176 



General Analytical Theorems 



[CH. VII 



Let axes be taken so that the boundary is the plane of xy, while the 

 lines of force at the point under consideration lie 

 in the plane of xz. Let the components of 

 intensity in the first medium be (X lf 0, Z-^), while 

 the corresponding quantities in the second medium 

 are (X 2) 0, Z 2 ). The boundary conditions ob- 

 tained in 137 require that 



X 



FIG. 55. 



where h is the normal component of polarisation. 



In view of a later physical interpretation of 

 the forces, it will be convenient to regard these forces as divided up into 

 the two systems mentioned in 199, and to consider the contributions from 

 these systems separately. 



As regards the contribution from the first system, the force per unit area 

 acting on the dielectric from the first medium has components 



while that from the second medium has components 



Since KJL^ K 2 X Z Z 2 , it follows that the resultant force on the 

 boundary is parallel to Oz i.e. is normal to the surface. Its amount, 

 measured as a tension dragging the surface in the direction from medium 1 

 to medium 2 



77- 77- 



which after simplification can be shewn to be equal to 



( l ~ * 



This is always positive if KI > K 2 . Thus this force invariably tends to 

 drag the surface from the medium in which K is greater, to that in which 

 K is less i.e. to increase the region in which K is large at the expense of 

 the region in which K is small. This normal force is exactly similar to the 

 normal force on the surface of a conductor, which tends to increase the 

 volume of the region enclosed by the conducting surface. 



On Maxwell's Theory, the forces which have now been considered are the only ones in 

 existence, so that according to this theory the total mechanical force is that just found, 

 and the boundary forces ought always to tend to increase the region in which K is large. 

 This theory, as we have said, is incomplete, so that it is not surprising that the result just 

 stated is not confirmed by experiment. 



