plane of the circuit, it is clear that now we add only ncos marked lines per second to the circuit.

Resistance expressed as a velocity. We have seen in Chapter XVIII. § 5 that, in the electro-magnetic system, R is of the dimensions of a velocity. It happens that the above simple apparatus enables us to show how R can in fact be measured by a velocity. (Compare also Chapter XVIII. § 4, note on method (ii.) of determining resistance.)

Let us assume for simplicity that the other quantities concerned are each unity; i.e. that the field is of unit strength and is perpendicular to the plane of the apparatus, so that we have one line of force piercing each I sq. cm. ; that the rail is of 1 cm. length, and that the current is of unit strength.

The rail must move with such a velocity that, in the formula given above, ʼn may be numerically equal to R. This means that we must add R sq. cms. to the area of the circuit each second; or, since the rail is 1 cm. long, this must move with a velocity expressed by the number R, or with R cms. per second.

Hence, under the above unit conditions' we can measure R by the velocity of the rail required to give unit current.

The velocity that would thus measure the absolute unit of resistance is that of 1 cm. per second, while 10 cms. per second measures 1 ohm.

§ 8. Changes that give Induced Currents. In considering the question as to whether in any particular case there will be an induced current, we have to remember two facts.

(a) That when a conductor cuts across lines of force there is always an induced E. M.F.

(b) That when there is a change in the number of lines piercing a circuit there is a resultant E.M F. induced in the circuit.

We will consider several cases.

(I.) A circuit moving in a uniform field.—If the circuit move parallel to itself (in such a way that the direction of movement causes the circuit to cut the lines of force), the number of marked lines embraced will not alter; and there will be no resultant E.M.F., or current, induced. The two sides of the circuit, it is true, do cut the lines; and so, by principle (a), there is an E.M.F. induced in each side. But these E.M.F.s are equal and opposed. There is no resultant E.M.F. in the circuit, but the top and bottom of the circuit, where the wire does not cut the lines of force, will be maintained at different potentials, as could be demonstrated by connecting them with a quadrant electrometer.

If, however, there be any movement of rotation, then the number of marked lines embraced is altered, and there is an E.M. F. induced. It is, in fact, a very usual way of obtaining induced currents to rotate a circuit or coil in a field that is more or less uniform.

(II.) Movement of a circuit in a non-uniform field.—Where the field is not uniform, movements of a circuit will in general produce a change in the number of marked lines embraced, and so there will be an E.M.F. induced. In certain cases the two movements of translation and rotation respectively may give as a result no change in the number of lines embraced; in which case there is no E.M.F. induced in the circuit as a whole.

(III.) Movements of an incomplete circuit; e.g. of a rectilinear wire. In considering the movements of an incomplete circuit, such, e.g., as a rectilinear piece of wire, we may adopt two courses. We may consider it to form part of a closed circuit, the rest of which is at an infinite distance, and is indefinitely remote from the magnetic field in question; or we may consider the piece of wire by itself. It may be stated as a general law that there will or will not be an E.M.F. induced according as the wire cuts or does not cut the lines of force, as strings are cut by a knife.

Note. If this were absolutely true we should never have an E.M.F. induced when the wire either moved in its own direction, piercing the field endforemost as a needle, or if it moved in any way in a plane in which lay the lines of force. Now, it is certainly true that in the latter case we never have an induced current. But in the former case we might have a current; for the wire might, so to speak, 'tunnel its way' end-on along the axis into a cylindrical system of lines of force; the self-induction in a rectilinear current, mentioned in the note to § 10, being a case of this nature.

While, therefore, the statement that

When a conductor moves in a magnetic field there will or will not be an induced current according as it does or does not cut the lines of force

is a good general rule, covering all cases of importance, still it is well to seck for a rule that shall cover all cases. Now, such a rule can be found from Lenz's law, if we remember the nature of the field given by a rectilinear current, and so by a current of any shape. We may say that

When a conductor moves in a magnetic field there will or will not be an induced E. M.F. in it according as the field due to a current produced by such an E.M.F. can or can not oppose the change in field due to the movement.

(IV.) Cases where the conductor (or circuit) is stationary, the field being altered.-In such cases there is no obvious cutting of lines

of force. But both theory and experiment tell us that all such cases of alteration in the field about a conductor, or circuit, can be regarded as equivalent to cases where the same change has been produced by movements on the part of the conductor from a weaker to a stronger part of the field, or conversely. We may, therefore, consider that all cases included under this head have been already discussed above.

§ 9. Coefficient of Mutual Induction, or of Mutual Potential. -Let us consider two simple circuits, A and A', carrying currents C and C' respectively. Each gives a magnetic field, and each is placed in the field due to the other. If n and n' are the number of marked lines due to A' embraced by A, and due to A embraced by A', respectively, then the potentials on each circuit due to the other are Cn and C'n', respectively. That is, it would require Cn ergs to bring up A from infinity to its present position, A' remaining stationary; or C'n' ergs to bring up A', A remaining stationary. Now, if A be moved to infinity, we do Cn ergs work; and if A' be moved after it into the same relative position as initially, we do + C'n' ergs work. But, on the whole, since things are exactly as they were, we must, by 'Conservation of energy,' have done no work. Hence it follows that Cn and C'n' must be equal; and, therefore, two circuits exert on one another a mutual potential.

Again, the number n of marked lines of force due to A' that pierce A are, cæteris paribus, directly proportional to the current C' of A'; this following from the fact that in our system of units we measure currents by the field-strength produced at constant distance. And the number n' of marked lines that pierce A' are in like manner directly proportional to C.

Hence, since we have shown the potential to be 'mutual,' this 'mutual potential' must be measured by some expression of the form CC'M; where M depends upon the shapes and positions of, and distance between, the two circuits, and not on the currentstrengths.

Thus we have Cn C'n' = CC'M.

If we make C and C' both unity, we find that Mn or n'. Hence we see that when unit current flows in each circuit, the number of marked lines due to the other, piercing each circuit respectively, is the same.

And we see, further, that the symbol M, whose meaning we had not given exactly, represents this number. The reader will notice that, for unit current, this number is, as we said, something depending upon the shapes and positions of, and distance between, the two circuits.

It only remains to state that M is called the coefficient of mutual induction or of mutual potential.

Hence, the coefficient M of mutual induction, or of mutual potential, between two circuits is measured by the number of marked lines due to either that are embraced by the other when the currents are both unity. And when the currents are C and C' respectively, then the mutual potential will be expressed by CC' M.

If we consider a current C sent through the one circuit, it will in consequence send C M marked lines of force through the other If it take t seconds to establish the current (where may be a small fraction of a

CM

second) then there are lines added per second to the second cir

cuit. Hence the induced E.M.F. e =

CM. This shows us (1) the

t

reason for M being called the coefficient of mutual induction as well as that of mutual potential; and (2) how, in such an arrangement, the magnitude of the induced E.M.F. e is inversely proportional to the time taken to establish or destroy the primary or inducing current.

§ 10. Self-Induction. The Extra Current.'-Let us consider a circuit comprising a coil of many turns and a battery; this circuit being so arranged that it can be made or broken at will.

When the circuit is made the current does not rise to a maximum at once. If we consider only the positive current, or the flow of electricity from the + pole of the battery to the pole, we find that it takes time to rise to a maximum; and that in so rising the turns of wire that are more remote from the + pole of the battery always lag behind those that are nearer to it. Each turn of wire, as the current in it increases, thrusts an increasing number of marked lines through the adjacent turns of wire. It, therefore, acts inductively on these, and, by Lenz's law, the induced E.M.F. is in such a direction as to oppose the rise in currentstrength. There is, in fact, an inverse induced current. Thus the coil, as a whole, offers an inductive obstruction to the rise in current that is quite distinct from resistance; and that will, as far as it is

due to the above given cause, disappear if the coil be unwound and laid out as a straight wire.

Note. In a straight wire there is also induction, but to a less degree. In this case each bit of wire gives a field of circular lines of force, not only about itself, but also, to a much smaller degree, about the wire in front and behind; and any increase in this field is also opposed by induction (see § 8 (III.), note).

When the circuit is broken, induction in the inverse direction, i.e. so as to oppose the cessation of the current, ensues; and there is an induced current added to the original current just as the latter is ceasing. Now, the primary current can be broken much more abruptly than it can be made; and hence, since e is inversely proportional to the time taken to effect the change in field, the E.M.F. of this direct induced current is much greater than is that of the inverse induced current; it may, indeed, be made very great indeed.

This direct induced current that occurs in the coil when the circuit is broken is called the extra current.

Experiments.—(i.) In the figure B is a coil in circuit with a battery whose poles are E and E'; this circuit can be broken at E. The two points A and C are connected by a short circuit that includes a galvanometer G.

When this circuit is complete the current flows as shown, and the galvanometer is deflected as indicated. When the circuit is broken at E, it is easy to see that a direct extra current in the coil would now pass round the galvano. meter in the contrary direction to that in which it before passed, although in the coil B it flows in the same direction as before. Hence the extra current