# Page:AbrahamMinkowski1.djvu/24

With respect to rule (5) of § 1, the sum of these two terms is

${\displaystyle {\frac {1}{c}}[{\mathfrak {JB}}]+{\frac {1}{c}}\left\{{\frac {\delta }{\delta t}}[{\mathfrak {DB}}]+\left([{\mathfrak {DB}}]\nabla \right){\mathfrak {w}}+\left[[{\mathfrak {DB}}]\mathrm {curl} {\mathfrak {w}}\right]\right\}}$

The ponderomotive force is given by addition of the forces ${\displaystyle {\mathfrak {K}}_{1}}$ and ${\displaystyle {\mathfrak {K}}_{2}}$; the emerging expression becomes simplified, when one introduces the vector defined in (22)

 (59) ${\displaystyle {\mathfrak {W=[DB]}}-c{\mathfrak {g}}}$,

and if one uses the denotations

 (59a) ${\displaystyle {\mathfrak {q}}={\frac {\mathfrak {w}}{c}},\ l=ct.}$

Furthermore, the density of true electricity may be set to

 (59b) ${\displaystyle \mathrm {div} {\mathfrak {D}}=\rho }$

and the density of true magnetism shall be assumed as equal to zero

 (59c) ${\displaystyle \mathrm {div} {\mathfrak {B}}=0}$;

also instead of the electrostatic measure of current strength, the electromagnetic measure shall be introduced

 (59d) ${\displaystyle {\mathfrak {J}}=ci}$.

Then the expression for the ponderomotive force, which acts upon the unit volume of moving matter, reads

 (60) ${\displaystyle {\mathfrak {K}}={\mathfrak {E}}'\rho +[{\mathfrak {iB}}]-\zeta \nabla \epsilon -\eta \nabla \mu +{\frac {\delta {\mathfrak {W}}}{\delta l}}+({\mathfrak {W}}\nabla ){\mathfrak {q}}+[{\mathfrak {W}}\mathrm {curl} {\mathfrak {q}}]}$

The first term represents the force acting upon the moving electricity, the second term represents the force acting upon the electric conduction current; the third and fourth term account for the influence of inhomogeneity of the body. While these four terms already have to be accounted for at static or stationary fields in resting bodies, the latter terms (containing the vector ${\displaystyle {\mathfrak {W}}}$) only play a role at non-stationary processes, or in moving bodies.

In the obtained expression for the ponderomotive force, the differences between the individual theories only become of importance by the fact – when one neglects the extremely small deviation in the meaning of quantities ${\displaystyle \zeta }$ and ${\displaystyle \eta }$ (eq. 54ab) –, that the vector ${\displaystyle {\mathfrak {W}}}$ assumes different values.

If ${\displaystyle {\mathfrak {K}}}$ gives the momentum exerted by the electromagnetic field, then the energy converted into non-electromagnetic forms, is given by the sum of Joule-heat and work of the ponderomotive force. For the Joule-heat, according to main equation (III) and (59d), it is given

${\displaystyle Q=cq={\mathfrak {JE}}'=ci{\mathfrak {E}}'}$

while the performance of work of force ${\displaystyle {\mathfrak {K}}}$, is given from (60)

${\displaystyle {\mathfrak {qK}}={\mathfrak {E}}'\rho {\mathfrak {q}}-i[{\mathfrak {qB}}]-\zeta ({\mathfrak {q}}\nabla )\epsilon -\eta (q\nabla )\mu +{\mathfrak {q}}\left\{{\frac {\delta {\mathfrak {W}}}{\delta l}}+({\mathfrak {W}}\nabla ){\mathfrak {q}}+[{\mathfrak {W}}\mathrm {curl} {\mathfrak {q}}]\right\}}$