1903.] Electric Waves round a Conducting Obstacle. 255 



and therefore 



_CV 8 r cfi d_ sm K(Bo-VQ 



~ kcl> 9/x [} fl) Ko aEo Ko 



i a « 2 a cos k (e - 



f?}] <* 



k a« E oE E 



At points on the surface of the sphere for which E is great com- 

 pared with the wave-length, this becomes, retaining only the most 

 important terms, 



F = _ cvgi A _ i^M (1 _ ^ sin K (Ro _ v , 



or 



F= (l-^^) F i = (l-cos X )F 1 (4), 



where x is the angle subtended by OC, and Fi is the electric force 

 along the normal, which would be due to the oscillator if the sphere were 

 absent. From this it follows that the ratio F/Fi gradually diminishes 

 as [i decreases, until /x approaches the value - 1, when it becomes com- 

 parable with the terms which have been neglected. Hence, when 

 electric waves are incident on a perfectly conducting sphere, there is no 

 true shadow near the surface when the wave-length is small compared 

 with the radius of the sphere. It can, therefore, be inferred that, when 

 electric waves are incident on a perfectly conducting body whose 

 surface is convex, and has its radii of curvature everywhere great 

 compared with the wave-length, there is no true shadow near to the 

 surface. It is known that, when electric waves of small wave-length 

 are incident on a perfectly conducting wedge, the disturbance does not 

 sensibly creep round the corner, but shoots out so that there is a shadow 

 which coincides the more closely with the geometrical shadow as the 

 wave-length diminishes.* It therefore appears that the condition for 

 the formation of a distinct shadow near the surface of a perfectly 

 conducting body, whose surface is convex, when electric waves are 

 incident on it, is that there should be a line on that part of the surface 

 inside the geometrical shadow along which the radius of curvature of 

 the surface in the plane of incidence of the waves is small compared 

 with the wave-length. 



3. The electric force normal to the surface of a sphere, which is a 

 fairly good conductor, may be obtained by an analysis similar to that 

 given above. The result is 



i df i l-te.a.r.. . i a „ r] 



a 1 cp r =a a " 1 + ik oa J 



* Sommerfeld, * Math. Annalen,' toI. 47, 1896; or Macdonald, ' Electric Waves,' 

 1902, p. 187. 



