9f5 
MESSRS. G. r. C. SEARLE AND T. G. BEDFORD 
This assumption is thus equivalent to the assumption that the second, third, . . . 
terms in (5) are negligible in comj)arison with the first term. In this case we find for 
the rate at which heat is generated by eddy currents 
o 
7r«~' 
(JX 
(It 
a u"2TTrdr 
Jo 
an-cj} fcmV- 
1287r \ cU j ’ 
where dX/dt has the meaning assigned to it in § 8. Here, since dJl/dt is constant 
over the section, we may put g = cZB/r/H, and thus 
^7X _ A /dBy _ QA /(IBV 
(It Sttct \ dti a \ cU / 
Hence, with the notation of (13) ^ 9, Q = I/Stt = •03979. 
The second term of (5) will he negligible in comparison with the first, provided 
that TTixa^lcr. d^Hfijdt^ is negligible in comparison with dH„ldt, and the third term 
will be negligible in comparison with the second if ir^cdlcT . ddYlnjdt'^ is negligible in 
comparison with d^Yiajdd'. 
Now, as in § 15, the characteristic of H„ is 
KdYi^jdl + KH„ = IttNE. 
Hence, supposing that K may l)e treated as constant, 
Kdm„idf^ + mujdt = 0 . 
Thus we see that the ratio of each term in (5) to the term before it is small, provided 
that Tr/xcdlcr is small compared with the “ time constant ” K/R. When this condition 
is satisfied, we may treat t/H/c/^ and also dBjdt as constant over the section of the 
rod, and may then calculate the eddy current from (1). 
On the Heat 2 ^Toduced hy Eddy Currents in a Rod of RectavgnJar Section. 
§ 73. The section of the rod is supposed so small that the current at any point may 
be calculated by Faraday’s law, on the assumption that has the same value at 
all points of the section. We see by the case of the circular rod that this assumption 
is legitimate, provided that Tryda is small com23ared with K/R, r being the radius of 
the largest circle inscribable in tJie section. 
Let, now, a, h be the sides of the rectangular section, and let the origin be at the 
centre of the section. Then, since the magnetic force is parallel to the axis of the rod, 
we liave, under the specified conditions 
du d V 
dy dx ^ 
M = 0 when x = 
du dv _ 
dr ^ dy~ ' ■ ■ 
't? z= 0 when y = ih 
where u, v are the components of the current, and qcr = clQjdt. 
