CHAP, iv.] ELECTRICITY AND MAGNETISM. 



207 



Required the Electric Force at point at any distance from a 

 plane of infinite extent charged to stirf ace- density p. 



Let P be the point, 

 and PX or a the normal 

 to the plane. Take any 

 small cone having its 

 apex at P. Let the 

 solid-angle of this cone 

 be w ; let its length be 

 r; and the angle its 

 axis makes with a. The 

 cone meets the surface 

 of the plane obliquely, 

 and if an orthogonal 

 section be made where Fig. 99. 



it meets the plane, the 

 angle between these sections will be = 6. 



XT ... . . , , ,. . . orthogonal area of section 

 Now solid-angle w is by definition = -^ ; 



Hence, area of oblique section = \ 

 . . charge on oblique section = 



I 



cos e 



cos 6 

 Hence if a + unit of electricity were placed at P, the force 



exerted on this by this small charge = x \-~-r 2 



wp 



or = ?- 

 cos 6 



Resolve this force into two parts, one acting along the plane, 

 the other along a, normal to the plane. The normal component 



along a is cos x Wf) = wp 

 cos 



But the whole surface of the plane may be similarly mapped 

 out into small surfaces, all forming small cones, with their summits 

 at P. If we take an infinite number of such small cones meeting 

 every part, and resolve their forces in a similar way, we shall 

 find that the components along the plane will neutralise one 

 another all round, while the normal components, or the resolved 

 forces along , will be equal to the sum of all their solid-angles 

 multiplied by the surface-density ; or 



Total resultant force along a Sw/>. 



