10 Bumstead — Heating Effects produced by Rontgen Rays. 



the abscissae represent time in minutes, the ordinates deflec- 

 tions of the radiometer in centimeters on the scale ; the posi- 

 tion of the wheel was such that a positive deflection means a 

 repulsion by the zinc of its vane, a negative deflection a repul- 

 sion by the lead of its vane. Between 2 ,n and 5 m , the zinc 

 strip was exposed to the rays ; the rays were then cut off and 

 the zinc allowed to cool for 5 minutes (5 m -10 m ), the lead was 

 then exposed for 3 minutes (10 m -13 m ) ; it, in turn, was allowed 

 to cool for 5 minutes (13 m -18 m ) and then both strips exposed 

 simultaneously for 3 minutes (18 m -21 m ) after which the rays 

 were again cut off. It is plain from the figure that neither 

 metal was exposed long enough to the rays for the steady 

 „ state to be attained ; but it is also plain that the great prepon- 

 derance of the lead over the zinc is not due to this cause. In 

 fact we may from these curves get an approximate idea of 

 what the steady deflections would have been if it had been safe 

 to continue running the bulb at the rate necessary for such 

 large deflections. 



If X is the coefficient of absorption of the metal for the rays 

 used, then the energy of the rays at any point in the interior 

 of the strip whose distance is x from the front face is I e-^ x and 

 the energy absorbed in an element of thickness, dx, is Xl e-^ x dx. 

 Let us assume that the heat generated in the element is propor- 

 tional to this, say a\I e-^ x dx. The equation of the flow of heat 

 under these circumstances is : 



dV rPV , T , 

 c — = k— — + a\\ e~ Ax , 

 dt dx 



with the boundarv conditions 



\ dx J x=0 \ dx } x=l 



where V is the temperature (measured above the surround- 

 ings), t the time, c the specific heat of unit volume, Jc the con- 

 ductivity, h the emissivity of the surface, and I the thickness 

 of the strip. The solution of this will have the form 



V=Y a ,-%Ae-y t 



where V^ is the steady value and is the solution of the differ- 

 ential equation with the left hand member put equal to zero ; 

 the successive values of 7 in the sum are determined by a cer- 

 tain transcendental equation, and the A's are functions of x 

 into which 7 enters. What is observed is the temperature of 

 the surface where x = l. The variation of this temperature 

 with the time is represented fairly well (except at the very 

 beginning) by a single term of the series of exponentials so that 

 we may write as a rough approximation 



v=v 00 (i-e-y«). 



