Vol. 8, 1922 
PHYSICS: H. A. LORENTZ 
337 
Now, since H does not depend on t, and B = 0 for / = 0, 
f (H.B)^i^ = (H.B); 
(15) therefore becomes 
- J(H.B)g?5 =-2 7 
and (14) 
A-W ^ 2{U -T) q. e. d. (16) 
Remarks, — 1. Both A and W depend on the choice of the time t the 
end of the (long) interval of time which we considered. But the differ- 
ence A — IF is independent of that choice. Indeed, in the steady state, 
the work of the impressed force per unit of time is equal to the generation 
of heat, equally per unit of time, so that, when the interval of time is length- 
ened, A and W increase by equal amounts. 
When, in the final state, there are no currents (but only electric charges) 
W = 0. In this case A is independent of the choice of t. 
2. W is not the quantity of heat that is really generated. This latter 
quantity W is given by 
W = ( dt f — dS. 
Jq J a 
3. In the interval of time during which the steady state is reached there 
will in general be a radiation of energy from the system outward. Let 
the total amount of the radiated energy (to be calculated by applying 
Poynting's rule to the flow through an infinitely great sphere) be R. 
Then, by the law of conservation of energy 
A = W'+U+T + R, 
Comparing this with (16) one finds 
W - W == ST - U -{- R. 
4. The theorem expressed by (16) can easily be verified for simple 
cases, when all quantities involved can be completely calculated. 
(a). A linear circuit with resistance r and self -inductance L, no ca- 
pacity. Let the electromotive force E be suddenly applied at ^ = 0, 
remaining constant ever afterwards 
The current is given by i = E/r. (1 — e~^^^^), and one finds 
Eidt — — t ~ — — {t very great)., 
0 r r^ 
