Now, if a galvanometer is placed in circuit with the electric - motor and the battery, it is found that when the motor is running it is impossible to force so strong a current through the wires as that which flows when the motor is standing still. There are only 2 causes that can stop such a current flowing in a circuit: either an obstructive resistance, or a counter - electromotive force. At first, the common idea was that, when the motor was spinning round, it offered a greater resistance to the passage of the electric current than when it stood still. Jacobi, however, discerned that the observed diminution of current was really due to the fact that the motor, by the act of spinning round, began to work as a dynamo on its own account, and tended to set up a current in the circuit in the opposite direction to that which was driving it. The faster it rotated, the greater was the counter - electromotive force (or "electro - motive force of reaction") developed, In fact, the theory of the conservation of energy requires that such a reaction should exist.

In the converse case, when employing mechanical power to generate currents by rotating a dynamo, directly we begin to generate currents - i.e. to do electric work - it requires much more power to turn the dynamo than when no electric work is being done. In other words, there is an opposing reaction to the mechanical force applied in order to do electric work. An opposing reaction to a mechanical force may be termed a "counter - force." When, on the other hand, we apply (by means of a voltaic battery, for example) an electromotive force to do mechanical work, there is an opposing reaction to an electromotive force, or a " counter - electromotive force."

The existence of this counter - electromotive force is of the utmost importance, in considering the action of the dynamo as a motor, because upon its existence and magnitude depends the degree to which a motor enables us to utilize energy supplied to it in the form of an electric current. In discussing the dynamo as a generator, were pointed out some considerations whose observance would improve their efficiency; many of these - e.g. the avoidance of useless resistances, unnecessary iron masses in cores - apply to motors. The freer a motor is from such objections, the more efficient will it be; but its efficiency in utilizing the energy of a current depends not only on its efficiency in itself, but on the relation between the electromotive force which it generates when rotating, and the electromotive force (or " electric pressure ") at which the current is supplied to it. A motor which itself in running generates only a low electromotive force, cannot, however well designed,, be an efficient or economical motor when supplied with currents at a high electromotive force.

Dynamos used as motors must be supplied with currents; at electromotive forces adapted to them.Even a perfect motor - one without friction or resistance of any kind--cannot give an "efficient" or economical result, if the law of efficiency is not observed in the conditions under which the electric current is supplied to it.

The efficiency with which a perfect motor utilizes the electric energy of the current depends upon the ratio between this counter-electromotive force and the electro-motive force of the current that is supplied by the battery. No motor ever turns into useful work the whole of the currents that feed it, for it is impossible to construct machines without resistance, and whenever resistance is offered to a current, part of the energy of the current is wasted in heating the resisting wire. Let the symbol W stand for the whole electric energy of a current, and let to stand for that part of the energy which the motor takes up as useful work from the circuit. All the rest of the energy of the current, or W-w, will be wasted in useless heating of the resistances. But to work a motor under the conditions of greatest economy, there must be as little heat-waste as possible; or, in symbols, to must be as nearly as possible equal to W. The ratio between the useful energy thus appropriated and the total energy spent, is equal to the ratio between the counter-electromotive force of the motor, and the whole electromotive force of the battery that feeds the motor.

Let this whole electromotive force with which the battery feeds the motor be E, and let the counter-electromotive force be e: then the rule is w: W =e: E; or, expressed as a fraction, w/W = e/E If the resistances of the circuit are constant, the current c, observed when the motor is running, will be less than C, the current while the motor is standing still. But from Ohm's law we know that c = E-e / R.

Hence

C - c /C = e/E=w/W.

From which, it appears that we can calculate the efficiency at which the motor is working by observing the ratio between the fall in the strength of the current and the original strength. This mathematical law of efficiency has been strangely misapprehended. Another law, discovered by Jacobi, not a law of efficiency at all, but a law of maximum work in a given time, has usually been given instead. Jacobi's law is as follows:- The mechanical work given out by a motor is a maximum when the motor is geared to run at such a speed that the current is reduced to half the strength that it would have if the motor was stopped. This implies that the counter-electromotive force of the motor is equal to half of the electromotive force furnished by the battery or generator. Under these circumstances, only half the energy furnished by the external source is utilized, the other half being wasted in heating the circuit. If Jacobi's law were indeed the law of efficiency, no motor, however perfect in itself, could convert more than 50 per cent. of the electric energy supplied to it into actual work. Siemens has shown that a dynamo can be, in practice, so used as to give out more than 50 per cent. of the energy of the current.