Moving Charged Spheres. 161 



in two parallel circles, distant b from each other. The plane 

 of revolution is taken perpendicular to the plane of the paper 

 (fig. 3). The needles lie in one of the planes of revolution. 

 The force at P due to the sphere at A is required. 



P = PA. 

 b =OB. 

 d = PB. 



c = OC = OA = radius of revolution. 



e = angle between p and tangent at A. 



= angle between vertical radius and radius to A. 



p * = d 2 + b 2 + c' i -2cVd 2 + b 2 coscp 

 cos *=cos -^= 



p* = d* + b 2 + c 2 -2dc cos 



d sin 



cos e= 



P 



J (duosO-cY + b 2 

 sin €=4/ 



d' + b* + c* — 2dc cos 



The force acts in a direction perpendicular to p and the 

 tangent at A. The component of this force in the direction of 

 the normal to the plane of revolution is required. Let ^r be 

 the angle between the direction of the force and the normal to 

 the plane of revolution. 



d cos 0—c 

 cos vf'^: — — 



Videos O-cf + V 



v = 2ttcN 



where X is the number of revolutions per second. 

 Hence : 



„_ 2ir~Ncq (d cos 0— c) 



~V[d* + b* + c*-2de cos 0]i 



X is the component of the force at P in the direction of the 

 axle due to the sphere at A. Y is the ratio of the units. The 

 capacity of the spheres and their potential are measured in 

 electrostatic units. 



Figure 4 is plotted from this expression, and shows how the 

 force varies with the position of the spheres. The upper curve 

 gives the resultant force at the lower needle due to both sets 

 of spheres, and the lower curve, which is nearly a straight line, 

 gives the force at the upper needle. Let the mean value of 



