Page:PoyntingTransfer.djvu/6

But from the values of P', Q', R' in (5) we see that

${\begin{array}{ll}{\frac {dQ'}{dz}}-{\frac {dR'}{dy}}&=-{\frac {d^{2}G}{dt\ dz}}-{\frac {d^{2}\psi }{dx\ dz}}+{\frac {d^{2}H}{dt\ dy}}-{\frac {d^{2}\psi }{dz\ dx}}\\\\&={\frac {d}{dt}}\left({\frac {dH}{dy}}-{\frac {dG}{dz}}\right)\\\\&={\frac {da}{dt}}=\mu {\frac {d\alpha }{dt}}\ (\mathrm {Maxwell,\ vol.\ ii.\ p} .\ 216)\end{array}}$

similarly

${\begin{array}{c}{\frac {dR'}{dx}}-{\frac {dP'}{dz}}={\frac {db}{dt}}=\mu {\frac {d\beta }{dt}}\\\\{\frac {dP'}{dy}}-{\frac {dQ'}{dx}}={\frac {dc}{dt}}=\mu {\frac {d\gamma }{dt}}\end{array}}$

Whence the triple integral in (6) becomes

$-{\frac {\mu }{4\pi }}\iiint \left(\alpha {\frac {d\alpha }{dt}}+\beta {\frac {d\beta }{dt}}+\gamma {\frac {d\gamma }{dt}}\right)dx\ dy\ dz$

Transposing it to the other side we obtain

${\begin{array}{r}{\frac {K}{4\pi }}\iiint \left(P{\frac {dP}{dt}}+Q{\frac {dQ}{dt}}+R{\frac {dR}{dt}}\right)dx\ dy\ dz+{\frac {\mu }{4\pi }}\iiint \left(\alpha {\frac {d\alpha }{dt}}+\beta {\frac {d\beta }{dt}}+\gamma {\frac {d\gamma }{dt}}\right)dx\ dy\ dz\\\\+\iiint (X{\dot {x}}+Y{\dot {y}}+Z{\dot {z}})dx\ dy\ dz+\iiint (Pp+Qq+Rr)dx\ dy\ dz\\\\={\frac {1}{4\pi }}\iint \left\{l(R'\beta -Q'\gamma )+m(P'\gamma -R'\alpha )+n(Q'\alpha -P'\beta )\right\}dS\end{array}}$

(7)

The first two terms of this express the gain per second in electric and magnetic energies as in (2). The third term expresses the work done per second by the electromagnetic forces, that is, the energy transformed by the motion of the matter in which currents exist. The fourth term expresses the energy transformed by the conductor into heat, chemical energy, and so on; for P, Q, R are by definition the components of the force acting at a point per unit of positive electricity, so that $Ppdxdydz$ or $Pdxpdydz$ is the work done per second by the current flowing parallel to the axis of $x$ through the element of volume $dxdydz$. So for the other two components. This is in general transformed into other forms of energy, heat due to resistance, thermal effects at thermoelectric surfaces, and so on.

The left side of (7) thus expresses the total gain in energy per second within the closed surface, and the equation asserts that this energy comes through the bounding surface, each element contributing the amount expressed by the right side.

This may be put in another form, for if ${\mathfrak {E'}}$ be the resultant of P', Q', R', and $\theta$ the