Page:A history of the theories of aether and electricity. Whittacker E.T. (1910).pdf/335

This page has been proofread, but needs to be validated.

Models of the Aether.

315

If a thin ring, for which the circulation is zero, is introduced into the fluid, it will experience no ponderomotive forces; but if a ring initially carrying no current introduced into a magnetic field, it will experience ponderomotive forces, owing to the electric currents induced in it by its motion.

Imperfect though the analogy is, it is not without interest. A bar-magnet, being equivalent to a current circulating in a wire wound round it, may be compared (as W. Thomson remarked) to a straight tube immersed in a perfect fluid, the fluid entering at one end and flowing out by the other, so that the particles of fluid follow the lines of magnetic force. If two such tubes are presented with like ends to each other, they attract; with unlike ends, they repel. The forces are thus diametrically opposite in direction to those of magnets; but in other respects the laws of mutual action between these tubes and between magnets are precisely the same.^[1]

↑ The mathematical analysis in this case is very simple. I narrow table through which water is flowing may be regarded as equivalent to a source at one end of the tube and a sink at the other; and the problem may therefore be reduced to the consideration of sinks in an unlimited fluid. If there are two sinks in such a fluid, of strengths $m$ and $m'$ , the velocity-potential is

$m / r + m' / r'$ ,

where $r$ and $r'$ denote distance from the sinks. The kinetic energy per unit volume of the fluid is

${\tfrac {1}{2}}\rho \left[\left\{{\frac {\partial }{\partial x}}\left({\frac {m}{r}}+{\frac {m^{\prime }}{r^{\prime }}}\right)\right\}^{2}+\left\{{\frac {\partial }{\partial y}}\left({\frac {m}{r}}+{\frac {m^{\prime }}{r^{\prime }}}\right)\right\}^{2}+\left\{{\frac {\partial }{\partial z}}\left({\frac {m}{r}}+{\frac {m^{\prime }}{r^{\prime }}}\right)\right\}^{2}\right]$ ,

where $ρ$ denotes the density of the fluid; whence it is easily seen that she total energy of the fluid, when the two sinks are at a distance $l$ apart, exceeds the total energy when they are at an infinite distance apart by an amount

$\rho mm^{\prime }\iiint \left\{{\frac {\partial }{\partial x}}\left({\frac {1}{r}}\right){\frac {\partial }{\partial x}}\left({\frac {1}{r^{\prime }}}\right)+{\frac {\partial }{\partial y}}\left({\frac {1}{r}}\right){\frac {\partial }{\partial y}}\left({\frac {1}{r^{\prime }}}\right)+{\frac {\partial }{\partial z}}\left({\frac {1}{r}}\right){\frac {\partial }{\partial z}}\left({\frac {1}{r^{\prime }}}\right)\right\}\ dx\ dy\ dz$ ,

the integration being taken throughout the whole volume of the fluid, except two small spheres $s$ , $s'$ , surrounding the sinks. By Green's theorem, this expression reduces at once to

$\rho mm^{\prime }\iint {\frac {1}{r^{\prime }}}{\frac {\partial }{\partial n}}\left({\frac {1}{r}}\right)dS$ ,

where the integration is taken over $s$ and $s'$ , and $n$ devotes the interior portal to $s$ or $s'$ . The integral taken over $s'$ vanishes; evaluating the remaining integral, we have

$-{\frac {\rho mm^{\prime }}{l}}\iint {\frac {\partial }{\partial r}}\left({\frac {1}{r}}\right)dS$ , or $4πρmm'/l$ .

The energy of the fluid is therefore greater ben sinks of strengths $m$ , $m'$ are at a

[p315-1] The mathematical analysis in this case is very simple. I narrow table through which water is flowing may be regarded as equivalent to a source at one end of the tube and a sink at the other; and the problem may therefore be reduced to the consideration of sinks in an unlimited fluid. If there are two sinks in such a fluid, of strengths $m$ and $m'$ , the velocity-potential is

$m / r + m' / r'$ ,

where $r$ and $r'$ denote distance from the sinks. The kinetic energy per unit volume of the fluid is

${\tfrac {1}{2}}\rho \left[\left\{{\frac {\partial }{\partial x}}\left({\frac {m}{r}}+{\frac {m^{\prime }}{r^{\prime }}}\right)\right\}^{2}+\left\{{\frac {\partial }{\partial y}}\left({\frac {m}{r}}+{\frac {m^{\prime }}{r^{\prime }}}\right)\right\}^{2}+\left\{{\frac {\partial }{\partial z}}\left({\frac {m}{r}}+{\frac {m^{\prime }}{r^{\prime }}}\right)\right\}^{2}\right]$ ,

where $ρ$ denotes the density of the fluid; whence it is easily seen that she total energy of the fluid, when the two sinks are at a distance $l$ apart, exceeds the total energy when they are at an infinite distance apart by an amount

$\rho mm^{\prime }\iiint \left\{{\frac {\partial }{\partial x}}\left({\frac {1}{r}}\right){\frac {\partial }{\partial x}}\left({\frac {1}{r^{\prime }}}\right)+{\frac {\partial }{\partial y}}\left({\frac {1}{r}}\right){\frac {\partial }{\partial y}}\left({\frac {1}{r^{\prime }}}\right)+{\frac {\partial }{\partial z}}\left({\frac {1}{r}}\right){\frac {\partial }{\partial z}}\left({\frac {1}{r^{\prime }}}\right)\right\}\ dx\ dy\ dz$ ,

the integration being taken throughout the whole volume of the fluid, except two small spheres $s$ , $s'$ , surrounding the sinks. By Green's theorem, this expression reduces at once to

$\rho mm^{\prime }\iint {\frac {1}{r^{\prime }}}{\frac {\partial }{\partial n}}\left({\frac {1}{r}}\right)dS$ ,

where the integration is taken over $s$ and $s'$ , and $n$ devotes the interior portal to $s$ or $s'$ . The integral taken over $s'$ vanishes; evaluating the remaining integral, we have

$-{\frac {\rho mm^{\prime }}{l}}\iint {\frac {\partial }{\partial r}}\left({\frac {1}{r}}\right)dS$ , or $4πρmm'/l$ .

The energy of the fluid is therefore greater ben sinks of strengths $m$ , $m'$ are at a

[1]