1903.] Expansion of some of the less Magnetic Metals. 375 



Note by G. A. SCHOTT, B.A., B.Sc. 



Cakulation of the Elongation, due to the Magnetic Field, of the 

 Cylindrical Bars. 



The dimensions of Dr. Shaw's apparatus are as follows : 



Length of solenoid and bar (21) = 19 cm. 



Mean radius of solenoid (a) = 2 -75 cm. 



Depth of winding (2c) = 2'40 cm. 



Number of layers (2p) = 22 



turns per layer (n) = 164 



Distance between layers (h) = 4/35 cm. 



Radius of bar (b) = from 0'3 to 0-6 cm. 



Use cylindrical co-ordinates (?, p) where z is the axis of the bar 

 and the origin is at the centre. 



Let the axial and radial components of the magnetic force (H), due 

 to the solenoid, be Z, R, and those due to the induced magnetisation 

 H', Z', K. 



R R' are of order />, and inside the bar are small compared with Z ; 

 they give terms in the strain of order /> 2 , which are less than 

 1 per cent, of the whole and may be neglected. In the same way 

 there are terms of order /> 2 in Z, which can be neglected. Thus we 

 may for our purpose regard the magnetic force inside the bar as axial 

 and constant over any cross section, but variable from end to end. 



The intensity of magnetisation is given by I = k (H + H'). The 

 volume density of magnetism is div. I = (H + H') div. k + k div. H', 

 since div. H = 0. 



For all the metals used by Dr. Shaw, k is exceedingly small (at 

 most X)~ 5 ), and its square may be neglected. Div. k involves the 

 differential coefficients of k with respect to the strains as well as 

 space-variations of the strains. The former, though they may be 

 large compared to k, yet are small absolutely ; the latter also are 

 small ; hence, div. k is negligible. Since H' is of order I, the volume 

 density of magnetism is of order k* and negligible. The induced 

 magnetism is confined to the surface of the bar, and the force H', 

 being of order k, may be neglected. 



Thus we need Only take account of the component J5. 



We shall require the mean values of Z and Z 2 , say Z and Z' J . 



The figure gives a meridian section. A' A is the axis, BCC'B' the 

 section of the upper half of the solenoid, QQ' the section of a layer of 

 wire of radius r. 



As we neglect variations of Z across the section of the bar, we need 

 only consider its value at a point P (z) on the axis. 



