Application of Cornus Spiral to Diffraction- Grating. 3 1 



Suppose the incident wave-front is parallel with the grating, 

 and that the direction of the point P, where the resultant ampli- 

 tude is sought, makes an angle 6 with the normal to the 

 grating, in a plane at right angles to the slits. Then if a is 

 the width of the slits, and h is the width of the opaque bars, 

 the angle ^ which expresses the difference in phase at P of 

 light from the two edges of a slit is 



27ra sin 6 



and the difference of phase of light from the two edges of 

 a bar is 



^, 27r?>sin<9 



The arc cd is proportional to a, hence (j) = k^ , where E. is the 

 radius of the circle cdef, and k is a constant. 



27ra sin 6 , a ^ k\ 



A: -,^ .'. K = 



\ R * • ^TTsin^* 



The length of the chord cd expressing the amplitude due to 

 a single slit may be represented by /, and w^e have 



. ira sin 



Z = 2Rsini<f) = /:a. / A^ 



\ Tra sin U 



^ X 



Construct a circle, radius r (in the above diagram smaller 

 than R), such that chords equal and parallel to chords cd, ef, 

 &c. will meet on its circumference. 



The angle between the normals to consecutive chords is 

 (f) H- <f>', and since they meet on its circumference we have 



2^. =sin i{4, + 4>'), bence ,•= 2,i„i(^-+-^/). 



But in the circle of radius r the n chords subtend an arc 

 n{<f> + cl>^)j and the resultant amplitude is therefore 



, ^ . n(0H-(/)') J 2 



A=^2rsm ^ ^ ^ =1 . ., 



2 . (t>-\-(j) 



^"^ 2 



and the intensity I = xV 



