Surface of a Needle-Point discharging in Air. 



269 



Fig. 2. 



Let A in fig. 2 represent the section of a hemispherical 

 point. Near its surface, discharge, when it occurs, will be 



approximately radial, and may be 

 thought of as filling the cone POQ 

 which has its apex at the centre of 

 curvature of the point. 



If /is the field in any spherical 

 layer centred at and of radius r 

 and thickness dr, the momentum 

 given to this layer per square centi- 

 metre per second will be 



dfi=fp f dr, 



assuming that ions of one sign only 

 are present, and that // is the volume density of the electricity 

 they carry. 



Also, if V is the specific velocity of these ions, C the 

 current from the point, and O the solid angle of the discharge 

 cone 



C=p'fYD,r*. 



Hence 



dp — 



_G_ dr 

 VXT r 2 ' 



Suppose now that the sides of the cone are impermeable to 

 gas. dp will result in a difference of pressure dp between 

 the two surfaces of the spherical layer such that 



dp = djjb ; 



and if r is the radius of the point and the ions are all supposed 

 to start from there, the pressure within the cone at the metal 

 surface will be less than that at a distance r from by the 

 amount 



J" 



dp 



JO /1__1 



" VX2\r ft r 



> 



With sharp points for which r is a small fraction of a 

 millimetre we may put r=oo, and obtain a value for the 

 integral which is not much greater than if r is a millimetre 

 or so, the result being an upper limit to the value of p 2 for 

 the conditions assumed, viz. 



p 2 =C/Vnr Q . 



