to see whether such a fluid as water will, or will not, flow from the one spot to the other when free to move. We call that the higher level from which the water flows. In electrical levels (or potentials) two points are at the same potential or different potentials, according as electricity is not, or is, urged from the one to the other point when free to move. That is called the higher potential from which the electricity is urged.

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Thus in gravity a spot has a level if water will run from the sea-level to it; in electricity a point is at - potential if tricity will flow from the earth to it.

In the dual nature of electricity, and in the opposite movements of + and electricity, we perceive the limitations of our rough analogy.

$ 3. Measurement of Differences of Electrical Level by Work-In Chapter X. we shall discuss this matter of measurement of potential more fully. Here we wish only to show in what way the measurement is made.

The measurement of gravity levels may be made by means of a tape, caused to hang vertically by being weighted at the end; or by some equivalent method in which the action of gravitation gives us either a vertical line to measure along, or a horizontal plane from which the vertical can be deduced. cases the measurement is made directly in feet.

In all such

But there is another way that is theoretically possible.

We shall in Chapter X. consider the meaning and measurement of work. Here we will only say that in lifting weights from a lower to a higher level we do work; that the work done is proportional to the weight raised, and also to the vertical height through which it is raised; and that it is independent of the route, whether direct or roundabout, by which the weight is raised from the lower to the higher level.

Now if we could in some way accurately register work done, then, by carrying some standard weight from one place to another, we could, by the amount of work done, estimate the difference of level of the two places. There has, however, been elaborated no exact method of this sort, because the former method is so very simple and direct.

In electricity, on the contrary, there is known no method analogous to the direct plumb-line measurement. The only

conceivable method of measuring the difference of electrical level between two points is by measuring the work done in moving a + unit of electricity from the one point to the other.

If we assume that the reader understands in what unit work is measured, we can say that we have now shown how to give an exact numerical meaning to that symbol 'V'which is used to express the potential of a point with respect to the earth as

zero.

Notes. —(i.) We are said to do + work when we move against the lines of force, or expend energy; we are said to do work when we move down the

lines of force, or have energy expended on us.

(ii.) The reader will understand from the above section, that the electrical potential of a point in space (or of a body) is +, zero, or —, according as we do +, zero, or work respectively in moving a + unit of electricity from the earth to the point in space (or body). The potential is higher as this work is greater.

The + unit of electricity was defined in § 1 of this chapter. Any work done in moving it from one place to another is due to the repulsive and attractive actions of other electrical quantities on this unit. [As regards sign, see Chapter IV. § 7.]

§ 4. Elementary Ideas on ‘Capacity.'—A certain quantity of water will fill a certain vessel to a definite level, this level depending on the quantity of water and on the dimensions of the vessel. The more the water, and the less the horizontal section of the vessel, the higher the level to which the latter will be filled.

In electricity, when we charge a conductor with a quantity Q of electricity, we raise the conductor to an electrical potential V. This potential will be greater or less according to a property of the conductor which we may naturally call, by analogy, its electrical capacity. We call the capacity greater or smaller according as the potential V, to which a given charge Q of electricity will raise it, is smaller or greater respectively. Now we have already indicated how Q and V are defined and measured. Since capacity (K) is a new term we may define it, and express it numerically, by the relation-

which is equivalent to saying the capacity of a conductor is directly proportional to the quantity of electricity required to raise it to a

certain potential, and inversely proportional to the potential to which it is raised by a certain quantity.

Of the units in which V and K are measured we shall treat more fully in Chapter X. We see, however, from the above definition of K, that—

A conductor has unit capacity when unit quantity raises it to unit potential, or to unit electrical level above the earth.

§ 5. Lines of Force, and Equipotential Surfaces.—We have in Chapter III. sufficiently explained what are meant by lines of force. The reader can easily, mutatis mutandis, define for himself lines of force in a gravitation field of force, or in an electrical field of force.

Thus in an electrical field a line of force is a line along which a particle charged with + electricity, and free from all other forces, would move under the influence of electrical forces; it being assumed that the movement is indefinitely slow, so that there is not any 'centrifugal' desertion of the line of force when this is curved.

It is along these lines of force that electrical level is measured; just as it is along the lines of gravitation force (i.e. vertical lines) that gravitation levels are measured with the plumb-line.

It is in movement of electricity along the lines of electrical force, as in movement of masses along the lines of gravitation force, that we do work.

Surfaces over which we can move our + unit of electricity without doing work are called equipotential surfaces.

In the case of gravitation these are clearly horizontal surfaces; a mass moved over these surfaces always cuts at right angles the lines of force of gravitation, and hence no work is done, or these horizontal surfaces are equipotential.

In electrostatics let us consider the simple case of a single isolated charged particle. Since a + unit would be urged straight from it or straight towards it, according as the charge on it is + or —, it follows that the lines of force are straight lines radiating from the particle as centre.

What are the equipotential surfaces, i.e. level surfaces, or surfaces of no work? They must be those over which our + unít always cuts the lines of force at right angles, thus doing no work.

That is, they must be spheres having the particle as centre

and the lines of force as radii. In more complicated systems of electrical charges we may have lines of force curved about in any way whatever. But still the equipotential surfaces, over which our + unit moves without doing any work, will be those that cut at right angles the lines of force.

From what we have said the reader will see that wherever are lines of force, there is rise or fall of potential; and conversely.

A region where are no lines of force must be all at one potential; since there can be no work done in moving our + unit against zero force, and therefore there can be no differences of electrical level. And conversely, where a region is all at one potential, there are no lines of force.

§ 6. Induction; from a 'Potential' Point of View. We will now consider briefly what light the above considerations throw upon the matter of electrostatic induction.

We mean

Definition of a conductor. -We must first, however, define what we mean by the often-used term conductors. bodies on which electrical charges are perfectly free to move, bodies which can resist no electrical stress due to differences of electrical potential.

By our definitions then it follows that when any conductor is left until the electrical charge has had time to arrange itself, it will be all at one potential; for were two parts at different potentials, the electrification would readjust itself until the two places were at the same potential. This readjustment is so rapid that we can consider it to be practically instantaneous.

The reader may object that one might conceivably have a conductor of which two parts remain at different potentials for the reason that there is no electrical charge to be readjusted. The answer to this is that experiment shows us the universal presence of unlimited quantities of and electrifications in all bodies, these being equal in amount if the body be in what we call an uncharged condition. If two parts of the conductor be at different potentials, then there is unlimited electricity ready to flow from the higher level to the lower, and unlimited electricity ready to flow from the lower to the higher level; and a flow takes place until all is at one potential.

Hydrostatic analogy to electrostatic inductions.-We may with advantage discuss, in connection with the matter of electrical induction, a hydrostatic analogy. We shall consider the sea-level

as analogous to the electrical level or potential of the earth; a hillside as analogous to a region through which there is a fall of electric potential, or what we may term an electric hill; and a trough of water as (very roughly indeed) analogous to an electrical conductor. In a still rougher sense we may sometimes take a tower as analogous to a + electrical charge; and a well as analogous to a trical charge. We may now note the following facts.

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(a) The water in the trough will always be at one level, or water will have no more tendency to flow in one direction than in the other, in whatever position the trough be placed.

(b) If the trough be placed upon a hill-side, the whole will be above the level of the sea; and water would flow from any part of the trough down to the sea, if connected therewith by a pipe.

(c) Further, the water will rise at one end and will sink at the other; still remaining level. It will be shallower at the end that lies uphill, and deeper at the end that lies downhill.

(d) If instead of a trough we have a trench of unlimited depth, and connected this with the sea-level, water will flow out until all is at the sea-level. The water will then be at the sea-level, but will be deep below the surface of the hill.

(e) If the hill-side now sank, with the trench still cut in it, to the sea-level, the water in the trench would be now below the sealevel; and water would flow from the sea into the trench until the level was again raised to sea-level.

(f) Let there be a trough of water whose surface is at the sealevel. Now let this trough be raised up on to a hill-side and there be put in a sloping position. The water will dispose itself as in (c) above. Now let a partition be put in the middle of the trough, parallel to the ends, and let the whole be replaced in its original position. It is obvious that in the one half we shall now have the surface above, and in the other half the surface below, the sea-level. On removing the partition the original state of things is restored, and the surface is once more all at sea-level.

(g) Let us build a tower on the sea bottom so that its top reaches the sea level. If the sea-bottom were to rise up, the top of the tower would now be above sea-level.

(h) Let us dig a well on the hill so that its bottom reaches sea-level. If the hill sinks to sea-level, the bottom of the well will be below this.