ON THE MEASUREMENT OF MAGNETIC HYSTERESIS. 
95 
Appendix I, 
On the Heat produced by Eddy Currents in a Rod of Circular Section. 
§ 72, The problem cannot be completely solved unless tbe jiermeability, p,, is 
independent of the magnetic force, a condition not fulfilled with actual specimens 
of iron. Though this is so, we can obtain useful information from the complete 
solution when p is constant. 
If the magnetic force be parallel to the axis of the rod, and if u be the current at 
a distance r from the axis, and if cr be the specific resistance, then the field within 
the rod has the characteristics 
(rd{ru)/dr ~ — prc/H/d^ , . . (l), dYijdr = — irru .... (2), 
so that - d (i'd}l/dr)/dr = ATrp/cr . diL/dt = ydYL/dt .(3). 
Expressing H in the form 
H = Iiq -f- h^r + lip'' +.(4), 
where Hq, h.^, . . . are functions of t only, we see that = 0, since u = 0 when r = 0. 
Using this value of H in (3), and comparing coefficients of powers of r, we find 
ko, , and thus obtain 
o' ’ 
H = 
(/ff 1 
2' cU 
Hence for H^, the magnetic force at the surface where r — a, 
Tx — I 7 
\ ^ 2 ^ dt^ 2K^'^dd ■ ■ • T 0’ 
sothat H={l+fj'|+ . . .Ul+^f| + . . 
= H. - {«= - r=) (3a* - + r*) 
Hence 
L 
Itt dr 
dt ' 64 
1 J gr (?H<^ g' 
47r 1 2 "(7^ 16 
dm, 
df~ 
/o 2 3\ I 
{2ah' - r3) ^ . 
(5). 
Thus when H„ is known as a function of t, u can be found at any point of the section. 
Now, if we had assumed that <iH/cZ^ is constant over the section, we should have ' 
found from (1), since u = 0 when ?’ = 0, 
