A Treatise on Electricity and Magnetism/Part I/Chapter VII

by James Clerk Maxwell

CHAPTER VII.

FORMS OF THE EQUIPOTENTIAL SURFACES AND LINES OF INDUCTION IN SIMPLE CASES.

117.] We have seen that the determination of the distribution of electricity on the surface of conductors may be made to depend on the solution of Laplace’s equation

${\frac {d^{2}V}{dx^{2}}}+{\frac {d^{2}V}{dy^{2}}}+{\frac {d^{2}V}{dz^{2}}}=0,$

$V$ being a function of x, y, and z, which is always finite and continuous, which vanishes at an infinite distance, and which has a given constant value at the surface of each conductor.

It is not in general possible by known mathematical methods to solve this equation so as to fulfil arbitrarily given conditions, but it is always possible to assign various forms to the function $V$ which shall satisfy the equation, and to determine in each case the forms of the conducting surfaces, so that the function $V$ shall be the true solution.

It appears, therefore, that what we should naturally call the inverse problem of determining the forms of the conductors from the potential is more manageable than the direct problem of determining the potential when the form of the conductors is given.

In fact, every electrical problem of which we know the solution has been constructed by an inverse process. It is therefore of great importance to the electrician that he should know what results have been obtained in this way, since the only method by which he can expect to solve a new problem is by reducing it to one of the cases in which a similar problem has been constructed by the inverse process.

This historical knowledge of results can be turned to account in two ways. If we are required to devise an instrument for making electrical measurements with the greatest accuracy, we may select those forms for the electrified surfaces which correspond to cases of which we know the accurate solution. If, on the other hand, we are required to estimate what will be the electrification of bodies whose forms are given, we may begin with some case in which one of the equipotential surfaces takes a form somewhat resembling the given form, and then by a tentative method we may modify the problem till it more nearly corresponds to the given case. This method is evidently very imperfect considered from a mathematical point of view, but it is the only one we have, and if we are not allowed to choose our conditions, we can make only an approximate calculation of the electrification. It appears, therefore, that what we want is a knowledge of the forms of equipotential surfaces and lines of induction in as many different cases as we can collect together and remember. In certain classes of cases, such as those relating to spheres, we may proceed by mathematical methods. In other cases we cannot afford to despise the humbler method of actually drawing tentative figures on paper, and selecting that which appears least unlike the figure we require.

This latter method I think may be of some use, even in cases in which the exact solution has been obtained, for I find that an eye-knowledge of the forms of the equipotential surfaces often leads to a right selection of a mathematical method of solution.

I have therefore drawn several diagrams of systems of equipotential surfaces and lines of force, so that the student may make himself familiar with the forms of the lines. The methods by which such diagrams may be drawn will be explained as we go on, as they belong to questions of different kinds.

118.] In the first figure at the end of this volume we have the equipotential surfaces surrounding two points electrified with quantities of electricity of the same kind and in the ratio of 20 to 5.

Fig. I

Here each point is surrounded by a system of equipotential surfaces which become more nearly spheres as they become smaller, but none of them are accurately spheres. If two of these surfaces, one surrounding each sphere, be taken to represent the surfaces of two conducting bodies, nearly but not quite spherical, and if these bodies be charged with the same kind of electricity, the charges being as 4 to 1, then the diagram will represent the equipotential surfaces, provided we expunge all those which are drawn inside the two bodies. It appears from the diagram that the action between the bodies will be the same as that between two points having the same charges, these points being not exactly in the middle of the axis of each body, but somewhat more remote than the middle point from the other body.

The same diagram enables us to see what will be the distribution of electricity on one of the oval figures, larger at one end than the other, which surround both centres. Such a body, if electrified with a charge 25 and free from external influence, will have the surface-density greatest at the small end, less at the large end, and least in a circle somewhat nearer the smaller than the larger end.

There is one equipotential surface, indicated by a dotted line, which consists of two lobes meeting at the conical point $P$. That point is a point of equilibrium, and the surface-density on a body of the form of this surface would be zero at this point.

The lines of force in this case form two distinct systems, divided from one another by a surface of the sixth degree, indicated by a dotted line, passing through the point of equilibrium, and some what resembling one sheet of the hyperboloid of two sheets.

This diagram may also be taken to represent the lines of force and equipotential surfaces belonging to two spheres of gravitating matter whose masses are as 4 to 1.

119.] In the second figure we have again two points whose charges are as 4 to 1, but the one positive and the other negative. In this case one of the equipotential surfaces, that, namely, corresponding to potential zero, is a sphere. It is marked in the diagram by the dotted circle $Q$. The importance of this spherical surface will be seen when we come to the theory of Electrical Images.

Fig. II

We may see from this diagram that if two round bodies are charged with opposite kinds of electricity they will attract each other as much as two points having the same charges but placed some what nearer together than the middle points of the round bodies.

Here, again, one of the equipotential surfaces, indicated by a dotted line, has two lobes, an inner one surrounding the point whose charge is 5 and an outer one surrounding both bodies, the two lobes meeting in a conical point $P$ which is a point of equilibrium.

If the surface of a conductor is of the form of the outer lobe, a roundish body having, like an apple, a conical dimple at one end of its axis, then, if this conductor be electrified, we shall be able to determine the superficial density at any point. That at the bottom of the dimple will be zero.

Surrounding this surface we have others having a rounded dimple which flattens and finally disappears in the equipotential surface passing through the point marked $M$.

The lines of force in this diagram form two systems divided by a surface which passes through the point of equilibrium.

If we consider points on the axis on the further side of the point $B$, we find that the resultant force diminishes to the double point $P$, where it vanishes. It then changes sign, and reaches a maximum at $M$, after which it continually diminishes.

This maximum, however, is only a maximum relatively to other points on the axis, for if we draw a surface perpendicular to the axis, $M$ is a point of minimum force relatively to neighbouring points on that surface.

120.] Figure III represents the equipotential surfaces and lines of force due to an electrified point whose charge is 10 placed at $A$, and surrounded by a field of force, which, before the introduction of the electrified point, was uniform in direction and magnitude at every part. In this case, those lines of force which belong to $A$ are contained within a surface of revolution which has an asymptotic cylinder, having its axis parallel to the undisturbed lines of force.

Fig. III

The equipotential surfaces have each of them an asymptotic plane. One of them, indicated by a dotted line, has a conical point and a lobe surrounding the point $A$. Those below this surface have one sheet with a depression near the axis. Those above have a closed portion surrounding $A$ and a separate sheet with a slight depression near the axis.

If we take one of the surfaces below $A$ as the surface of a conductor, and another a long way below $A$ as the surface of another conductor at a different potential, the system of lines and surfaces between the two conductors will indicate the distribution of electric force. If the lower conductor is very far from $A$ its surface will be very nearly plane, so that we have here the solution of the distribution of electricity on two surfaces, both of them nearly plane and parallel to each other, except that the upper one has a protuberance near its middle point, which is more or less prominent according to the particular equipotential line we choose for the surface.

121.] Figure IV represents the equipotential surfaces and lines of force due to three electrified points $A$, $B$ and $C$, the charge of $A$ being 15 units of positive electricity, that of $B$ 12 units of negative electricity, and that of $C$ 20 units of positive electricity. These points are placed in one straight line, so that

$AB=9,\ BC=16,\ AC=25.$

Fig. IV

In this case, the surface for which the potential is unity consists of two spheres whose centres are $A$ and $C$ and their radii 15 and 20. These spheres intersect in the circle which cuts the plane of the paper in $D$ and $D'$, so that $B$ is the centre of this circle and its radius is 12. This circle is an example of a line of equilibrium, for the resultant force vanishes at every point of this line.

If we suppose the sphere whose centre is $A$ to be a conductor with a charge of 3 units of positive electricity, and placed under the influence of 20 units of positive electricity at $C$, the state of the case will be represented by the diagram if we leave out all the lines within the sphere $A$. The part of this spherical surface within the small circle $DD'$ will be negatively electrified by the influence of $C$. All the rest of the sphere will be positively electrified, and the small circle $DD'$ itself will be a line of no electrification.

We may also consider the diagram to represent the electrification of the sphere whose centre is $C$, charged with 8 units of positive electricity, and influenced by 15 units of positive electricity placed at $A$.

The diagram may also be taken to represent the case of a conductor whose surface consists of the larger segments of the two spheres meeting in $DD'$, charged with 23 units of positive electricity.

We shall return to the consideration of this diagram as an illustration of Thomson’s Theory of Electrical Images. See Art. 168.

122.] I am anxious that these diagrams should be studied as illustrations of the language of Faraday in speaking of ‚lines of force,‘ the ‚forces of an electrified body‘, &c.

In strict mathematical language the word Force is used to signify the supposed cause of the tendency which a material body is found to have towards alteration in its state of rest or motion. It is indifferent whether we speak of this observed tendency or of its immediate cause, since the cause is simply inferred from the effect, and has no other evidence to support it.

Since, however, we are ourselves in the practice of directing the motion of our own bodies, and of moving other things in this way, we have acquired a copious store of remembered sensations relating to these actions, and therefore our ideas of force are connected in our minds with ideas of conscious power, of exertion, and of fatigue, and of overcoming or yielding to pressure. These ideas, which give a colouring and vividness to the purely abstract idea of force, do not in mathematically trained minds lead to any practical error.

But in the vulgar language of the time when dynamical science was unknown, all the words relating to exertion, such as force, energy, power, &c., were confounded with each other, though some of the schoolmen endeavoured to introduce a greater precision into their language.

The cultivation and popularization of correct dynamical ideas since the time of Galileo and Newton has effected an immense change in the language and ideas of common life, but it is only within recent times, and in consequence of the increasing importance of machinery, that the ideas of force, energy, and power have become accurately distinguished from each other. Very few, however, even of scientific men, are careful to observe these distinctions ; hence we often hear of the force of a cannon-ball when either its energy or its momentum is meant, and of the force of an electrified body when the quantity of its electrification is meant.

Now the quantity of electricity in a body is measured, according to Faraday’s ideas, by the number of lines of force, or rather of induction, which proceed from it. These lines of force must all terminate somewhere, either on bodies in the neighbourhood, or on the walls and roof of the room, or on the earth, or on the heavenly bodies, and wherever they terminate there is a quantity of electricity exactly equal and opposite to that on the part of the body from which they proceeded. By examining the diagrams this will be seen to be the case. There is therefore no contradiction between Faraday’s views and the mathematical results of the old theory, but, on the contrary, the idea of lines of force throws great light on these results, and seems to afford the means of rising by a continuous process from the somewhat rigid conceptions of the old theory to notions which may be capable of greater expansion, so as to provide room for the increase of our knowledge by further researches.

123.] These diagrams are constructed in the following manner :—

First, take the case of a single centre of force, a small electrified body with a charge $E$. The potential at a distance $r$ is $V={\frac {E}{r}}$; hence, if we make $r={\frac {E}{V}}$, we shall find $r$, the radius of the sphere for which the potential is $V$. If we now give to $V$ the values 1, 2, 3, &c., and draw the corresponding spheres, we shall obtain a series of equipotential surfaces, the potentials corresponding to which are measured by the natural numbers. The sections of these spheres by a plane passing through their common centre will be circles, which we may mark with the number denoting the potential of each. These are indicated by the dotted circles on the right hand of Fig. 6.

Fig. 6

If there be another centre of force, we may in the same way draw the equipotential surfaces belonging to it, and if we now wish to find the form of the equipotential surfaces due to both centres together, we must remember that if $V_{1}$ be the potential due to one centre, and $V_{2}$ that due to the other, the potential due to both will be $V_{1}+V_{2}=V$. Hence, since at every intersection of the equipotential surfaces belonging to the two series we know both $V_{1}$ and $V_{2}$, we also know the value of $V$. If therefore we draw a surface which passes through all those intersections for which the value of $V$ is the same, this surface will coincide with a true equipotential surface at all these intersections, and if the original systems of surfaces be drawn sufficiently close, the new surface may be drawn with any required degree of accuracy. The equipotential surfaces due to two points whose charges are equal and opposite are represented by the continuous lines on the right hand side of Fig. 6.

This method may be applied to the drawing of any system of equipotential surfaces when the potential is the sum of two potentials, for which we have already drawn the equipotential surfaces.

The lines of force due to a single centre of force are straight lines radiating from that centre. If we wish to indicate by these lines the intensity as well as the direction of the force at any point, we must draw them so that they mark out on the equipotential surfaces portions over which the surface-integral of induction has definite values. The best way of doing this is to suppose our plane figure to be the section of a figure in space formed by the revolution of the plane figure about an axis passing through the centre of force. Any straight line radiating from the centre and making an angle $\theta$ with the axis will then trace out a cone, and the surface-integral of the induction through that part of any surface which is cut off by this cone on the side next the positive direction of the axis, is $2\pi E(1-\cos \theta )$.

If we further suppose this surface to be bounded by its inter section with two planes passing through the axis, and inclined at the angle whose arc is equal to half the radius, then the induction through the surface so bounded is

$E(1-\cos \theta )=2\Psi$ say ;

and

$\theta =\cos ^{-1}\left(1-2{\frac {\Psi }{E}}\right).$

If we now give to $\Psi$ a series of values 1, 2, 3 ... $E$, we shall find

Fig. 6.

a corresponding series of values of $\theta$, and if $E$ be an integer, the number of corresponding lines of force, including the axis, will be equal to $E$.

We have therefore a method of drawing lines of force so that the charge of any centre is indicated by the number of lines which converge to it, and the induction through any surface cut off in the way described is measured by the number of lines of force which pass through it. The dotted straight lines on the left hand side of Fig. 6 represent the lines of force due to each of two electrified points whose charges are 10 and -10 respectively.

If there are two centres of force on the axis of the figure we may draw the lines of force for each axis corresponding to values of $\Psi _{1}$ and $\Psi _{2}$, and then, by drawing lines through the consecutive intersections of these lines, for which the value of $\Psi _{1}+\Psi _{2}$ is the same, we may find the lines of force due to both centres, and in the same way we may combine any two systems of lines of force which are symmetrically situated about the same axis. The continuous curves on the left hand side of Fig. 6 represent the lines of force due to the two electrified points acting at once.

After the equipotential surfaces and lines of force have been constructed by this method the accuracy of the drawing may be tested by observing whether the two systems of lines are every where orthogonal, and whether the distance between consecutive equipotential surfaces is to the distance between consecutive lines of force as half the distance from the axis is to the assumed unit of length.

In the case of any such system of finite dimensions the line of force whose index number is $\Psi$ has an asymptote which passes through the centre of gravity of the system, and is inclined to the axis at an angle whose cosine is $1-2{\frac {\Psi }{E}}$, where $E$ is the total electrification of the system, provided $\Psi$ is less than $E$. Lines of force whose index is greater than $E$ are finite lines.

The lines of force corresponding to a field of uniform force parallel to the axis are lines parallel to the axis, the distances from the axis being the square roots of an arithmetical series.

The theory of equipotential surfaces and lines of force in two dimensions will be given when we come to the theory of conjugate functions^[1].

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1. See a paper ‚On the Flow of Electricity in Conducting Surfaces‘, by Prof. W. R. Smith, Proc. R. S. Edin., 1869-70, p. 79.