364 ELECTROSTATICS AND MAGNETISM. [PT. II. CH. IX. 



the condition will be satisfied. The surface density induced on 

 the surface of separation is 



4?r l r 2 2?r fa + fa r 3 



so that the distribution is proportional to the distribution on a 

 conducting plane, as in 152. The lines of force in the medium 2 

 are straight, and the refraction on crossing the plane is shown in 

 the diagram, Fig. 72, in which fa/ fa = 4. The greater /x 2 the less 

 is the force in the medium 2, and if we make /* 2 infinite, the force 

 in the medium 2 vanishes, and the surface density becomes 



, e a 

 " -2SH' 

 as in the case of a conductor. 



188. Slab in Uniform Field. Suppose a slab of induc- 

 tivity /x 2 with parallel faces of infinite extent is placed in a uniform 

 field parallel to the equipotential surfaces, and let the indue tivity 

 of the surrounding homogeneous medium be fa. Then the potential 

 satisfies Laplace's equation in the slab as well as outside it. Ac- 

 cordingly the solution for either of the three parts of the field, in, 

 above or below the slab, is a linear function of the single co- 

 ordinate perpendicular to the equipotential planes, and the force 

 has values which are constant, but different, in the three regions. 



v 



V 



FIG. 73. 



If Fi and V 2 are any two equipotential planes outside the slab at 

 distances d t and d 2 from its faces, F 3 and F 4 the potentials of the 

 faces of the slab respectively facing them, and d the thickness of 

 the slab, we have the conditions at the surfaces V 3 , V 4 



(1) faF l= =faF, ^F 

 that is 



(2) F^Ft-^ 



