536 
PROFESSOR H. LAMB OR ELECTRICAL 
Writing £=Jcr=(l—i)qr, and keeping only the most important term, we find 
«H = (-)*3.5...2»+1.^^2.(70). 
Hence the factor 
& 2 ^nQcr) 
47 T i|r„_i(^E) 
which occurs in the expressions for tlie induced currents, becomes, after several 
reductions, 
K 
r 
(2n + l)q / 
2a/27tR V 
£ + 1 
—E)+; £ q( r — E)+J- j 
(71). 
It appears from this that the disturbance inside the sphere consists of a series of 
waves propagated inwards from the surface with rapidly decreasing amplitude. Thus 
at a depth equal to the wavedength (y, say), the amplitude is only 1/535 of what it is 
at the surface. The currents are therefore almost entirely confined to a superficial 
stratum of thickness comparable with v. It appears from (58) that v, =2-n-jq, 
— \/(p!p)- As a numerical example let p= 1642 (copper), y?=4000; we find 
n='64 centimetre. 
The condition of the applicability of our approximation is that 27 tR must be large in 
comparison with vP 
Since, by (70), is of the order 1/XTt, it appears from (60) and (38) 
that the disturbance in the field caused by the currents in the sphere is given by 
1 ax 
K= -nW» +1 f- X„r -2 " -1 
dy 
c x — —nW- n+l ~ X ;i r -2 " -1 
1 dz n 
(72). 
The magnitude of the disturbance depends therefore on the size of the sphere, but 
is independent of the conductivity, so long as the fundamental condition of our 
approximation is satisfied. The reason of this is not far to seek. The greater the 
conductivity the greater will be the intensity of the currents at the surface of the 
sphere, but the more rapid will be the rate of diminution as we pass inwards; and it 
is easily seen from (71) that one cause will exactly compensate the other. 
* The above results enable us to estimate what ought to be the thickness of a sheet of a given metal 
in order that it should act as a screen against a periodic electromagnetic action of given frequency. See 
the paper by Lord Rayleirh, cited below. 
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