June 19, 1921 swann: earth's size; and magnetization 275 



TERRESTRIAL MAGNETISM.— 7/^(7 bcarmg of the earth's size 

 upon changes in its magnetization ^ W. F. G. Swann, University 

 of Minnesota. 



H. Lamb- has shown that if a system of currents is started in a 

 sphere of the size of the earth, endowed with a conductivity equal to 

 that of copper at normal temperature, a period of ten million 3'^ears 

 will elapse before the magnetic field becomes reduced to 1/e of its 

 original value as a result of the decay of the currents. The physical 

 reason for this result lies in the very large effect of self induction in a 

 body of the size of the Earth, as compared with the effect of resistance. 

 The purpose of this note is to examine the bearing of this matter upon 

 one or two questions relating to the Earth's magnetism. For the pur- 

 pose in hand, it is unnecessary to give an exact mathematical analysis, 

 and it wdll sufhce to handle the problem in a way which does not claim 

 more than a determination of the orders of magnitude of the elements 

 involved. Such a simplified procedure serves, moreover, to keep the 

 physical principles more prominently to the fore. In the first place, 

 it may be of interest to verify Lamb's result by this method. 



Decay of currents in a sphere. — If B is the average magnetic flux 

 through a section of the sphere containing the equator of the axis of 

 magnetic flux, the e. m. f . around the equator is ira-dB/dt, so that, con- 

 sidering an equatorial ring of unit cross section and specific resistance p 

 we have : 



Tta- h27rapz = (l) 



dt 



where i is the current density. 



Now, at each instant, B is of the order of magnitude of the magnetic 

 field at the equator of a sphere of radius a due to a current density 

 which may be taken for convenience as proportional to the distance 

 from the axis, and such as to have the value i on the equator. For 

 this case, it may readily be shownHhat the field at the equator is 

 4.Tria/\b, so that, as regards order of magnitude, we may write 



B = 0.25Tria (2) 



for a sphere of unit permeability. Substituting in (1) we have, as 



^ Presented at the Washington meeting of the American Physical Society, April 29-30» 

 1921. Received May 25, 1921. 



^ Quoted by A. Schuster in A critical examination of the causes of terrestrial magnetism. 

 Proc. Phys. Soc. Lond. 24: 124. 1911-1912. 



3 See for example, W. F. G. Swann, The Earth's magnetic field, Phil. Mag. (6) 24: 

 97. 1912. 



