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PHYSICS: H. A. LORENTZ 
Proc. N. a. S. 
(b). A dielectric with the constant e fills all space. At time t = 0 
an electromotive force Fi is suddenly applied. It has the same direction 
and magnitude at all points inside a sphere and is limited to that part of 
space. The steady state is now easily calculated; likewise the amount 
of energy R, that is radiated. 
5. The theorem mentioned in G. W. Pierce, Electrical Oscillations, . 
p. 40, is closely connected with Heaviside's general theorem. 
Consider a circuit containing a condenser and in series with it a self- 
induction and resistance. In this case T = 0, 1^ = 0, so that (16) be- 
comes A = 2U. But, by the law of energy {R = 0), A = W -\- U. 
Therefore W = U, which is Pierce's statement. 
^ This general theorem stated without proof by Heaviside, was useful to Mr. Charles 
Manneback in his thesis for the doctorate in science at the Massachusetts Institute of 
Technology, June 1922: Radiation from Transmission Lines fas yet unpublished). 
As Dr. Manneback could not locate, in the textbook or periodical literature, a satis- 
factory proof of the theorem in all its generality, he appealed for help to Professor 
Lorentz who kindly sent the demonstration. It has seemed as though the treatment 
of the theorem by Dr. Lorentz with his comments upon it might be of wide interest, 
particularly on account of possible practical applications, and he has assented to its pub- 
lication . — Editor . 
2 One can just as well (somewhat greater complication of the formulae) consider 
crystalline substances. 
2 One could add a "magneto-motive" force, but for the sake of simplicity we shall 
not do so. 
4 (A. B) is the scalar product of the vectors A and B; (A. [B -|- C]) the scalar pro- 
duct of the vectors A and B -|- C. Of course when two vectors have the same direction 
(as B and H) we can as well take the product of their magnitudes. 
^ What is said here of D does not apply to the electric force. When, during the inter- 
val !\t, D is infinitely small (practically 0), eg. (1) shows that E is nearly — Fi, a finite 
value. Note that we have drawn our conclusions from the fact that the equations 
contain B and D. They do not contain E. 
^ We need not write Fi and F2, because Fi and F2 are constant from t = 0 onward. 
This formula expresses the fact that in the final state, when both U and T remain 
constant, the Joule heat is equal to the work of F2. (The work of Fi is 0, because in 
the final state D = 0.) One can deduce one form from the other by using C == 
<T (E + F2). J\E.C)dS = 0, because div C = 0 and curl E = 0. 
^ Since (F2.C) is independent of t we could just as well write t.J^ (F2.C)dS, but in 
what follows, the form (9) is to be preferred. 
^ First extend the integration to the space within a sphere S around O, the radius 
r of which ultimately increases indefinitely. Since E depends on a potential <p, we 
find by partial integration (putting D+C — C = Q)J^ (E.Q)dS = — J* (grad 
<p.Q)dS = — J (pQnd'^ S ^ (^dS. (Qn normal component.) The surface 
integral vanishes for r = 00 because (when the total charge of the system is 0) ^ decreases 
at least as 1/r^, Whereas Qn becomes 0. 
There the integration by parts leads to the surface integral 
c y { [H3'2(Ez — Ez) ~ Hz(Ey — E3;)]cos a -{- !f^S(cos a direction constants of 
normal) to S and this vanishes for similar reasons as the surface integral of the pre- 
ceding note. 
