Question

Consider the arrangement shown in the figure (17-E4). The distance D is large compared to the separation d between the slits. (a) Find the minimum value of d so that there is a dark fringe at O. (b) Suppose d has this value. Find the distance x at which the next bright fringe is formed. (c) Find the fringe-width.

Answer 1

Answer:

what is fringe

Answer 2

From the figure,

AB=BO and AC=CO.

Path difference of the wave front reaching O,

∆x=(AB+BO)-(AC+CO )

 =2 AB-AC    

=2√( D2+d2) - D

For dark fringe to be formed at O, path difference should be an odd multiple of  λ/2.

∆x=(2n+1 )λ/2

⇒2( √D²+d² -D)=(2n+1) λ/2

√D²+d²=D+(2n+1 )λ/4

⇒D²+d²=D²+(2n+1)² x λ²/ 16    +(2n+1 )λD/2

Neglecting, as (2n+1)² x λ²/ 16  it is very small, we get:

d=√(2n+1 )λD/2

For minimum d, putting, n=0

dmin=√λD/2, we get:

Thus, for

dmin=√λD/ 2 there is a dark fringe at O.