XVII
279
FUNDAMENTAL EQUATIONS OF ELECTROMAGNETICS
actual case.
In the calculation we use the following special
notation. u is to denote a function U for which throughout
infinitely extended space V2U-4πu; hence generally
U==
=
- f = dr.
the integral being taken throughout all space.
As regards the electric currents, let u, v, w be the com-
ponents. As we only consider closed circuits we have
Further let U₁ =ū,
1
Ju av dw
+ +
dx dy dz
дубъ
V₁ =ï, W₁ = w.
1
1
0.
Then the components
L M, N, of the magnetic force exerted by the currents
are, according to the usual electromagnetics, given by the
equations
L-A (3,-2)
=
Α
M₁ = A
Əz
ᎧᎳ
dy
aw au
(a) +1-0 (1).
av
N₁ = A (JU, ǝv.)
ду Эх
++
ax მყ
=
From the existence of these forces, and from the principle of
the conservation of energy, it may be, and has been, concluded
that changes of u, v, w produce electrical forces whose
components X, Y, Z₁ are
1
X₁
=
-
A²
dU1
dt
Y₁
=
-
A2
dV₁
Z₁
=
-
- A2
dt
A² dW ₁
dt
(2).
These expressions hold good inside the conductors conveying
the currents u, v, w as well as for the space outside. The
forces (2) have been deduced from the forces (1) on the
assumption that the latter were due to electric currents. But
on account of our premises we may affirm that even if the
forces (1) are caused by any system whatever of variable
currents and variable magnets, then their variation must
equally give rise to the forces (2).
which produces the forces (1).
Let A denote the system
We superpose a system B