Page:Electromagnetic effects of a moving charge.djvu/9

which is not so well known. (First, I believe, given by me in the third Section of "Electromagnetic Induction and its Propagation," The Electrician, January 24, 1885 [vol. I., p. 446]; again, obtained in a different way in Section XXIL, January 15, 1886 [vol. I., p. 546]; see also Phil. Mag., August, 1886 [vol. II., Art. L.], and an example of the use of (4) and (5) in The Electrician, April 12, 1889, p. 683 [vol. II., Art. LI.].)

The mechanical force called by Maxwell the "electromagnetic force" is VCB, where C is the true current and B the induction. It is the force on the matter supporting electric current. Let it move. If w is its velocity, the activity of the force is

$\scriptstyle {\mathbf {wVCB} =\mathbf {CVBw} =-\mathbf {eC} .\quad \cdots \cdots \cdots \cdots \cdots \cdots \cdots \cdots \cdots \cdots \cdot \cdot {\text{(6)}}}$

Similarly, as I obtained in Section XXII. above referred to, there is a mechanical force (the magneto-electric) on matter supporting magnetic current G=μpH/4π, expressed by 4πVDG, and its activity is

$\scriptstyle {4\pi \mathbf {wVDG} =4\pi \mathbf {GVwD} =-\mathbf {hG} .\quad \cdots \cdots \cdots \cdots \cdots \cdots \cdots \cdots {\text{(7)}}}$

Of course e and h. are reckoned as impressed forces, which is the reason of the change of sign. Their activities are eC and hG.

It should be remarked further, that the above expressions for e and h are not certain. For I have shown that the sources of all disturbances are the lines of curl of the impressed forces (Phil. Mag., Dec., 1887) [vol. II., p. 362], and that the fluxes produced depend solely upon the curls of e and h, both as regards the steady fluxes and the variable ones leading to them. We may, therefore, use any other expressions for e and h which have the same curls as the above. And, in fact, we see that equations (1) and (2) only contain their curls.

Equations (1) and (3), with e and h defined by (4) and (5), therefore enable us to determine the effect of the moving medium. Prof. Thomson also arrives at (4) and (5), and at the "magneto-electric force," in his paper to which I have referred, by an entirely different method. And to show how well things fit together, he concludes, from the consideration of the moving medium, that a moving electrified surface is a current-sheet, which is another way of saying that a convection current is a part of the true current, as expressed in (3). I must, however, disagree with Prof. Thomson's assumption that the motion must be irrotational. It would appear, by the above, that this limitation is unnecessary.

As an example, and to introduce a new point, take the case of a charge q moving at speed u along the axis of z. It will come to the same thing if we keep the charge at rest, and move the medium the other way. We then use the equations (1) and (2), and in them use (4) and (5) with w=-u. Now when the steady state is arrived at, we have p=0, so (1) and (2) become

${\begin{aligned}\scriptstyle {\mathrm {curl} (\mu \mathbf {VHu} -\mathbf {E} )=0,\quad \cdots \cdots \cdots \cdots \cdots \cdots \cdots \cdots \cdots \cdots \cdots \cdots \cdots \cdots }&\scriptstyle {\text{(8)}}\\\scriptstyle {\mathrm {curl} (\mathbf {H} -c\mathbf {VuE} )=0.\quad \cdots \cdots \cdots \cdots \cdots \cdots \cdots \cdots \cdots \cdots \cdots \cdots \cdots \cdots }&\scriptstyle {\text{(9)}}\\\end{aligned}}$