506 MR. 0. W. RICHARDSON ON THE ELECTRICAL CONDUCTIVITY 
the volume density ^ at any point x being 
1 dW 
Att dx" 
/277?2;^No 
V ne 
-Kim 
— iu>/R0 
( 18 ). 
It is evident that ^dx = cr, since dY/dx = 0 for x = co. Thus, as we should 
J 0 
expect, the charge on the surface is equal and opposite to the total charge in the 
space outside the metal. 
As a numerical illustration we may calculate the potential at a point distant 
10 centims. from a plane surface of platinum which is put to earth and maintained at 
a tenqDerature of, say, 1500° absolute. Taking the number of molecules in a cubic 
centimetre at 0° and 7G0 millims. as 2 X 10^®, the charge on an ion as 6'5 X 10"^° 
and the value of z/j/R, which has been determined experimentally, to be 4'OS X 10b 
we find the potential at a point 10 centims. from the surface to be 1'5 volts, while at 
a point 1 centim. distant it would be about 1’2 volts. 
The experiments in the sequel were not intended to test this part of the theory, 
but they show, as we should expect, that practically the whole of the current is 
stojoped by a fall of potential of the order of one volt when it tends to drive the 
coriDuscles back to the hot metal. 
It will be seen by inspection of formula (15) that even at the highest temperatures 
we can attain the jDotential diflerences at small distances from the hot surface never 
become very great. For instance, at the temperature of the sun (6000° C.) the 
difference of potential between the surface and a point 1 centim. distant from it 
would be only about sixteen times its value at 1300° C. On the other hand, the 
surface density increases very quickly with the temperature, as will be seen from 
formula (17). In the case of carbon at 6000° C., taking 10"^ as a probable maximum 
value of 71 and 7'8 X 10^ as the value of ir/R, we find that cr has the enormous value 
of 300 electrostatic units, whereas at 1300° C. cr would have been less than this in the 
ratio of 1 to 3 X 10®. 
These numbers are to be taken as purely illustrative. It is not supposed that any 
conductor could 230ssibly exist in a vacuum at 6000° C. 
It will be noticed that the preceding theory of the equilibrium of corpuscles near a 
surface where they are being emitted is quite independent of any hypothesis as to the 
nature of the mechanism by which they are set free. The results are therefore of 
Interest even if the hypothesis, that the negative ions from hot conductors are the 
same as those which carry the current inside the metal, is ultimately found to be 
untrue. 
