACT.

The Potentials of Different Substances in Contact.

246.] If we define the potential of a hollow conducting vessel as the potential of the air inside the vessel, we may ascertain this potential by means of an electrometer as described in Part I, Art, 222.

If we now take two hollow vessels of different metals, say copper and zinc, and put them in metallic contact with each other, and then test the potential of the air inside each vessel, the potential of the air inside the zinc vessel will be positive as compared with that inside the copper vessel. The difference of potentials depends on the nature of the surface of the insides of the vessels, being greatest when the zinc is bright and when the copper is coated with oxide.

It appears from this that when two different metals are in contact there is in general an electromotive force acting from the one to the other, so as to make the potential of the one exceed that of the other by a certain quantity. This is Volta's theory of Contact Electricity.

If we take a certain metal, say copper, as the standard, then if the potential of iron in contact with copper at the zero potential is $I$, and that of zinc in contact with copper at zero is $Z$, then the potential of zinc in contact with iron at zero will be $Z-I$.

It appears from this result, which is true of any three metals, that the differences of potential of any two metals at the same temperature in contact is equal to the difference of their potentials when in contact with a third metal, so that if a circuit be formed of any number of metals at the same temperature there will be electrical equilibrium as soon as they have acquired their proper potentials, and there will be no current kept up in the circuit.

247.] If, however, the circuit consist of two metals and an electrolyte, the electrolyte, according to Volta's theory, tends to reduce the potentials of the metals in contact with it to equality, so that the electromotive force at the metallic junction is no longer balanced, and a continuous current is kept up. The energy of this current is supplied by the chemical action which takes place between the electrolyte and the metals.

248.] The electric effect may. however, be produced without chemical action if by any other means we can produce an equalization of the potentials of two metals in contact. Thus, in an experiment due to Sir W. Thomson^[1], a copper funnel is placed in contact with a vertical zinc cylinder, so that when copper filings are allowed to pass through the funnel, they separate from each other and from the funnel near the middle of the zinc cylinder, and then fall into an insulated receiver placed below. The receiver is then found to be charged negatively, and the charge increases as the filings continue to pour into it. At the same time the zinc cylinder with the copper funnel in it becomes charged more and more positively.

If now the zinc cylinder were connected with the receiver by a wire, there would be a positive current in the wire from the cylinder to the receiver. The stream of copper filings, each filing charged negatively by induction, constitutes a negative current from the funnel to the receiver, or, in other words, a positive current from the receiver to the copper funnel. The positive current, therefore, passes through the air (by the filings) from zinc to copper, and through the metallic junction from copper to zinc, just as in the ordinary voltaic arrangement, but in this case the force which keeps up the current is not chemical action but gravity, which causes the filings to fall, in spite of the electrical attraction between the positively charged funnel and the negatively charged filings.

249.] A remarkable confirmation of the theory of contact electricity is supplied by the discovery of Peltier, that, when a current of electricity crosses the junction of two metals, the junction is heated when the current is in one direction, and cooled when it is in the other direction. It must be remembered that a current in its passage through a metal always produces heat, because it meets with resistance, so that the cooling effect on the whole conductor must always be less than the heating effect. We must therefore distinguish between the generation of heat in each metal, due to ordinary resistance, and the generation or absorption of heat at the junction of two metals. We shall call the first the frictional generation of heat by the current, and, as we have seen, it is proportional to the square of the current, and is the same whether the current be in the positive or the negative direction. The second we may call the Peltier effect, which changes its sign with that of the current.

The total heat generated in a portion of a compound conductor consisting of two metals may be expressed by

$H={\frac {R}{J}}C^{2}t-\Pi Ct,$

where $H$ is the quantity of heat, $J$ the mechanical equivalent of unit of heat, $R$ the resistance of the conductor, $C$ the current, and $t$ the time; $\Pi$ being the coefficient of the Peltier effect, that is, the heat absorbed at the junction due to the passage of unit of current for unit of time.

Now the heat generated is mechanically equivalent to the work done against electrical forces in the conductor, that is, it is equal to the product of the current into the electromotive force producing it. Hence, if $E$ is the external electromotive force which causes the current to flow through the conductor,

$JH=CEt=RC^{2}t-J\Pi Ct,$

whence

$E=RC-J\Pi .$

It appears from this equation that the external electromotive force required to drive the current through the compound conductor is less than that due to its resistance alone by the electromotive force $J\Pi$. Hence $J\Pi$ represents the electromotive contact force at the junction acting in the positive direction.

This application, due to Sir W. Thomson^[2], of the dynamical theory of heat to the determination of a local electromotive force is of great scientific importance, since the ordinary method of connecting two points of the compound conductor with the electrodes of a galvanometer or electroscope by wires would be useless, owing to the contact forces at the junctions of the wires with the materials of the compound conductor. In the thermal method, on the other hand, we know that the only source of energy is the current of electricity, and that no work is done by the current in a certain portion of the circuit except in heating that portion of the conductor. If, therefore, we can measure the amount of the current and the amount of heat produced or absorbed, we can determine the electromotive force required to urge the current through that portion of the conductor, and this measurement is entirely independent of the effect of contact forces in other parts of the circuit.

The electromotive force at the junction of two metals, as determined by this method, does not account for Volta's electromotive force as described in Art. 246. The latter is in general far greater than that of this Article, and is sometimes of opposite sign. Hence the assumption that the potential of a metal is to be measured by that of the air in contact with it must be erroneous, and the greater part of Volta's electromotive force must be sought for, not at the junction of the two metals, but at one or both of the surfaces which separate the metals from the air or other medium which forms the third element of the circuit.

250.] The discovery by Seebeck of thermoelectric currents in circuits of different metals with their junctions at different temperatures, shews that these contact forces do not always balance each other in a complete circuit. It is manifest, however, that in a complete circuit of different metals at uniform temperature the contact forces must balance each other. For if this were not the case there would be a current formed in the circuit, and this current might be employed to work a machine or to generate heat in the circuit, that is, to do work, while at the same time there is no expenditure of energy, as the circuit is all at the same temperature, and no chemical or other change takes place. Hence, if the Peltier effect at the junction of two metals $a$ and $b$ be represented by $\Pi _{ab}$ when the current flows from $a$ to $b$, then for a circuit of two metals at the same temperature we must have

$\Pi _{ab}+\Pi _{ba}=0,$

and for a circuit of three metals $a,b,c,$ we must have

$\Pi _{bc}+\Pi _{ca}+\Pi _{ab}=0.$

It follows from this equation that the three Peltier effects are not independent, but that one of them can be deduced from the other two. For instance, if we suppose $c$ to be a standard metal, and if we write $P_{a}=J\Pi _{ac}$ and $P_{b}=J\Pi _{bc}$, then

$J\Pi _{ab}=P_{a}-P_{b}.$

The quantity $P_{a}$ is a function of the temperature, and depends on the nature of the metal $a$.

251.] It has also been shewn by Magnus that if a circuit is formed of a single metal no current will be formed in it, however the section of the conductor and the temperature may vary in different parts.

Since in this case there is conduction of heat and consequent dissipation of energy, we cannot, as in the former case, consider this result as self-evident. The electromotive force, for instance, between two portions of a circuit might have depended on whether the current was passing from a thick portion of the conductor to a thin one, or the reverse, as well as on its passing rapidly or slowly from a hot portion to a cold one, or the reverse, and this would have made a current possible in an unequally heated circuit of one metal.

Hence, by the same reasoning as in the case of Peltier's phenomenon, we find that if the passage of a current through a conductor of one metal produces any thermal effect which is reversed when the current is reversed, this can only take place when the current flows from places of high to places of low temperature, or the math>reverse, and if the heat generated in a conductor of one metal in flowing from a place where the temperature is $x$ to a place where it is $y,$ is $H,$ then

$JH=RC^{2}t-S_{xy}Ct,$

and the electromotive force tending to maintain the current will be $S_{xy}$.

If $x,y,z$ be the temperatures at three points of a homogeneous circuit, we must have

$S_{yz}+S_{zx}+S_{xy}=0,$

according to the result of Magnus. Hence, if we suppose $z$ to be the zero temperature, and if we put

$Q_{x}=S_{xz},\quad \quad$ and $\quad \quad Q_{y}=S_{yz},$

we find

$S_{xy}=Q_{x}-Q_{y},$

where $Q_{x}$ is a function of the temperature $x$, the form of the function depending on the nature of the metal.

If we now consider a circuit of two metals $a$ and $b$ in which the temperature is $x$ where the current passes from $a$ to $b$, and $y$ where it passes from $b$ to $a$, the electromotive force will be

$F=P_{ax}-P_{bx}+Q_{bx}-Q_{by}+P_{by}-P_{ay}+Q_{ay}-Q{az},$

where $P_{ax}$ signifies the value of $P$ for the metal $a$ at the temperature $x$, or

$F=P_{ax}-Q_{ax}-(P_{ay}-Q_{ay})-(P_{bx}-Q_{bx})+P_{by}-Q_{by}.$

Since in unequally heated circuits of different metals there are in general thermoelectric currents, it follows that $P$ and $Q$ are in genera] different for the same metal and same temperature.

252.] The existence of the quantity $Q$ was first demonstrated by Sir W. Thomson, in the memoir we have referred to, as a deduction from the phenomenon of thermoelectric inversion discovered by Cummming^[3], who found that the order of certain metals in the thermoelectric scale is different at high and at low temperatures, so that for a certain temperature two metals may be neutral to each other. Thus, in a circuit of copper and iron if one junction be kept at the ordinary temperature while the temperature of the other is raised, a current sets from copper to iron through the hot junction, and the electromotive force continues to increase till the hot junction has reached a temperature $T$, which, according to Thomson, is about 284°C. When the temperature of the hot junction is raised still further the electromotive force is reduced, and at last, if the temperature be raised high enough, the current is reversed. The reversal of the current may be obtained more easily by raising the temperature of the colder junction. If the temperature of both junctions is above $T$ the current sets from iron to copper through the hotter junction, that is, in the reverse direction to that observed when both junctions are below $T$.

Hence, if one of the junctions is at the neutral temperature $T$ and the other is either hotter or colder, the current will set from copper to iron through the junction at the neutral temperature.

253.] From this fact Thomson reasoned as follows:—

Suppose the other junction at a temperature lower than $T$. The current may be made to work an engine or to generate heat in a wire, and this expenditure of energy must be kept up by the transformation of heat into electric energy, that is to say, heat must disappear somewhere in the circuit. Now at the temperature $T$ iron and copper are neutral to each other, so that no reversible thermal effect is produced at the hot junction, and at the cold junction there is, by Peltier's principle, an evolution of heat. Hence the only place where the heat can disappear is in the copper or iron portions of the circuit, so that either a current in iron from hot to cold must cool the iron, or a current in copper from cold to hot must cool the copper, or both these effects may take place. By an elaborate series of ingenious experiments Thomson succeeded in detecting the reversible thermal action of the current in passing between parts of different temperatures, and he found that the current produced opposite effects in copper and in iron^[4].

When a stream of a material fluid passes along a tube from a hot part to a cold part it heats the tube, and when it passes from cold to hot it cools the tube, and these effects depend on the specific capacity for heat of the fluid. If we supposed electricity, whether positive or negative, to be a material fluid, we might measure its specific heat by the thermal effect on an unequally heated conductor. Now Thomson's experiments shew that positive electricity in copper and negative electricity in iron carry heat with them from hot to cold. Hence, if we supposed either positive or negative electricity to be a fluid, capable of being heated and cooled, and of communicating heat to other bodies, we should find the supposition contradicted by iron for positive electricity and by copper for negative electricity, so that we should have to abandon both hypotheses.

This scientific prediction of the reversible effect of an electric current upon an unequally heated conductor of one metal is another instructive example of the application of the theory of Conservation of Energy to indicate new directions of scientific research. Thomson has also applied the Second Law of Thermodynamics to indicate relations between the quantities which we have denoted by $P$ and $Q$, and has investigated the possible thermoelectric properties of bodies whose structure is different in different directions. He has also investigated experimentally the conditions under which these properties are developed by pressure, magnetization, &c.

254.] Professor Tait^[5] has recently investigated the electromotive force of thermoelectric circuits of different metals, having their junctions at different temperatures. He finds that the electromotive force of a circuit may be expressed very accurately by the formula

$E=a(t_{1}-t_{2})[t_{0}-{\frac {1}{2}}(t_{1}+t_{2})],$

where $t_{1}$ is the absolute temperature of the hot junction, $t_{2}$ that of the cold junction, and $t_{0}$ the temperature at which the two metals are neutral to each other. The factor $a$ is a coefficient depending on the nature of the two metals composing the circuit. This law has been verified through considerable ranges of temperature by Professor Tait and his students, and he hopes to make the thermoelectric circuit available as a thermometric instrument in his experiments on the conduction of heat, and in other cases in which the mercurial thermometer is not convenient or has not a sufficient range.

According to Tait's theory, the quantity which Thomson calls the specific heat of electricity is proportional to the absolute temperature in each pure metal, though its magnitude and even its sign vary in different metals. From this he has deduced by thermodynamic principles the following results. Let $k_{a}t,\,k_{b}t,\,k_{c}t$ be the specific heats of electricity in three metals $a,\,b,\,c,$ and let $T_{bc},\,T_{ca},\,T_{ab}$ be the temperatures at which pairs of these metals are neutral to each other, then the equations

$(k_{b}-k_{c})T_{bc}+(k_{c}-k_{a})T_{ca}+(k_{a}-k_{b})T_{ab}=0$,

$J\Pi _{ab}=(k_{a}-k_{b})t(T_{ab}-t)$,

$E_{ab}=(k_{a}-k_{b})(t_{1}-t_{2})[T_{ab}-{\frac {1}{2}}(t_{1}+t_{2})]$

express the relation of the neutral temperatures, the value of the Peltier effect, and the electromotive force of a thermoelectric circuit.

1. North British Review, 1864, p. 353; and Proc. R. S., June 20, 1867.
2. Proc. R. S. Edin., Dec. 15, 1851; and Trans. R. S. Edin., 1854.
3. Cambridge Transactions, 1823.
4. 'On the Electrodynamic Qualities of Metals,' Phil Trans., 1856.
5. Proc. R. S. Edin., Session 1870–71, p. 308, also Dec. 18, 1871.