PEIRCE. — BEHAVIOR OF THE CORE OF AN ELECTROMAGNET. 179 



When there are eddy currents the vahie of Hs is given with suffi- 

 cient accuracy by the first term of (83) very soon after the electromo- 

 tive force has been shunted out of the circuit, that is by the equation, 



and the ratio of ^ to irb'^trixHs is practically equal to the constant 

 2051/2000, for it is easy to find a very convergent geometrical series 

 every term of which is greater than the corresponding term of the 

 series which begins with the second term of (85), and the sum of this 

 geometrical series is extremely small except for very small values of t. 



According to this analysis, the current in the solenoid will ha%'e 

 fallen in the first second to the fraction 0.002025 or to the fraction 

 0.001777 of its original value according as there are or are not eddy 

 currents in the iron. 



If the ten centimeter square iron core of the solenoid were built up of 

 straight rods only one millimeter in diameter, we should have b = 1/20, 

 n = 100, and A. = 1/1000 ; the m's would need to be roots of the 

 equation 



J^ (mh) = 1000 mb ■ Ji {mb). (96) 



By using differences of the third order it is possible to show from 

 Meissel's table that the first root is approximately equal to 0.044715 + 

 and the second to 3.83. For the first, then. A- + m^b^ = 0.002000, 

 and /S^ = 6.33077. For the second root, /3^ = 46500, and the second 

 terms of the series (83) and (85) become negligible almost immedi- 

 ately after the electromotive force has been removed from the circuit. 



In this case 



<f>, = 2500 ttHo- e-6-3307'' (97) 



very nearly ; and 



-^, = Hs = H,- e-6-33077^ , (98) 



^ttJS 



so that the disturbing effects of the eddy currents are comparatively 

 slight. At the end of one second, the current will have fallen to the 

 fraction 0.001777 of its original value or to the fraction 0.001781, 

 according as eddy currents were absent or existent. These differ by 

 only about one two hundred and fifty thousandth part of the original 

 current strength. We may note in passing that a very approximate 

 value (correct to four significant figures) of the first root of the equa- 

 tion might be found by equating to unity the coefficient of the first 

 term of the series (83). 



