and
sin α ¼
x
ffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi
x 2 þ L
2
p
By combining both expressions we find that
Δ
d
¼
x
ffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi
x 2 þ L
2
p
that relates the paths’ length difference to the X-point of the screen.
The interference will be constructive where the path difference is a multiple of the
wavelength:
Δ ¼ nλ
In our case, we will work only with n ¼ 1, therefore:
λ
d
¼
x max
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
x 2
max þ L
2
q
So the positions of the maximums will be:
x max ¼
λL
ffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi
d
2
À λ
2
p
Fig. 21.3 Diffraction grid containing slits at a distance d. At a distance L from the grid, we set a
screen on which the light spectrum is observed. We consider an X-point where constructive
interference occurs. Red and green lines represent the paths followed by the waves from two
consecutive slits to X-point and Δ represents the difference in both paths’ length
21 Design, Construction and Use of a Quantitative Spectroscope for. . .
267