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dw cos <f»' dx y VO^4 — «V)'
On the surface A therefore, in this example, the .general value of p is _
and the potential function for any point p\ exterior to A, is
Making now a; and y both infinite, in order that p may be at an infinite distance, there results
and thus the condition determining A, in Q, the quantity of electricity upon the surface, is, since B may be supposed equal to +
These results of our analysis agree with what has been long known concerning the law of the distribution of electric fluid on the surface of a spheroid, when in a state of equilibrium.
(13.) In what has preceded, we have confined ourselves to the consideration of perfect conductors. We will now give an example of the application of our general method, to a body that is supposed to conduct electricity imperfectly, and which will, moreover, be interesting, as it serves to illustrate the magnetic phenomena, produced by the rotation of bodies under the influence of the earth's magnetism.
If any solid body whatever of revolution, turn about its axis, it is required to determine what will take place, when the matter of this solid is not perfectly conducting, supposing it under the influence of a constant electrical force, acting parallel to any given right line fixed in space, the body being originally in a natural state.
Let /3 designate the coercive force of the body, which we will suppose analogous to friction in its operation, so that as lung as the total force acting upon any particle within the body is less than ft, its electrical state shall remain unchanged, but when it begins to exceed fi, a change shall ensue.
In the first place, suppose the constant electrical force, which we will designate by b, to act in a direction parallel to a line passing through the centre of the body, and perpendicular to its axis of revolution; and let us consider this line as the axis of x, that of revolution being the axis of z, and y the other rectangular co-ordinate of a point p, within the body and fixed in space. Thus, if V be the value of the total potential function for the same point p, at any instant of time, arising from the electricity of the body and the exterior force,
bx + V
will be the part due to the body itself at the same instant: since - bx is that due to the constant force b, acting in the direction of x, and tending to increase it. If now we make
z = r cos 6, x = r sin 6 cos w, y = r sin 6 sin «r;
the angle w being supposed to increase in the direction of the body's revolution, the part due to the body itself becomes
br sin 6 cos «r + V.
Were we to suppose the value of the potential function V given at any instant, we might find its value at the next instant, by conceiving, that whilst the body moves forward through the infinitely small angle d% the electricity within it shall remain fixed, and then be permitted to move, until it is in equilibrium with the coercive force.
Now the value of the potential function at p, arising from the body itself, after having moved through the angle dor (the electricity being fixed), will evidently be obtained by changing tar into «. — rfor*in the expression just given, and is therefore
br sin 8 cos *r + V+ br sin 6 sin <a du> — -r- dtar.
adding now the part — bx = — br sin 6 cos a, due to the exterior bodies, and restoring x, y, &c. we have, since
for the value of the total potential function at the end of the next instant, the electricity being still supposed fixed. We have now only to determine what this will become, by allowing the electricity to move forward until the total forces acting on points within the body, which may now exceed the coercive force by an infinitely small quantity, are again reduced to an equilibrium with it. If this were done, we should, when the initial state of the body was given, be able to determine, successively, its state for every one of the following instants. But since it is evident from the nature of the problem, that the body, by revolving, will quickly arrive at a permanent state, in which the value of Fwill afterwards remain unchanged and be independent of its initial value, we will here confine ourselves to the determination of this permanent state. It is easy to see, by considering the forces arising from the new total potential function, whose value has just been given, that in this case the electricity will be in motion over the whole interior of the body, and consequently
which equation expresses that the total force to move any particle p, within the body, is just equal to /S, the coercive force. Now if we can assume any value for V, satisfying the above, and such, that it shall reproduce itself after the electricity belonging to the new total potential function (Art. 7), is allowed to find its equilibrium with the coercive force, it is evident this will be the required value, since the rest of the electricity is exactly in equilibrium with the exterior force b, and may therefore be here neglected. To be able to do this the more easily, conceive two new axes X', Y', in advance of the old ones X, Y, and making the angle 7 with them; then the value of the new potential function, before given, becomes
rr J U' 1 • • _,dV ,dV\
V + aa. ^oy cos 7 + ox sin 7 + y — x , j,
which, by assuming V= /3i/, and determining 7 by the equation
0 = b sin 7 — /8,
reduces itself to
y (fi + b cosydtlf}.
Considering now the symmetrical distribution of the electricity belonging to this potential function, with regard to the plane whose equation is 0=y', it will be evident that, after the electricity has found its equilibrium, the value of V at this plane must be equal to zero: a condition which, combined with the partial differential equation before given, will serve to determine, completely, the value of V at the next instant, and this value of Fwill be
We thus see that the assumed value of V reproduces itself at the end of the following instant, and is therefore the one required belonging to the permanent state.
If the body had been a perfect conductor, the value of V would evidently have been equal to zero, seeing that it was supposed originally in a natural state: that just found is therefore due to the rotation combined with the coercive force, and we thus see that their effect ia to polarise the body in the direction of y' positive, making the angle + 7 with the direction of
the constant force b; and the degree of polarity will be the same as would be produced by a force equal to j3, acting in this direction on a perfectly conducting body of the same dimensions.
We have hitherto supposed the constant force to act in a direction parallel to the equatorial plane of the body, but whatever may be its direction, we may conceive it decomposed into two; one equal to b as before, and parallel to this plane, the other perpendicular to it, which last will evidently produce no effect on the value of V, as this is due to the coercive force, and would still be equal to zero under the influence of the new force, if the body conducted electricity perfectly.
Knowing the value of the potential function at the surface of the body, due to the rotation, its value for all the exterior space may be considered as determined (Art. 5), and if the body be a solid sphere, may easily be expressed analytically; for it is evident (Art. 7), from the value of V just given, that even in the present case all the electricity will be confined to the surface of the solid; and it has been shown (Art. 10), that when the value of the potential function for the point p within a spherical surface, whose radius is a, is represented by
the value of the same function for a point p', situate without this sphere, on the prolongation of r, and at the distance r from its centre, will be
But we have seen that the value of V due to the rotation, for the point p, is
V - ft ~/8r cos ff;
ff being the angle formed by the ray r and the axis of y; the corresponding value for the pointy will therefore be
y-, _ j8a8 cos ff