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

by James Clerk Maxwell

CHAPTER VII.

CONDUCTION IN THREE DIMENSIONS.

Notation of Electric Currents.

285.] At any point let an element of area $dS$ be taken normal to the axis of $x$, and let $Q$ units of electricity pass across this area from the negative to the positive side in unit of time, then, if ${\frac {Q}{dS}}$ becomes ultimately equal to $u$ when $dS$ is indefinitely diminished, $u$ is said to be the Component of the electric current in the direction of $x$ at the given point.

In the same way we may determine $v$ and $w$, the components of the current in the directions of $y$ and $z$ respectively.

286.] To determine the component of the current in any other direction $OR$ through the given point $O$.

Fig. 22.

Let $l,m,n$ be the direction-cosines of $OR$, then cutting off from the axes of $x,y,z$ portions equal to

${\frac {r}{l}},{\frac {r}{m}},\;\;{\mbox{ and }}\;\;{\frac {r}{n}}$

respectively at $A,B,\mathrm {and} C,$ the triangle $ABC$ will be normal to $OR.$

The area of this triangle $ABC$ will be

$ds={\frac {1}{2}}{\frac {r^{2}}{lmn}}$

and by diminishing $r$ this area may be diminished without limit.

The quantity of electricity which leaves the tetrahedron $ABCO$ by the triangle $ABC$ must be equal to that which enters it through the three triangles $OBC,OCA,$ and $OAB.$

The area of the triangle $OBC$ is ${\frac {1}{2}}{\frac {r^{2}}{mn}},$ and the component of the current normal to its plane is $u,$ so that the quantity which enters through this triangle is ${\frac {1}{2}}r^{2}{\frac {u}{mn}}.$

The quantities which enter through the triangles $OCA$ and $OAB$ respectively are

${\frac {1}{2}}r^{2}{\frac {v}{nl}},\quad \quad {\mbox{and}}\quad \quad {\frac {1}{2}}r^{2}{\frac {w}{lm}}.$

If $\gamma$ is the component of the velocity in the direction $OR,$ then the quantity which leaves the tetrahedron through $ABC$ is

${\frac {1}{2}}r{\frac {\gamma }{lmn}}.$

Since this is equal to the quantity which enters through the three other triangles,

${\frac {1}{2}}{\frac {r^{2}\gamma }{lmn}}={\frac {1}{2}}r^{2}\left\lbrace {\frac {u}{mn}}+{\frac {v}{nl}}+{\frac {w}{lm}}\right\rbrace ;$

multiplying by ${\frac {2lmn}{r^{2}}},$ we get

$\gamma =lu+mv+nw$

(1)

If we put

$u^{2}+v^{2}+w^{2}=\Gamma ^{2}$

and make $l',m',n'$ such that

$u=l'\Gamma ,\quad v=m'\Gamma ,\quad {\mbox{and}}\quad w=n'\Gamma ;$

then

$\gamma =\Gamma (ll'+mm'+nn').$

(2)

Hence, if we define the resultant current as a vector whose magnitude is $\Gamma ,$ and whose direction-cosines are $l',m',n'$ and if $\gamma$ denotes the current resolved in a direction making an angle $\theta$ with that of the resultant current, then

$\gamma =\Gamma \cos \theta ;$

(3)

shewing that the law of resolution of currents is the same as that of velocities, forces, and all other vectors.

287.] To determine the condition that a given surface may be a surface of flow.

Let

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

(4)

be the equation of a family of surfaces any one of which is given by making $\lambda$ constant, then, if we make

$\left.{\frac {\overline {d\lambda }}{dx}}\right|^{2}+\left.{\frac {\overline {d\lambda }}{dy}}\right|^{2}+\left.{\frac {\overline {d\lambda }}{dz}}\right|^{2}={\frac {1}{N^{2}}},$

(5)

the direction-cosines of the normal, reckoned in the direction in which $\lambda$ increases, are

$l=N{\frac {d\lambda }{dx}},\quad \quad m=N{\frac {d\lambda }{dy}},\quad \quad n=N{\frac {d\lambda }{dz}}.$

(6)

Hence, if $\gamma$ is the component of the current normal to the surface,

$\gamma =N\left\{u{\frac {d\lambda }{dx}}+v{\frac {d\lambda }{dy}}+w{\frac {d\lambda }{dz}}\right\}.$

(7)

If $\gamma =0$ there will be no current through the surface, and the surface may be called a Surface of Flow, because the lines of motion are in the surface.

288.] The equation of a surface of flow is therefore

$u{\frac {d\lambda }{dx}}+v{\frac {d\lambda }{dy}}+w{\frac {d\lambda }{dz}}=0$

(8)

If this equation is true for all values of $\lambda ,$ all the surfaces of the family will be surfaces of flow.

289.] Let there be another family of surfaces, whose parameter is $\lambda ',$ then, if these are also surfaces of flow, we shall have

$u{\frac {d\lambda '}{dx}}+v{\frac {d\lambda '}{dy}}+w{\frac {d\lambda '}{dz}}=0$

(9)

If there is a third family of surfaces of flow, whose parameter is $\lambda '',$ then

$u{\frac {d\lambda ''}{dx}}+v{\frac {d\lambda ''}{dy}}+w{\frac {d\lambda ''}{dz}}=0$

(10)

Eliminating between these three equations, $u,v,$ and $w$ disappear together, and we find

${\begin{vmatrix}{\frac {d\lambda }{dx}},&{\frac {d\lambda }{dy}},&{\frac {d\lambda }{dz}}\\{\frac {d\lambda '}{dx}},&{\frac {d\lambda '}{dy}},&{\frac {d\lambda '}{dz}}\\{\frac {d\lambda ''}{dx}},&{\frac {d\lambda ''}{dy}},&{\frac {d\lambda ''}{dz}}\end{vmatrix}}=0;$

(11)

$\mathrm {or} \quad \quad \lambda ''=\phi (\lambda ,\lambda ');$

(12)

that is, $\lambda ''$ is some function of $\lambda$ and $\lambda '.$

290.] Now consider the four surfaces whose parameters are $\lambda ,\lambda +\delta \lambda ,\lambda ',$ and $\lambda '+\delta \lambda '.$ These four surfaces enclose a quadrilateral tube, which we may call the tube $\delta \lambda \cdot \delta \lambda '.$ Since this tube is bounded by surfaces across which there is no flow, we may call it a Tube of Flow. If we take any two sections across the tube, the quantity which enters the tube at one section must be equal to the quantity which leaves it at the other, and since this quantity is therefore the same for every section of the tube, let us call it $L\delta \lambda \cdot \delta \lambda '$ where $L$ is a function of $\lambda$ and $\lambda ',$ the parameters which determine the particular tube.

291.] If $dS$ denotes the section of a tube of flow by a plane normal to $x,$ we have by the theory of the change of the independent variables,

$\delta \lambda \cdot \delta \lambda '=\delta S\left({\frac {d\lambda }{dy}}{\frac {d\lambda '}{dz}}-{\frac {d\lambda }{dz}}{\frac {d\lambda '}{dy}}\right),$

(13)

and by the definition of the components of the current

$u\,\delta S=L\,\delta \lambda \cdot \delta \lambda '.$

(14)

$\left.{\begin{matrix}{\mbox{Hence}}&&u=L\left({\frac {d\lambda }{dy}}{\frac {d\lambda '}{dz}}-{\frac {d\lambda }{dz}}{\frac {d\lambda '}{dy}}\right).\\{\mbox{Similarly}}&&v=L\left({\frac {d\lambda }{dz}}{\frac {d\lambda '}{dx}}-{\frac {d\lambda }{dx}}{\frac {d\lambda '}{dz}}\right),\\&&w=L\left({\frac {d\lambda }{dx}}{\frac {d\lambda '}{dy}}-{\frac {d\lambda }{dy}}{\frac {d\lambda '}{dx}}\right).\end{matrix}}\right\}$

(15)

292.] It is always possible when one of the functions $\lambda$ or $\lambda$ is known, to determine the other so that $L$ may be equal to unity. For instance, let us take the plane of $yz,$ and draw upon it a series of equidistant lines parallel to $y,$ to represent the sections of the family $\lambda '$ by this plane. In other words, let the function $\lambda '$ be determined by the condition that when $x=0\;\lambda '=z.$ If we then make $L=1,$ and therefore (when $x=0$)

$\lambda =\int u\,dy;$

then in the plane $(x=0)$ the amount of electricity which passes through any portion will be

$\iint u\,dy\,dz=\iint d\lambda \,d\lambda '.$

(16)

Having determined the nature of the sections of the surfaces of flow by the plane of $yz,$ the form of the surfaces elsewhere is determined by the conditions (8) and (9). The two functions $\lambda$ and $\lambda '$ thus determined are sufficient to determine the current at every point by equations (15), unity being substituted for $L.$

On Lines of Flow.

293.] Let a series of values of $\lambda$ and of $\lambda ;$ be chosen, the successive differences in each series being unity. The two series of surfaces defined by these values will divide space into a system of quadrilateral tubes through each of which there will be a unit current. By assuming the unit sufficiently small, the details of the current may be expressed by these tubes with any desired amount of minuteness. Then if any surface be drawn cutting the system of tubes, the quantity of the current which passes through this surface will be expressed by the number of tubes which cut it, since each tube carries unity of current.

The actual intersections of the surfaces may be called Lines of Flow. When the unit is taken sufficiently small, the number of lines of flow which cut a surface is approximately equal to the number of tubes of flow which cut it, so that we may consider the lines of flow as expressing not only the direction of the current but its strength, since each line of flow through a given section corresponds to a unit current.

On Current-Sheets and Current-Functions.

294.] A stratum of a conductor contained between two consecutive surfaces of flow of one system, say that of $\lambda ',$ is called a Current-Sheet. The tubes of flow within this sheet are determined by the function $\lambda .$ If $\lambda _{A}$ and $\lambda _{P}$ denote the values of $\lambda$ at the points $A$ and $P$ respectively, then the current from right to left across any line drawn on the sheet from $A$ to $P$ is $\lambda _{P}-\lambda _{A}.$ If $AP$ be an element, $ds,$ of a curve drawn on the sheet, the current which crosses this element from right to left is

${\frac {d\lambda }{ds}}ds.$

This function $\lambda ,$ from which the distribution of the current in the sheet can be completely determined, is called the Current-Function.

Any thin sheet of metal or conducting matter bounded on both sides by air or some other non-conducting medium may be treated as a current-sheet, in which the distribution of the current may be expressed by means of a current-function. See Art. 647.

Equation of 'Continuity.'

295.] If we differentiate the three equations (15) with respect to $x,y,z$ respectively, remembering that $L$ is a function of $\lambda$ and $\lambda ',$ we find

${\frac {du}{dx}}+{\frac {dv}{dy}}+{\frac {dw}{dz}}=0$

(17)

The corresponding equation in Hydrodynamics is called the Equation of 'Continuity.' The continuity which it expresses is the continuity of existence, that is, the fact that a material substance cannot leave one part of space and arrive at another, without going through the space between. It cannot simply vanish in the one place and appear in the other, but it must travel along a continuous path, so that if a closed surface be drawn, including the one place and excluding the other, a material substance in passing from the one place to the other must go through the closed surface. The most general form of the equation in hydrodynamics is

${\frac {d(\rho u)}{dx}}+{\frac {d(\rho v)}{dy}}+{\frac {d(\rho w)}{dz}}+{\frac {d\rho }{dt}}=0;$

(18)

where $\rho$ signifies the ratio of the quantity of the substance to the volume it occupies, that volume being in this case the differential element of volume, and $(\rho u),\,(\rho v),$ and $(\rho w)$ signify the ratio of the quantity of the substance which crosses an element of area in unit of time to that area, these areas being normal to the axes of $x,\,y,$ and $z$ respectively. Thus understood, the equation is applicable to any material substance, solid or fluid, whether the motion be continuous or discontinuous, provided the existence of the parts of that substance is continuous. If anything, though not a substance, is subject to the condition of continuous existence in time and space, the equation will express this condition. In other parts of Physical Science, as, for instance, in the theory of electric and magnetic quantities, equations of a similar form occur. We shall call such equations 'equations of continuity' to indicate their form, though we may not attribute to these quantities the properties of matter, or even continuous existence in time and space.

The equation (17), which we have arrived at in the case of electric currents, is identical with (18) if we make $\rho =1,$ that is, if we suppose the substance homogeneous and incompressible. The equation, in the case of fluids, may also be established by either of the modes of proof given in treatises on Hydrodynamics. In one of these we trace the course and the deformation of a certain element of the fluid as it moves along. In the other, we fix our attention on an element of space, and take account of all that enters or leaves it. The former of these methods cannot be applied to electric currents, as we do not know the velocity with which the electricity passes through the body, or even whether it moves in the positive or the negative direction of the current. All that we know is the algebraical value of the quantity which crosses unit of area in unit of time, a quantity corresponding to $(\rho u)$ in the equation (18). We have no means of ascertaining the value of either of the factors $\rho$ or $u,$ and therefore we cannot follow a particular portion of electricity in its course through the body. The other method of investigation, in which we consider what passes through the walls of an element of volume, is applicable to electric currents, and is perhaps preferable in point of form to that which we have given, but as it may be found in any treatise on Hydrodynamics we need not repeat it here.

Quantity of Electricity which passes through a given Surface.

296.] Let $\Gamma$ be the resultant current at any point of the surface. Let $dS$ be an element of the surface, and let $\epsilon$ be the angle between $\Gamma$ and the normal to the surface, then the total current through the surface will be

$\iint \Gamma \cos \epsilon \;ds,$

the integration being extended over the surface.

As in Art. 21, we may transform this integral into the form

$\iint \Gamma \cos \epsilon \;ds=\iiint \left({\frac {du}{dz}}+{\frac {dv}{dy}}+{\frac {dw}{dz}}\right)dx\;dy\;dz$

(19)

in the case of any closed surface, the limits of the triple integration being those included by the surface. This is the expression for the total efflux from the closed surface. Since in all cases of steady currents this must be zero whatever the limits of the integration, the quantity under the integral sign must vanish, and we obtain in this way the equation of continuity (17).