applied to Electric Currents. 287
in unit of volume is
where C is a constant to be determined.
Let us take the case in which
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(35)
|
Let
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(36)
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and let
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(37)
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then is the potential at any point due to the magnetic system , and that due to the distribution of magnetism represented by . The actual energy of all the vortices is
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(38)
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the integration being performed over all space.
This may be shown by integration by parts (see Green's 'Essay on Electricity,' p. 10) to be equal to
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(39)
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Or since it has been proved (Green's 'Essay,' p. 10) that
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(40)
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Now let the magnetic system remain at rest, and let be moved parallel to itself in the direction of through a space ; then, since depends on only, it will remain as before, so that will be constant; and since depends on only, the distribution of about will remain the same, so that will be the same as before the change. The only part of E that will be altered is that depending on , because becomes on account of the displacement. The variation of actual energy due to the displacement is therefore
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(41)
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But by equation (12), the work done by the mechanical forces on during the motion is
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(42)
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and since our hypothesis is a purely mechanical one, we must