Horizontal Cylinders 



Suppose the body to be a horizontal cylinder of uniform cross section 

 whose length is very great compared to the dimensions of its cross section 

 and that we wish to find the value of A z at points along a line at the surface 

 perpendicular to the axis of the cylinder. As in the case of the mass particle we 

 choose a set of coordinate axes with the origin at the field point on the surface 

 and with the z axis vertically downward. The y axis (fig. 27-2) is turned 

 parallel to the axis of the cylinder so that the x axis lies along the line where 

 we wish to compute the values of A e . We suppose the density p of the cylinder 

 to be a constant. 



Figure 27-2. 

 Equation (16) may now be written 



A ° = k 'ffi 



wdudvdw 



oo (u 2 + v 2 + w 2 ) 312 



(17) 



where the limits — co to oo refer to integration with respect to v. When the 

 integration is carried out, equation (17) becomes 



A ' =k Pffl? 



2wdudw 



+ w< 



(18) 



The field of integration is now the cross section of the cylinder. 



Our problem at this point is to find a method of evaluating (18) which 

 will work for a cylinder of any cross section. To do this we first consider the 

 problem of evaluating it over the rectangle such as that shown in Figure 27-3. 



We now have 



%2 r z 2wdudw 



L « 



Xi ^ Z l 



r x 2 r x 2 



= kp I log (u 2 + z 2 2 ) du - kp J log (u 2 + z t 2 ) du 

 J x 1 J %i 



589 



