may likewise be conceived as made up of a number of elementary small cones, each of solid-angle : Let r1 and r2 be the distances from P to the two faces of the element: Let a section be made across the small cone orthogonally, or at right angles to r1, and call the area of this section a: Let the angle between the surfaces s and a be called angle ẞ: then s = = a/cos B. Let i be the "strength" of the shell (i.e. = its surface-density of magnetism X its thickness); then i/t = surface-density of magnetism, and si/t strength of either pole of the little magnet = m.

But the potential at P of the magnet whose pole is m will be

because 71 and r2 may be made as nearly equal as we please. And since 2-r1 = t cos B

=

the strength

or the potential due to the element of the shell of the shell the solid-angle subtended by the element of the shell. Hence, if V be the sum of all the values of v for all the different elements, and if o be the whole solid-angle (the sum of all the small solid-angles such as ),

Vp = wi,

or the potential due to a magnetic shell at a point is equal to the strength of the shell multiplied by the solid-angle subtended by the whole of the shell at that point.

Hence wi represents the work that would have to be done on or by a unit-pole, to bring it up from an infinite distance to the point P, where the shell subtends the solid-angle w. At a point Q

where the solid-angle subtended by the shell is different, the potential will be different, the difference of potential between P and Q being

VQ - VP = i (wq- wp).

If a magnet-pole whose strength is m were brought up to P, m times the work would have to be done, or the mutual potential would be = mwi.

349. Potential of a Magnet-pole on a Shell. It is evident that if the shell of strength i is to be placed where it subtends a solid-angle w at the pole m, it would require the expenditure of the same amount of work to bring up the shell from an infinite distance on the one hand, as to bring up the magnet-pole from an infinite distance on the other; hence moi represents both the potential of the pole on the shell and the potential of the shell on the pole. Now the lines of force from a pole may be regarded as proportional in number to the strength of the pole, and from a single pole they would radiate out in all directions equally. Therefore, if a magnet-pole was placed at P, at the apex of the solid-angle of a cone, the number of lines of force which would pass through the solid-angle would be proportional to that solid-angle. It is therefore convenient to regard mo as representing the number of lines of force of the pole which pass through the shell, and we may call the number so intercepted N. Hence the potential of a magnet-pole on a magnetic shell is equal to the strength of the shell multiplied by the number of lines of force (due to the magnet-pole) which pass through the shell; or V Ni. If either the shell or the pole were moved to a point where a different number of lines of force were cut, then the difference of potential would be

VqVpi (Ną — Np).

To bring up a N-seeking (or +) pole against the repelling force of the N-seeking face of a magnetic shell requires a positive amount of work to be done; and their mutual reaction would enable work to be done afterwards by virtue of their position: in this case then the potential is +. But in moving a N-seeking pole up to the S-seeking face of a shell work will be done by the pole, for it is attracted up; and as work done by the pole may be regarded as our doing negative work, the potential here will have a negative value.

Again, suppose we could bring up a unit N-seeking pole against the repulsion of the N-seeking face of a shell of strength i, and should push it right up to the shell; when it actually reached the plane of the shell the shell would occupy a whole

horizon, or half the whole space around the pole, the solid-angle it subtended being therefore 2, and the potential will be +2πi. If we had begun at the S-seeking face the potential at that face would be-2i. It appears then that the potential alters its value by 4i on passing from one side of the shell to the other.

There is a reaction between pole and shell similar to that (Art. 121) between pole and pole.

If a N-seeking pole be brought up to the N-seeking face of a shell none of the lines of force of the magnet will cut the shell, but will be repelled out as in Fig. 72; whereas if a N-seeking pole be brought up to the S-seeking face of a shell, large numbers of the lines will be run into one another; and the pole, as a matter of fact, will be attracted up to the shell, where as many lines of force as possible are cut by the shell. We may formulate this action by saying that a magnetic shell and a magnet-pole react on one another and urge one another in such a direction as to make the number of lines of force that are cut by the shell a maximum (Maxwell's Rule, Art. 204). Outside the attracting face of the shell the potential is wi, and the pole moves so as to make this negative quantity as great as possible, or to make the potential a minimum. Which is but another way of putting the matter as a particular case of the general proposition that bodies tend to move so that the energy they possess in virtue of their position tends to run down to a minimum.

350. Magnetic Potential due to Current. The propositions concerning magnetic shells given in the preceding paragraphs derive their great importance because of the fact laid down in Art. 203 that circuits, traversed by currents of electricity, behave like magnetic shells. Adopting the electromagnetic unit of current (Art. 353), we may at once go back to Art. 347, and take the theorems about magnetic shells as being also true of closed voltaic circuits.

(a) Potential due to closed circuit (compare Art. 348).

The potential V due to a closed voltaic circuit (traversed by a current) at a point P near it, is equal to the strength of the current multiplied by the solid-angle w subtended by the circuit at that point. If C be the strength of the current in electromagnetic units, then

VPwC.

ω

(b) At a point Q, where the solid-angle subtended by the circuit is wQ instead of wp, the potential will have a different value, the difference of potential being

*See note on Ways of Reckoning Angles, Art. 144 and Appendix A.

(c) Mutual Potential of a Magnet-pole and a Circuit. - If a magnet-pole of strength m were brought up to P, m times as much work will be done as if the magnet-pole had been of unit strength, and the work would be just as great whether the pole m were brought up to the circuit, or the circuit up to the pole. Hence, the mutual potential will be

- MoC.

But, as in Art. 349, we may regard mo as representing the number of lines of force of the pole which are intercepted by and pass through the circuit, and we may write N for that number, and say

CN,

or the mutual potential of a magnet-pole and a circuit is equal to the strength of the current multiplied by the number of the magnet-pole's lines of force that are intercepted by the circuit, taken with reversed sign.

(d) As in the case of the magnetic shell, so with the circuit, the value of the potential changes by 4 πC from a point on one side of the circuit to a point just on the other side; that is to say, being 2 C on one side and +2 C on the other side work equal to 4 C must be done in carrying a unit-pole from one side to the other round the outside of the circuit. The work done in thus threading the circuit along a path looped S times round it would be 4 #SC.

351. (e) Mutual Potential of two Circuits. Two closed circuits will have a mutual potential, depending on the strengths of their respective currents, on their distance apart, and on their form and position. If their currents be respectively C and C', and if the distance between two elements ds and ds' of the circuits be called r, and the angle between the elements, it can be shown that their mutual potential is

COS €

CC'S S
'S Scose as as'.

This expression represents the work that would have to be done to bring up either of the circuits from an infinite distance to its present position near the other, and is a negative quantity if they attract one another. Now, suppose the strength of current in each circuit to be unity; their mutual potential will in that case be ds ds', a quantity which depends purely upon the geometrical form and position of the circuits, and for which we may substitute the single symbol M, which we will call the coefficient of mutual potential”: we may now write the mutual potential of the two circuits when the currents are C and C'as == CC'M.

66

But we have seen in the case of a single circuit that we may represent the potential between a circuit and a unit-pole as the product of the strength of the current C into the number N of the magnet-pole's lines of force intercepted by the circuit. Hence the symbol M must represent the number of each other's lines of force mutually intercepted by both circuits, if each carried unit current. If we call the two circuits A and B, then, when each carries unit current, A intercepts M lines of force belonging to B, and B intercepts M lines of force belonging to A.

Now suppose both currents to run in the same (clock-wise) direction; the front or S-seeking face of one circuit will be opposite to the back or N-seeking face of the other circuit, and they will attract one another, and will actually do work as they approach one another, or (as the negative sign shows) negative work will be done in bringing up one to the other. When they have attracted one another up as much as possible the circuits will coincide in direction and position as nearly as can ever be. Their potential energy will have run down to its lowest minimum, their mutual potential being a negative maximum, and their coefficient of mutual potential M, having its greatest possible value. Two circuits, then, are urged so that their coefficient of mutual potential M shall have the greatest possible value. This justifies Maxwell's Rule (Art. 204), because M represents the number of lines of force mutually intercepted by both circuits. And since in this position each circuit induces as many lines of magnetic force as possible through the other, the coefficient of mutual potential M is also called the coefficient of mutual induction (Art. 454).

LESSON XXVII.—The Electromagnetic System of Units

352. Magnetic Units. All magnetic quantities, strength of poles, intensity of magnetization, etc., are expressed in terms of special units derived from the fundamental units of length, mass, and time, explained in the Note on Fundamental and Derived Units (Art. 280). Most of the following units have been directly explained in the preceding Lesson, or in Lesson XI.; the others follow from them.

Unit Magnet-pole. - The unit magnetic pole is one of such a strength, that when placed at a distance of 1 centimetre (in air) from a similar pole of equal strength, repels it with a force of 1 dyne (Art. 141).

Magnetic Potential. Magnetic potential being measured by work done in moving a unit magnetic pole against the