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

by James Clerk Maxwell

CHAPTER II.

ELEMENTARY MATHEMATICAL THEORY OF STATICAL ELECTRICITY.

Definition of Electricity as a Mathematical Quantity.

63.] We have seen that the actions of electrified bodies are such that the electrification of one body may be equal to that of another, or to the sum of the electrifications of two bodies, and that when two bodies are equally and oppositely electrified they have no electrical effect on external bodies when placed together within a closed insulated conducting vessel. We may express all these results in a concise and consistent manner by describing an electrified body as charged with a certain quantity of electricity, which we may denote by $e$. When the electrification is positive, that is, according to the usual convention, vitreous, $e$ will be a positive quantity. When the electrification is negative or resinous, $e$ will be negative, and the quantity $e$ may be interpreted either as a negative quantity of vitreous electricity or as a positive quantity of resinous electricity.

The effect of adding together two equal and opposite charges of electricity, $+e$ and $-e$, is to produce a state of no electrification expressed by zero. We may therefore regard a body not electrified as virtually charged with equal and opposite charges of indefinite magnitude, and an electrified body as virtually charged with unequal quantities of positive and negative electricity, the algebraic sum of these charges constituting the observed electrification. It is manifest, however, that this way of regarding an electrified body is entirely artificial, and may be compared to the conception of the velocity of a body as compounded of two or more different velocities, no one of which is the actual velocity of the body. When we speak therefore of a body being charged with a quantity $e$ of electricity we mean simply that the body is electrified, and that the electrification is vitreous or resinous according as $e$ is positive or negative.

ON ELECTRIC DENSITY.

Distribution in Three Dimensions.

64.] Definition. The electric volume-density at a given point in space is the limiting ratio of the quantity of electricity within a sphere whose centre is the given point to the volume of the sphere, when its radius is diminished without limit.

We shall denote this ratio by the symbol $\rho$, which may be positive or negative.

Distribution on a Surface.

It is a result alike of theory and of experiment, that, in certain cases, the electrification of a body is entirely on the surface. The density at a point on the surface, if defined according to the method given above, would be infinite. We therefore adopt a different method for the measurement of surface-density.

Definition. The electric density at a given point on a surface is the limiting ratio of the quantity of electricity within a sphere whose centre is the given point to the area of the surface contained within the sphere, when its radius is diminished without limit.

We shall denote the surface-density by the symbol $\sigma$.

Those writers who supposed electricity to be a material fluid or a collection of particles, were obliged in this case to suppose the electricity distributed on the surface in the form of a stratum of a certain thickness $\theta$, its density being $\rho _{0}$ , or that value of $\rho$ which would result from the particles having the closest contact of which they are capable. It is manifest that on this theory

$\rho _{0}\,\theta =\sigma \,\!$

When $\sigma$ is negative, according to this theory, a certain stratum of thickness $\theta$ is left entirely devoid of positive electricity, and filled entirely with negative electricity.

There is, however, no experimental evidence either of the electric stratum having any thickness, or of electricity being a fluid or a collection of particles. We therefore prefer to do without the symbol for the thickness of the stratum, and to use a special symbol for surface-density.

Distribution along a Line.

It is sometimes convenient to suppose electricity distributed on a line, that is, a long narrow body of which we neglect the thickness. In this case we may define the line-density at any point to be the limiting ratio of the electricity on an element of the line to the length of that element when the element is diminished without limit.

If $\lambda$ denotes the line-density, then the whole quantity of electricity on a curve is $e=\int \lambda \,ds$, where dS is the element of the curve.

Similarly, if $\rho$ is the surface-density, the whole quantity of electricity on the surface is

$e=\iint \sigma \,dS$,

where dS is the element of surface.

If $\rho$ is the volume-density at any point of space, then the whole electricity within a certain volume is

$e=\iiint \rho \;dx\;dy\;dz$

where $dxdydz$ is the element of volume. The limits of integration in each case are those of the curve, the surface, or the portion of space considered.

It is manifest that e, $\lambda$, $\sigma$ and $\rho$ are quantities differing in kind, each being one dimension in space lower than the preceding, so that if $a$ be a line, the quantities e, $a\lambda ,a^{2}\sigma$ and $a^{3}\rho$ will be all of the same kind, and if a be the unit of length, and $\lambda ,\sigma ,\rho$ each the unit of the different kinds of density, $a\lambda ,a^{2}\sigma$ and $a^{3}\rho$will each denote one unit of electricity.

Definition of the Unit of Electricity.

65.] Let $A$ and $B$ be two points the distance between which is the unit of length. Let two bodies, whose dimensions are small compared with the distance $AB$, be charged with equal quantities of positive electricity and placed at $A$ and $B$ respectively, and let the charges be such that the force with which they repel each other is the unit of force, measured as in Art. 6. Then the charge of either body is said to be the unit of electricity. If the charge of the body at $B$ were a unit of negative electricity, then, since the action between the bodies would be reversed, we should have an attraction equal to the unit of force.

If the charge of $A$ were also negative, and equal to unity, the force would be repulsive, and equal to unity.

Since the action between any two portions of electricity is not affected by the presence of other portions, the repulsion between $e'$ units of electricity at $A$ and $e'$ units at $B$ is $ee'$, the distance $AB$ being unity. See Art. 39.

Law of Force between Electrified Bodies.

66.] Coulomb shewed by experiment that the force between electrified bodies whose dimensions are small compared with the distance between them, varies inversely as the square of the distance. Hence the actual repulsion between two such bodies charged with quantities $e$ and $e'$ and placed at a distance $r$ is

${\frac {ee'}{r^{2}}}$

We shall prove in Art. 74 that this law is the only one consistent with the observed fact that a conductor, placed in the inside of a closed hollow conductor and in contact with it, is deprived of all electrical charge. Our conviction of the accuracy of the law of the inverse square of the distance may be considered to rest on experiments of this kind, rather than on the direct measurements of Coulomb.

Resultant Force between Two Bodies.

67.] In order to find the resultant force between two bodies we might divide each of them into its elements of volume, and consider the repulsion between the electricity in each of the elements of the first body and the electricity in each of the elements of the second body. We should thus get a system of forces equal in number to the product of the numbers of the elements into which we have divided each body, and we should have to combine the effects of these forces by the rules of Statics. Thus, to find the component in the direction of $x$ we should have to find the value of the sextuple integral

$\iiint \!\!\!\iiint {\frac {\rho \rho '(x-x')\;dx\;dy\;dz\;dx'\;dy'\;dz'\;}{\{(x-x')^{2}+(y-y')^{2}+(z-z')^{2}\}^{3 \over 2}}}$,

where $x,y,z$ are the coordinates of a point in the first body at which the electrical density is $\rho$, and $x',y',z'$, and $\rho '$ are the corresponding quantities for the second body, and the integration is extended first over the one body and then over the other.

Resultant Force at a Point.

68.] In order to simplify the mathematical process, it is convenient to consider the action of an electrified body, not on another body of any form, but on an indefinitely small body, charged with an indefinitely small amount of electricity, and placed at any point of the space to which the electrical action extends. By making the charge of this body indefinitely small we render insensible its disturbing action on the charge of the first body.

Let $e$ be the charge of this body, and let the force acting on it when placed at the point $(x,y,z)$ be $Re$, and let the direction-cosines of the force be $l,m,n$, then we may call $R$ the resultant force at the point $(x,y,z)$.

In speaking of the resultant electrical force at a point, we do not necessarily imply that any force is actually exerted there, but only that if an electrified body were placed there it would be acted on by a force $Re$, where $e$ is the charge of the body.

Definition. The Resultant electrical force at any point is the force which would be exerted on a small body charged with the unit of positive electricity, if it were placed there without disturbing the actual distribution of electricity.

This force not only tends to move an electrified body, but to move the electricity within the body, so that the positive electricity tends to move in the direction of $R$ and the negative electricity in the opposite direction. Hence the force $R$ is also called the Electromotive Force at the point $(x,\,y,\,z)$.

When we wish to express the fact that the resultant force is a vector, we shall denote it by the German letter ${\mathfrak {C}}$. If the body is a dielectric, then, according to the theory adopted in this treatise, the electricity is displaced within it, so that the quantity of electricity which is forced in the direction of ${\mathfrak {C}}$ across unit of area fixed perpendicular to ${\mathfrak {C}}$ is

${\mathfrak {D}}={\frac {1}{4\,\pi }}\,K{\mathfrak {C}}$

where ${\mathfrak {D}}$ is the displacement, ${\mathfrak {C}}$ the resultant force, and $K$ the specific inductive capacity of the dielectric. For air, $K=1$.

If the body is a conductor, the state of constraint is continually giving way, so that a current of conduction is produced and maintained as long as the force ${\mathfrak {C}}$ acts on the medium.

Components of the Resultant Force.

If $X,\,Y,\,Z$ denote the components of R, then

$X=Rl,{\color {White}xxxx}Y=Rm,{\color {White}xxxx}Z=Rn;$

where $l,m,n$ are the direction-cosines of $R$.

Line-Integral of Electric Force, or Electromotive Force along an Arc of a Curve.

69.] The Electromotive force along a given arc $AP$ of a curve is numerically measured by the work which would be done on a unit of positive electricity carried along the curve from the beginning, $A$, to $P$, the end of the arc.

If $s$ is the length of the arc, measured from $A$, and if the resultant force $R$ at any point of the curve makes an angle $c$ with the tangent drawn in the positive direction, then the work done on unit of electricity in moving along the element of the curve ${ds}$ will be

$R\cos \epsilon \,ds$

,

and the total electromotive force $V$ will be

$V=\int _{0}^{s}R\cos \epsilon \,ds$,

the integration being extended from the beginning to the end of the arc.

If we make use of the components of the force $R$, we find

$V\int _{0}^{s}(X{\frac {dx}{dy}}+Y{\frac {dy}{ds}}+Z{\frac {dz}{ds}})\,ds.$.

If $X,Y,$ and $Z$ are such that $X{dx}+Y{dy}+Z{dz}$ is a complete differential of a function of$x,y,z,$ then

$V=\int _{A}^{P}(X\,dx+Y\,dy+Z\,dz)=V_{A}-V_{P}$;

where the integration is performed in any way from the point $A$ to the point $P$, whether along the given curve or along any other line between $A$ and $P$.

In this case $V$ is a scalar function of the position of a point in space, that is, when we know the coordinates of the point, the value of $V$ is determinate, and this value is independent of the position and direction of the axes of reference. See Art. 16.

On Functions of the Position of a Point.

In what follows, when we describe a quantity as a function of the position of a point, we mean that for every position of the point the function has a determinate value. We do not imply that this value can always be expressed by the same formula for all points of space, for it may be expressed by one formula on one side of a given surface and by another formula on the other side.

On Potential Functions.

70.] The quantity $Xdx+Ydy+Zdz$ is an exact differential whenever the force arises from attractions or repulsions whose in tensity is a function of the distance only from any number of points. For if $r_{1}$ be the distance of one of the points from the point $(x,y,z)$, and if $R_{1}$ be the repulsion, then

$X_{1}=R_{1}l=R_{1}{\frac {dr_{1}}{dx}}$,

with similar expressions for $Y_{1}$ and $Z_{1}$, so that

$X_{1}\,dx+Y_{1}\,dy+Z_{1}\,dz=R_{1}\,dr_{1}$;

and since $R_{l}$ is a function of $r_{l}$ only, $R_{l}dr_{1}$ is an exact differential of some function of $r_{1}$, say $V_{1}$.

Similarly for any other force $R_{2}$, acting from a centre at distance $r_{2}$,

$X_{2}\,dx+Y_{2}\,dy+Z_{2}\,dz=R_{2}\,dr_{2}=dV_{2}$.

But $X=X_{1}+X_{2}+{\mbox{etc. and }}Y$ and $Z$ are compounded in the same way, therefore

$X\,dx+Y\,dy+Z\,dz=dV_{1}+dV_{2}+\And \!\!c.=dV$.

$V$, the integral of this quantity, under the condition that $V=0$ at an infinite distance, is called the Potential Function.

The use of this function in the theory of attractions was introduced by Laplace in the calculation of the attraction of the earth. Green, in his essay 'On the Application of Mathematical Analysis to Electricity' gave it the name of the Potential Function. Gauss, working independently of Green, also used the word Potential. Clausius and others have applied the term Potential to the work which would be done if two bodies or systems were removed to an infinite distance from one another. We shall follow the use of the word in recent English works, and avoid ambiguity by adopting the following definition due to Sir W. Thomson.

Definition of Potential. The Potential at a Point is the work which would be done on a unit of positive electricity by the electric forces if it were placed at that point without disturbing the electric distribution, and carried from that point to an infinite distance.

71.] Expressions for the Resultant Force and its components in terms of the Potential.

Since the total electromotive force along any arc $AB$ is

$V_{A}-V_{B}\,\!$,

if we put $ds$ for the arc $AB$ we shall have for the force resolved in the direction of $ds$,

$R\,cos\epsilon \,=-{\frac {dV}{ds}}$;

whence, by assuming $ds$ parallel to each of the axes in succession, we get

$X=-{\frac {dV}{dx}},{\color {White}xxxx}Y=-{\frac {dV}{dy}},{\color {White}xxxx}Z=-{\frac {dV}{dx}}$;

$R={\left({\left.{\frac {\overline {dV}}{dx}}\right|}^{2}+{\left.{\frac {\overline {dV}}{dy}}\right|}^{2}+{\left.{\frac {\overline {dV}}{dz}}\right|}^{2}\right)}^{1 \over 2}$

We shall denote the force itself, whose magnitude is $R$ and whose components are $X,Y,Z$, by the German letter ${\mathfrak {C}}$, as in Arts. 17 and 68.

The Potential at all Points within a Conductor is the same.

72.] A conductor is a body which allows the electricity within it to move from one part of the body to any other when acted on by electromotive force. When the electricity is in equilibrium there can be no electromotive force acting within the conductor. Hence $R=0$ throughout the whole space occupied by the conductor. From this it follows that

${\frac {dV}{dx}}=0,{\color {White}xxx}{\frac {dV}{dy}}=0,{\color {White}xxx}{\frac {dV}{dz}}=0$;

and therefore for every point of the conductor

$V\;=\;C$,

where $C$ is a constant quantity.

Potential of a Conductor.

Since the potential at all points within the substance of the conductor is C, the quantity C is called the Potential of the conductor. C may be defined as the work which must be done by external agency in order to bring a unit of electricity from an infinite distance to the conductor, the distribution of electricity being supposed not to be disturbed by the presence of the unit.

If two conductors have equal potentials, and are connected by a wire so fine that the electricity on the wire itself may be neglected, the total electromotive force along the wire will be zero, and no electricity will pass from the one conductor to the other.

If the potentials of the conductors $A$ and $B$ be $V_{A}$ and $V_{B}$ then the electromotive force along any wire joining $A$ and $B$ will be

$V_{A}\;-\;V_{B}$

in the direction $AB$, that is, positive electricity will tend to pass from the conductor of higher potential to the other.

Potential, in electrical science, has the same relation to Electricity that Pressure, in Hydrostatics, has to Fluid, or that Temperature, in Thermodynamics, has to Heat. Electricity, Fluids, and Heat all tend to pass from one place to another, if the Potential, Pressure, or Temperature is greater in the first place than in the second. A fluid is certainly a substance, heat is as certainly not a substance, so that though we may find assistance from analogies of this kind in forming clear ideas of formal electrical relations, we must be careful not to let the one or the other analogy suggest to us that electricity is either a substance like water, or a state of agitation like heat.

Potential due to any Electrical System.

73.] Let there be a single electrified point charged with a quantity $e$ of electricity, and let $r$ be the distance of the point $x',y',z'$ from it, then

$V=\int _{r}^{\infty }R\,dr=\int _{r}^{\infty }{\frac {e}{r^{2}}}dr={\frac {e}{r}}$.

Let there be any number of electrified points whose coordinates are $(x_{1},y_{1},z_{1})$,$(x_{2},y_{2},z_{2})$ &c. - and their charges $e_{1},e_{2}$ &c., and let their distances from the point $(x',y',z')$ be $r_{1}$, $r_{2}$, &c., then the potential of the system at $x',y',z'$' will be

$V=\sum {\left({\frac {e}{r}}\right)}$.

Let the electric density at any point $(x,y,z)$ within an electrified body be p, then the potential due to the body is

$V=\iiint {\frac {\rho }{r}}dx\,dy\,dz$;

where

$r=\{(x-x')^{2}+(y-y')^{2}+(z-z')^{2}\}^{1 \over 2}$

the integration being extended throughout the body.

On the Proof of the Law of the Inverse Square.

74.] The fact that the force between electrified bodies is inversely as the square of the distance may be considered to be established by direct experiments with the torsion-balance. The results, however, which we derive from such experiments must be regarded as affected by an error depending on the probable error of each experiment, and unless the skill of the operator be very great, the probable error of an experiment with the torsion-balance is considerable. As an argument that the attraction is really, and not merely as a rough approximation, inversely as the square of the distance, Experiment VII (p. 34) is far more conclusive than any measurements of electrical forces can be.

In that experiment a conductor $B$, charged in any manner, was enclosed in a hollow conducting vessel $C$, which completely surrounded it. $C$ was also electrified in any manner.

$B$ was then placed in electric communication with $C$, and was then again insulated and removed from $C$ without touching it, and examined by means of an electroscope. In this way it was shewn that a conductor, if made to touch the inside of a conducting vessel which completely encloses it, becomes completely discharged, so that no trace of electrification can be discovered by the most delicate electrometer, however strongly the conductor or the vessel has been previously electrified.

The methods of detecting the electrification of a body are so delicate that a millionth part of the original electrification of $B$ could be observed if it existed. No experiments involving the direct measurement of forces can be brought to such a degree of accuracy.

It follows from this experiment that a non-electrified body in the inside of a hollow conductor is at the same potential as the hollow conductor, in whatever way that conductor is charged. For if it were not at the same potential, then, if it were put in electric connexion with the vessel, either by touching it or by means of a wire, electricity would pass from the one body to the other, and the conductor, when removed from the vessel, would be found to be electrified positively or negatively, which, as we have already stated, is not the case.

Hence the whole space inside a hollow conductor is at the same potential as the conductor if no electrified body is placed within it. If the law of the inverse square is true, this will be the case what ever be the form of the hollow conductor. Our object at present, however, is to ascertain from this fact the form of the law of attraction.

For this purpose let us suppose the hollow conductor to be a thin spherical shell. Since everything is symmetrical about its centre, the shell will be uniformly electrified at every point, and we have to enquire what must be the law of attraction of a uniform spherical shell, so as to fulfil the condition that the potential at every point within it shall be the same.

Let the force at a distance $r$ from a point at which a quantity $e$ of electricity is concentrated be $R$, where $R$ is some function of $r$. All central forces which are functions of the distance admit of a potential, let us write ${\tfrac {f(r)}{r}}$ for the potential function due to a unit of electricity at a distance $r$.

Let the radius of the spherical shell be $a$, and let the surface-density be $\sigma$. Let $P$ be any point within the shell at a distance $p$ from the centre. Take the radius through $P$ as the axis of spherical coordinates, and let $r$ be the distance from $P$ to an element $dS$ of the shell. Then the potential at $P$ is

$V=\iint \sigma \,{\frac {f(r)}{r}}dS$,

$V=\int _{0}^{2\pi }\int _{0}^{\pi }\sigma \,{\frac {f(r)}{r}}\,a^{2}\,sin\theta \,d\theta \,d\phi$.

Now

$r^{2}=a^{2}-2ap\,cos\theta +p^{2}$,

{{{2}}}

$V=2\pi \sigma {\frac {a}{p}}\int _{a-p}^{a+p}f(r)dr$;

and V must be constant for all values of $p$ less than $a$.

Multiplying both sides by $p$ and differentiating with respect to $p$,

$V=2\pi \sigma a{f(a+p)+f(a-p)}\,\!$.

Differentiating again with respect to $p$,

$0=f'(a+p)-f'(a-p)\,\!$

, Since a and p are independent,

$f'(r)=C\,\!$, a constant.

Hence

$f(r)=Cr+C'\,\!$

and the potential function is

${\frac {f(r)}{r}}=C+{\frac {C'}{r}}$

The force at distance $r$ is got by differentiating this expression with respect to $r$, and changing the sign, so that

$R={\frac {C'}{r^{2}}}$

or the force is inversely as the square of the distance, and this therefore is the only law of force which satisfies the condition that the potential within a uniform spherical shell is constant^[1]. Now this condition is shewn to be fulfilled by the electric forces with the most perfect accuracy. Hence the law of electric force is verified to a corresponding degree of accuracy.

Surface-Integral of Electric Induction, and Electric Displacement through a Surface.

75.] Let $R$ be the resultant force at any point of the surface, and $\epsilon$ the angle which R makes with the normal drawn towards the positive side of the surface, then $Rcos\epsilon$ is the component of the force normal to the surface, and if $dS$ is the element of the surface, the electric displacement through $dS$ will be, by Art. 68,

${\frac {1}{4\pi }}\,KR\,cos\epsilon \,dS$

Since we do not at present consider any dielectric except air, $K=1$ .

We may, however, avoid introducing at this stage the theory of electric displacement, by calling $Rcos\epsilon dS$ the Induction through the element $dS$. This quantity is well known in mathematical physics, but the name of induction is borrowed from Faraday. The surface-integral of induction is

$\iint Rcos\epsilon dS$;

and it appears by Art. 21, that if $X,Y,Z$ are the components of $R$, and if these quantities are continuous within a region bounded by a closed surface $S$, the induction reckoned from within outwards is

$\iint Rcos\epsilon dS=\iiint \left({\frac {}{}}+{\frac {}{}}+{\frac {}{}}\right)dx\,dy\,dz$,

the integration being extended through the whole space within the surface.

Induction through a Finite Closed Surface due to a Single Centre of Force.

76.] Let a quantity e of electricity be supposed to be placed at a point $0$, and let $r$ be the distance of any point $P$ from $0$, the force at that point is $R={\frac {e}{r^{2}}}$ in the direction $OP$.

Let a line be drawn from $O$ in any direction to an infinite distance. If $O$ is without the closed surface this line will either not cut the surface at all, or it will issue from the surface as many times as it enters. If $O$ is within the surface the line must first issue from the surface, and then it may enter and issue any number of times alternately, ending- by issuing from it.

Let $\epsilon$ be the angle between $OP$ and the normal to the surface drawn outwards where $OP$ cuts it, then where the line issues from the surface $cos\epsilon$ will be positive, and where it enters $cos\epsilon$ will be negative.

Now let a sphere be described with centre $O$ and radius unity, and let the line $OP$ describe a conical surface of small angular aperture about $O$ as vertex.

This cone will cut off a small element $d\omega$ from the surface of the sphere, and small elements $dS_{l}$, $dS_{2}$, &c. from the closed surface at the various places where the line $OP$ intersects it.

Then, since any one of these elements $dS$ intersects the cone at a distance $r$ from the vertex and at an obliquity $\epsilon$,

$dS=r^{2}\,\sec \epsilon \,d\omega$;

and, since $R=er^{-2}$, we shall have

$R\,\cos \epsilon \,dS=\pm e\,d\omega$;

the positive sign being taken when $r$ issues from the surface, and the negative where it enters it.

If the point $O$ is without the closed surface, the positive values are equal in number to the negative ones, so that for any direction of $r$,

$\sum R\,\cos \epsilon \;dS=0\,\!$,

and therefore

$\iint R\,\cos \epsilon \,dS=0$,

the integration being extended over the whole closed surface.

If the point $O$ is within the closed surface the radius vector $OP$ first issues from the closed surface, giving a positive value of $ed\omega$, and then has an equal number of entrances and issues, so that in this case

$\sum R\cos \epsilon \,dS=e\,d\omega$

.

Extending the integration over the whole closed surface, we shall include the whole of the spherical surface, the area of which is $4\pi$, so that

$\iint R\,cos\epsilon \,dS=e\,\iint d\omega =4\pi \,e$.

Hence we conclude that the total induction outwards through a closed surface due to a centre of force $e$ placed at a point is zero when is without the surface, and $4\pi e$ when $O$ is within the surface.

Since in air the displacement is equal to the induction divided by $4\pi$, the displacement through a closed surface, reckoned outwards, is equal to the electricity within the surface.

Corollary. It also follows that if the surface is not closed but is bounded by a given closed curve, the total induction through it is $\omega$, where $\omega$ is the solid angle subtended by the closed curve at $0$. This quantity, therefore, depends only on the closed curve, and not on the form of the surface of which it is the boundary.

On the Equations of Laplace and Poisson.

77.] Since the value of the total induction of a single centre of force through a closed surface depends only on whether the centre is within the surface or not, and does not depend on its position in any other way, if there are a number of such centres $e_{l}$, $e_{2}$ , &c. within the surface, and $e_{1}'$, $e_{2}'$, &c. without the surface, we shall have

$\iint R\,\cos \epsilon \,dS=\,4\pi e$;

where $e$ denotes the algebraical sum of the quantities of electricity at all the centres of force within the closed surface, that is, the total electricity within the surface, resinous electricity being reckoned negative.

If the electricity is so distributed within the surface that the density is nowhere infinite, we shall have by Art. 64,

$4\pi \,e=4\pi \iiint \rho \,dx\,dy\,dz$

and by Art. 75,

$\iint R\,\cos \epsilon \,dS=\iiint \left({\frac {dX}{dx}}+{\frac {dY}{dy}}+{\frac {dZ}{dz}}\right)\,dx\,dy\,dz$.

If we take as the closed surface that of the element of volume $dx$ $dy$ $dz$, we shall have, by equating these expressions,

${\frac {dX}{dx}}+{\frac {dY}{dy}}+{\frac {dZ}{dz}}=4\pi \,\rho$;

and if a potential $V$ exists, we find by Art. 71 ,

${\frac {d^{2}V}{dx^{2}}}+{\frac {d^{2}V}{dy^{2}}}+{\frac {d^{2}V}{dz^{2}}}+4\pi \,\rho =0$

This equation, in the case in which the density is zero, is called Laplace's Equation. In its more general form it was first given by Poisson. It enables us, when we know the potential at every point, to determine the distribution of electricity.

We shall denote, as at Art. 26, the quantity

${\frac {d^{2}V}{dx^{2}}}+{\frac {d^{2}V}{dy^{2}}}+{\frac {d^{2}V}{dz^{2}}}$ by $-\nabla ^{2}V$,

and we may express Poisson's equation in words by saying that the electric density multiplied by $4\pi$ is the concentration of the potential. Where there is no electrification, the potential . has no concentration, and this is the interpretation of Laplace's equation.

If we suppose that in the superficial and linear distributions of electricity the volume-density $\rho$ remains finite, and that the electricity exists in the form of a thin stratum or narrow fibre, then, by increasing $\rho$ and diminishing the depth of the stratum or the section of the fibre, we may approach the limit of true superficial or linear distribution, and the equation being true throughout the process will remain true at the limit, if interpreted in accordance with the actual circumstances.

On the Conditions to be fulfilled at an Electrified Surface.

78.] We shall consider the electrified surface as the limit to which an electrified stratum of density $\rho$ and thickness $v$ approaches when $\rho$ is increased and $v$ diminished without limit, the product $\rho \,v$ being always finite and equal to $\sigma$ the surface-density.

Let the stratum be that included between the surfaces

$F(x,\,y,\,z)=\,F\,=\,a$

(1)

and

and $F=a\,+\,h$

(2)

If we put

$R^{2}={\left.{\frac {\overline {dV}}{dy}}\right|}^{2}+{\left.{\frac {\overline {dV}}{dy}}\right|}^{2}+{\left.{\frac {\overline {dV}}{dz}}\right|}^{2}$,

(3)

and if $l,m,n$ are the direction-cosines of the normal to the surface,

$Rl={\frac {dF}{dx}},{\color {White}xxxxx}Rm={\frac {dF}{dy}},{\color {White}xxxxx}Rn={\frac {dF}{dz}}$

(4)

Now let $V_{1}$ be the value of the potential on the negative side of the surface $F=a,V'$ its value between the surfaces $F=a$ and $F=a+h$, and $V_{2}$ its value on the positive side of $F=a+h$.

Also, let $\rho _{1},\rho '$, and $\rho _{2}$ be the values of the density in these three portions of space. Then, since the density is everywhere finite, the second derivatives of $V$ are everywhere finite, and the first derivatives, and also the function itself, are everywhere continuous and finite.

At any point of the surface $F=a$ let a normal be drawn of length $v$, till it meets the surface $F=a+h$, then the value of $F$ at the extremity of the normal is

$a+v\left(l{\frac {dF}{dx}}+m{\frac {dF}{dy}}+n{\frac {dF}{dz}}\right)$+&c.,

(5)

or

$a+h=a+vR+\,\!$&c.

(6)

The value of $V$ at the same point is

$V_{2}=V_{1}+v\left(l{\frac {dV'}{dx}}+m{\frac {dV'}{dy}}+n{\frac {dV'}{dz}}\right)+$ &c,

(7)

or

$V_{2}-V_{1}={\frac {h}{R}}{\frac {dV'}{dv}}+$ &c.

(8)

Since the first derivatives of $V$ continue always finite, the second side of the equation vanishes when $h$ is diminished without limit, and therefore if $V_{2}$ and $V_{1}$ denote the values of $V$ on the outside and inside of an electrified surface at the point $x,y,z,$

$V_{1}\,=\,V_{2}$

(9)

If $x+dx$, $y+dy$, $z+dz$ be the coordinates of another point on the electrified surface, $F=a$ and $F=a+h$ at this point also ; whence

$0={\frac {dF}{dx}}dx+{\frac {dF}{dy}}dy+{\frac {dF}{dz}}dz+$ &c.,

(10)

$0=\left({\frac {dV_{2}}{dx}}-{\frac {dV_{1}}{dx}}\right)dx+\left({\frac {dV_{2}}{dy}}-{\frac {dV_{1}}{dy}}\right)dy+\left({\frac {dV_{2}}{dz}}-{\frac {dV_{1}}{dz}}\right)dz+$ &c.;

(11)

and when $dx$, $dy$, $dz$ vanish, we find the conditions

$\left.{\begin{matrix}{\dfrac {dV_{2}}{dx}}-{\dfrac {dV_{1}}{dx}}=Cl,\\\\{\dfrac {dV_{2}}{dy}}-{\dfrac {dV_{1}}{dy}}=Cm\\\\{\dfrac {dV_{2}}{dz}}-{\dfrac {dV_{1}}{dz}}=Cn\end{matrix}}\right\}$

(12)

where $C$ is a quantity to be determined.

Next, let us consider the variation of $F$ and ${\frac {dV}{dx}}$ along the ordinate parallel to $x$ between the surfaces $F=a$ and $F=a+h$.

We have

$F=a+{\frac {dF}{dx}}dx+{1 \over 2}{\frac {d^{2}F}{dx^{2}}}(dx)^{2}+$ &c,

(13)

and

${\frac {dV}{dx}}={\frac {dV'}{dx}}+{\frac {d^{2}V'}{dx^{2}}}dx+{\tfrac {1}{2}}{\frac {d^{3}V'}{dx^{3}}}(dx)^{2}+$ &c.,

(14)

Hence, at the second surface, where $F=a+h$, and $V$ becomes $V_{2}$,

${\frac {dV_{2}}{dx}}={\frac {dV_{1}}{dx}}+{\frac {d^{2}V'}{dx^{2}}}dx+$ &c.;

(15)

whence

${\frac {d^{2}V'}{dx^{2}}}dx+{\mbox{etc.}}=Cl$,

(16)

by the first of equations (12).

Multiplying by $Rl$, and remembering that at the second surface

$Rl\,dx=h$

(17)

we find

${\frac {d^{2}V'}{dx^{2}}}h=CR\,l^{2}$.

(18)

Similarly

${\frac {d^{2}V'}{dy^{2}}}h=CR\,m^{2}$

(19)

and

${\frac {d^{2}V'}{dz^{2}}}h=CR\,n^{2}$

(20)

Adding

$\left({\frac {d^{2}V'}{dx^{2}}}+{\frac {d^{2}V'}{dy^{2}}}+{\frac {d^{2}V'}{dz^{2}}}\right)\,h=CR$;

(21)

but

${\frac {d^{2}V'}{dx^{2}}}+{\frac {d^{2}V'}{dy^{2}}}+{\frac {d^{2}V'}{dz^{2}}}=-4\pi \rho ^{'}$ and $h=vR$;

(22)

hence

$C=-4\pi \rho ^{'}v\,\!=-4\pi \sigma$;

(23)

where $\sigma$ is the surface-density; or, multiplying the equations (12) by $l,m,n$ respectively, and adding,

$l\left({\frac {dV_{2}}{dx}}-{\frac {dV_{1}}{dx}}\right)++m\left({\frac {dV_{2}}{dy}}-{\frac {dV_{1}}{dy}}\right)+n\left({\frac {dV_{2}}{dz}}-{\frac {dV_{1}}{dz}}\right)+4\pi \sigma =0$.

(24)

This equation is called the characteristic equation of $V$ at a surface. This equation may also be written

${\frac {dV_{1}}{dv_{1}}}+{\frac {dV_{2}}{dv_{2}}}+4\pi \sigma =0$;

(25)

where $v_{1},v_{2}$ are the normals to the surface drawn towards the first and the second medium respectively, and $V_{1},V_{2}$ the potentials at points on these normals. We may also write it

$R_{2}\cos \epsilon _{2}+R_{1}\cos \epsilon _{1}+4\pi \sigma =0\,\!$;

(26)

where $R_{1},R_{2}$ are the resultant forces, and $\epsilon _{1},\epsilon _{2}$ the angles which they make with the normals drawn from the surface on either side.

79.] Let us next determine the total mechanical force acting on an element of the electrified surface.

The general expression for the force parallel to $x$ on an element whose volume is $dx\,dy\,dz$, and volume-density $\rho$, is

$dX=-{\frac {dV}{dx}}\rho \;dz\,dy\,dz$.

(27)

In the present case we have for any point on the normal $v$

${\frac {dV}{dx}}={\frac {dV_{1}}{dx}}+v{\frac {d^{2}V_{1}}{dx^{2}}}+\And \!\!\!{\text{c.}}$;

(28)

also, if the element of surface is $dS$, that of the volume of the element of the stratum may be written $dS\,dv$; and if $X$ is the whole force on a stratum of thickness $v$,

$X=-\iiint \left({\frac {dV_{1}}{dx}}+v{\frac {d^{2}V_{1}}{dx^{2}}}+etc.\right)\rho '\,dS\,dv$.

(29)

Integrating with respect to $v$, we find

$x=-\iint \rho '\,dS\left(v{\frac {dV_{1}}{dx}}+{\frac {v^{2}}{2}}{\frac {d^{2}V_{1}}{dx^{2}}}+\And \!\!\!{\text{c.}}\right)$

(30)

or, since

${\frac {dV_{2}}{dx}}={\frac {dV_{1}}{dx}}+v{\frac {d^{2}V_{1}}{dx^{2}}}+\And \!\!\!{\text{c.}}$;

(31)

$X=-\iint {1 \over 2}\,\rho '\,v\,dS\left({\frac {dV_{1}}{dx}}+{\frac {dV_{2}}{dx}}\right)+\And \!\!\!{\text{c.}}$;

(32)

When $v$ is diminished and $\rho '$ increased without limit, the product $\rho 'v$ remaining always constant and equal to $\sigma$, the expression for the force in the direction of $x$ on the electricity $\sigma \,dS$ on the element of surface $dS$ is

$X=-\sigma \,dS\,{1 \over 2}\left({\frac {dV_{1}}{dx}}+{\frac {dV_{2}}{dx}}\right)$;

(33)

that is, the force acting on the electrified element $\sigma \,dS$ in any given direction is the arithmetic mean of the forces acting on equal quantities of electricity placed one just inside the surface and the other just outside the surface close to the actual position of the element, and therefore the resultant mechanical force on the electrified element is equal to the resultant of the forces which would act on two portions of electricity, each equal to half that on the element, and placed one on each side of the surface and infinitely near to it.

80.] When a conductor is in electrical equilibrium, the whole of the electricity is on the surface.

We have already shewn that throughout the substance of the conductor the potential $V$ is constant. Hence $\nabla ^{2}V$ is zero, and therefore by Poisson's equation, $\rho$ is zero throughout the substance of the conductor, and there can be no electricity in the interior of the conductor.

Hence a superficial distribution of electricity is the only possible one in the case of conductors in equilibrium. A distribution throughout the mass can only exist in equilibrium when the body is a non-conductor.

Since the resultant force within a conductor is zero, the resultant force just outside the conductor is along the normal and is equal to $4\pi \sigma$, acting outwards from the conductor.

81.] If we now suppose an elongated body to be electrified, we may, by diminishing its lateral dimensions, arrive at the conception of an electrified line.

Let $ds$ be the length of a small portion of the elongated body, and let $c$ be its circumference, and $\sigma$ the superficial density of the electricity on its surface; then, if $\lambda$ is the electricity per unit of length, $\lambda =c\sigma$, and the resultant electrical force close to the surface will be

$4\pi \sigma =4\pi {\frac {\lambda }{c}}$

If, while $\lambda$ remains finite,$c$ be diminished indefinitely, the force at the surface will be increased indefinitely. Now in every dielectric there is a limit beyond which the force cannot be increased without a disruptive discharge. Hence a distribution of electricity in which a finite quantity is placed on a finite portion of a line is inconsistent with the conditions existing in nature.

Even if an insulator could be found such that no discharge could be driven through it by an infinite force, it would be impossible to charge a linear conductor with a finite quantity of electricity, for an infinite electromotive force would be required to bring the electricity to the linear conductor.

In the same way it may be shewn that a point charged with a finite quantity of electricity cannot exist in nature. It is convenient, however, in certain cases, to speak of electrified lines and points, and we may suppose these represented by electrified wires, and by small bodies of which the dimensions are negligible com pared with the principal distances concerned.

Since the quantity of electricity on any given portion of a wire diminishes indefinitely when the diameter of the wire is indefinitely diminished, the distribution of electricity on bodies of considerable dimensions will not be sensibly affected by the introduction of very fine metallic wires into the field, so as to form electrical connexions between these bodies and the earth, an electrical machine, or an electrometer.

On Lines of Force.

82.] If a line be drawn whose direction at every point of its course coincides with that of the resultant force at that point, the line is called a Line of Force.

If lines of force be drawn from every point of a line they will form a surface such that the force at any point is parallel to the tangent plane at that point. The surface-integral of the force with respect to this surface or any part of it will therefore be zero.

If lines of force are drawn from every point of a closed curve $L_{l}$ they will form a tubular surface $S_{0}$. Let the surface $S_{1}$ bounded by the closed curve $L_{1}$ be a section of this tube, and let $S_{2}$ be any other section of the tube. Let $Q_{0},Q_{1},Q_{2}$ be the surface-integrals over $S_{0},S_{1},S_{2}$, then, since the three surfaces completely enclose a space in which there is no attracting matter, we have

$Q_{0}\,+Q_{1}\,+Q_{2}\,=\,0.$

But $Q_{0}=0$, therefore $Q_{2}=-Q_{1}$, or the surface-integral over the second section is equal and opposite to that over the first: but since the directions of the normal are opposite in the two cases, we may say that the surface-integrals of the two sections are equal, the direction of the line of force being supposed positive in both.

Such a tube is called a Solenoid^[2], and such a distribution of force is called a Solenoidal distribution. The velocities of an in compressible fluid are distributed in this manner.

If we suppose any surface divided into elementary portions such that the surface-integral of each element is unity, and if solenoids are drawn through the field of force having these elements for their bases, then the surface-integral for any other surface will be re presented by the number of solenoids which it cuts. It is in this sense that Faraday uses his conception of lines of force to indicate not only the direction but the amount of the force at any place in the field.

We have used the phrase Lines of Force because it has been used by Faraday and others. In strictness, however, these lines should be called Lines of Electric Induction.

In the ordinary cases the lines of induction indicate the direction and magnitude of the resultant electromotive force at every point, because the force and the induction are in the same direction and in a constant ratio. There are other cases, however, in which it is important to remember that these lines indicate the induction, and that the force is indicated by the equipotential surfaces, being normal to these surfaces and inversely proportional to the distances of consecutive surfaces.

On Specific Inductive Capacity.

83.] In the preceding investigation of surface-integrals I have adopted the ordinary conception of direct action at a distance, and have not taken into consideration any effects depending on the nature of the dielectric medium in which the forces are observed.

But Faraday has observed that the quantity of electricity induced by a given electromotive force on the surface of a conductor which bounds a dielectric is not the same for all dielectrics. The induced electricity is greater for most solid and liquid dielectrics than for air and gases. Hence these bodies are said to have a greater specific inductive capacity than air, which is the standard medium.

We may express the theory of Faraday in mathematical language by saying that in a dielectric medium the induction across any surface is the product of the normal electric force into the coefficient of specific inductive capacity of that medium. If we denote this coefficient by $K$, then in every part of the investigation of surface-integrals we must multiply $X$, $Y$, and $Z$ by $K$, so that the equation of Poisson will become

${\frac {d}{dx}}\cdot \,K{\frac {dV}{dx}}+{\frac {d}{dy}}\cdot \,K{\frac {dV}{dy}}+{\frac {d}{dz}}\cdot \,K{\frac {dV}{dz}}+4\pi \,\rho =0$.

At the surface of separation of two media whose inductive capacities are $K_{1}$ and $K_{2}$ , and in which the potentials are $V_{1}$ and $V_{2}$ the characteristic equation may be written

$K_{2}{\frac {dV_{2}}{dv}}-K_{1}{\frac {dV_{1}}{dv}}+4\pi \,\rho =0$;

where $v$ is the normal drawn from the first medium to the second, and $\sigma$ is the true surface-density on the surface of separation; that is to say, the quantity of electricity which is actually on the surface in the form of a charge, and which can be altered only by conveying electricity to or from the spot. This true electrification must be distinguished from the apparent electrification $\rho '$, which is the electrification as deduced from the electrical forces in the neighbourhood of the surface, using the ordinary characteristic equation

${\frac {dV_{2}}{dv}}-{\frac {dV_{1}}{dv}}+4\pi \,\rho '=0$.

If a solid dielectric of any form is a perfect insulator, and if its surface receives no charge, then the true electrification remains zero, whatever be the electrical forces acting on it.

Hence

${\frac {dV_{2}}{dv}}={\frac {K_{1}}{K_{2}}}{\frac {dV_{1}}{dv}}$, and ${\frac {K_{1}-K_{2}}{K_{2}}}{\frac {dV_{1}}{dv}}+4\pi \rho '=0$,

${\frac {dV_{1}}{dv}}={\frac {4\pi \rho 'K_{2}}{K_{1}-K_{2}}}$, ${\color {White}xxxxx}{\frac {dV_{2}}{dv}}={\frac {4\pi \rho 'K_{1}}{K_{1}-K_{2}}}$.

The surface-density $\sigma '$ is that of the apparent electrification produced at the surface of the solid dielectric by induction. It disappears entirely when the inducing force is removed, but if during the action of the inducing force the apparent electrification of the surface is discharged by passing a flame over the surface, then, when the inducing force is taken away, there will appear an electrification opposite to $\sigma '$ ^[3].

In a heterogeneous dielectric in which $K$ varies continuously, if $\rho '$ be the apparent volume-density,

${\frac {d^{2}V}{dx^{2}}}+{\frac {d^{2}V}{dy^{2}}}+{\frac {d^{2}V}{dz^{2}}}+4\pi \rho '=0$

Comparing this with the equation above, we find

$4\pi (\rho -K\rho ')+{\frac {dK}{dx}}{\frac {dV}{dx}}+{\frac {dK}{dy}}{\frac {dV}{dy}}+{\frac {dK}{dz}}{\frac {dV}{dz}}=0$

The true electrification, indicated by $\rho$, in the dielectric whose variable inductive capacity is denoted by $K$, will produce the same potential at every point as the apparent electrification, indicated by $\rho '$, would produce in a dielectric whose inductive capacity is every where equal to unity.

CHAPTER III.

SYSTEMS OF CONDUCTORS.

On the Superposition of Electrical Systems.

84.] Let $E_{l}$ be a given electrified system of which the potential at a point $P$ is $V_{1}$, and let $E_{2}$ be another electrified system of which the potential at the same point would be $V_{2}$ if $E_{l}$ did not exist. Then, if $E_{1}$ and $E_{2}$ exist together, the potential of the combined system will be $V_{1}+V_{2}$.

Hence, if $V$ be the potential of an electrified system $E$, if the electrification of every part of $E$ be increased in the ratio of $n$ to 1 , the potential of the new system $nE$ will be $nV$.

Energy of an Electrified System.

85.] Let the system be divided into parts, $A_{1}$, $A_{2}$ , &c. so small that the potential in each part may be considered constant through out its extent. Let $e_{l}$ ,$e_{2}$ , &c. be the quantities of electricity in each of these parts, and let $V_{1}$, $V_{2}$ &c. be their potentials.

If now $e_{1}$ is altered to $ne_{1}$, $e_{2}$ to $ne_{2}$, &c., then the potentials will become $nV_{1}$, $nV_{2}$, &c.

Let us consider the effect of changing $n$ into $n+dn$ in all these expressions. It will be equivalent to charging $A_{1}$ with a quantity of electricity $e_{l}dn$, $A_{2}$ with $e_{2}dn$, &c. These charges must be supposed to be brought from a distance at which the electrical action of the system is insensible. The work done in bringing $e_{1}dn$ of electricity to $A_{1}$, whose potential before the charge is $nV_{1}$, and after the charge $(n+dn)V_{1}$, lf must lie between

$nV_{1}e_{1}\,dn\,\!$ and $(n+dn)V_{1}e_{1}\,dn\,\!$.

In the limit we may neglect the square of $dn$, and write the expression

$V_{1}e_{1}n\,dn\,\!$

---

1. See Pratt's Mechanical Philosophy, p. 144.
2. From $\sigma \omega \lambda \eta \nu$, a tube. Faraday uses (3271) the term Sphondyloid in the same sense.
3. See Faraday's 'Remarks on Static Induction', Proceedings of the Royal Institution, Feb. 12, 1858.