Energy in the Electromagnetic Field. 1& 



are given by 



so that y=-~p—a6 (16) 



From (15) and (16) it is seen that 

 and ^=-/~+ajS<9, 



and hence dy\dx = 6. At any point, therefore, the curves 

 bisect theTangle between the directions of: the incident and 

 the reflected rays which pass through that point. 



As might be expected, the formulae obtained above for 

 the case of diffraction by a cylinder reduce to the ordinary 

 formulae for diffraction by a straight edge on writing a = 0, 

 provided of course that the light from the other side of the 

 cylinder is cut off by a semi-infinite plane extending to the 

 left of the origin 0. For then equations (11) become 



x=2yd, 



x=ZyV, | 



2u6 2 = m\/2-e\/27r) 



X/2-eX/i 



rrr 



and e=j, so that x 2 =y\(Am—T)/4 :t This is Schuster's 



formula for diffraction of plane waves at a straight edge. 

 The results regarding the loci of maxima and minima of 

 illumination will also apply for diffraction at a straight edge 

 under the same conditions. 



5. The Flow of Energy in the Field. 



We will first take into account only the effects due to the 

 interference of the direct and the reflected rays. The effect 

 due to diffraction at the edge of the cylinder can be brought 

 in later as a correction. 



Let us assume, for simplicity, that the light is polarized 

 in the plane of incidence, so that the electric intensity is 

 perpendicular to that plane and the magnetic intensity lies 

 in it. Then at any point P (fig. 1) the resultant electric 

 intensitv is 



i=cos„( i -y) + .(^oo S {«( t -g + . 



i 

 r 



where r^PM, r 2 = PQ, and p is the radius of curvature of 



2 



