296 



Electric Time-constant of a Circular Dish [Apr. 21, 



JcD . sin ^ = /imD'e - "' 5 / 1 cos ** 



2"' 



whence JcB . tan — = pmB (16.) 



2 



If jx = 1, as we have supposed, and m$ is small, the principal root 

 of this equation in JcB is small, and the current-intensity, which varies 

 as cosfo/, will be nearly uniform throughout the thickness. The 

 equation (16) then gives — 



and therefore from (15) — 



t = AT 1 = (l-ima-f&c), 

 mp 



where p = p/h. For the purpose of a rough comparison with our 

 original problem we may suppose that 7r/m is comparable with R, the 

 radius of the disk. It follows that the effect of replacing the actual 

 disk, of finite thickness, by an infinitely thin disk of the same con- 

 ductivity (per unit area) is to increase the time-constant by the 

 fraction £/R of itself, about. 



In an iron plate, on the other hand, the current-intensity will fall 

 off considerably from the median plane to the surface, unless the 

 ratio 5/R be extremely small. For instance, if /imB = it/2, or say 

 g/R = 1/2/*, the principal root of (16) is ltd = sr/2, and the intensity 

 at the surface is only 0*71 of its value in the median plane, although 

 the thickness of the disk may perhaps not exceed one-thousandth of 

 the radius. Again if, mS being still small, /imd is moderately large, we 

 shall have kB = sr, nearly, so that the current-intensity almost vanishes 

 at the surface. In such a case — 



t = Atirjx\]^'p — 4<fib 2 /7rp, 



roughly. It will be seen that within certain limits (e.g., if = 500 

 and the lateral dimensions be not more than about 100 times the 

 thickness) this result is independent of the size and shape of the plate. 

 Under these circumstances, the value of t for an iron plate 

 (p = 10,000 C.Gr.S.) whose thickness is 2*5 mm. will be comparable 

 with 0-003 sec. 



