# A Treatise on Electricity and Magnetism/Volume 1/Part 2/Chapter 7

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 $O$.
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) |
$\Gamma ,$
$l',m',n'$
$\gamma$
$\theta$
| $\gamma =\Gamma \cos \theta ;$ | (3) |
shewing that the law of resolution of currents is the same as that of velocities, forces, and all other vectors.
| Let | $F(x,y,z)=\lambda$ | (4) |
$\lambda$
| $\left.{\frac {\overline {d\lambda }}{dx}}\right|^{2}+\left.{\frac {\overline {d\lambda }}{dy}}\right|^{2}+\left.{\frac {\overline {d\lambda }}{dz}}\right|^{2}={\frac {1}{N^{2}}},$ | (5) |
$\lambda$
| $l=N{\frac {d\lambda }{dx}},\quad \quad m=N{\frac {d\lambda }{dy}},\quad \quad n=N{\frac {d\lambda }{dz}}.$ | (6) |
$\gamma$
| $\gamma =N\left\{u{\frac {d\lambda }{dx}}+v{\frac {d\lambda }{dy}}+w{\frac {d\lambda }{dz}}\right\}.$ | (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.
| $u{\frac {d\lambda }{dx}}+v{\frac {d\lambda }{dy}}+w{\frac {d\lambda }{dz}}=0$ | (8) |
If this equation is true for all values of $\lambda ,$ all the surfaces of the family will be surfaces of flow.
$\lambda ',$
| $u{\frac {d\lambda '}{dx}}+v{\frac {d\lambda '}{dy}}+w{\frac {d\lambda '}{dz}}=0$ | (9) |
$\lambda '',$
| $u{\frac {d\lambda ''}{dx}}+v{\frac {d\lambda ''}{dy}}+w{\frac {d\lambda ''}{dz}}=0$ | (10) |
$u,v,$
$w$
| ${\begin{vmatrix}{\frac {d\lambda }{dx}},&{\frac {d\lambda }{dy}},&{\frac {d\lambda }{dz}}\\{\frac {d\lambda '}{dx}},&{\frac {d\lambda '}{dy}},&{\frac {d\lambda '}{dz}}\\{\frac {d\lambda ''}{dx}},&{\frac {d\lambda ''}{dy}},&{\frac {d\lambda ''}{dz}}\end{vmatrix}}=0;$ | (11) |
| $\mathrm {or} \quad \quad \lambda ''=\phi (\lambda ,\lambda ');$ | (12) |
that is, $\lambda ''$ is some function of $\lambda$ and $\lambda '.$
$\lambda ,\lambda +\delta \lambda ,\lambda ',$
$\lambda '+\delta \lambda '.$
$\delta \lambda \cdot \delta \lambda '.$
$L\delta \lambda \cdot \delta \lambda '$
$L$
$\lambda$
$\lambda ',$
​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.

$\lambda ',$
$\lambda .$
$\lambda _{A}$
$\lambda _{P}$
$\lambda$
$A$
$P$
$A$
$P$
$\lambda _{P}-\lambda _{A}.$
$AP$
$ds,$
* ${\frac {d\lambda }{ds}}ds.$
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.'

$x,y,z$
$L$
$\lambda$
$\lambda ',$
| ${\frac {du}{dx}}+{\frac {dv}{dy}}+{\frac {dw}{dz}}=0$ | (17) |
| ${\frac {d(\rho u)}{dx}}+{\frac {d(\rho v)}{dy}}+{\frac {d(\rho w)}{dz}}+{\frac {d\rho }{dt}}=0;$ | (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.

$\Gamma$
$dS$
$\epsilon$
$\Gamma$
* $\iint \Gamma \cos \epsilon \;ds,$
the integration being extended over the surface.
| $\iint \Gamma \cos \epsilon \;ds=\iiint \left({\frac {du}{dz}}+{\frac {dv}{dy}}+{\frac {dw}{dz}}\right)dx\;dy\;dz$ | (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).