Question

Light of wavelength O is passed through a diffraction grating with N lines/meter and then lands on a screen a distance L from the grating. What is the linear distance between adjacent fringes on this screen? The width of each slit in the grating is a. You can assume the small angle approximation applies.

Answer 1

Condition for diffraction

$dsin\theta = m\lambda$

Where

a = Distance between slits

m = Order of the fringes

$\lambda$ = Wavelength

$\theta$ = At the angle between the ray of light and the projected distance perpendicular between the two objects

For small angles

$sin\theta = \approx tan\theta$

Where

$tan\theta = \frac{Y}{L}$

Where L is the distance between the slits and Y the length of the light.

Replacing we have

$d\frac{Y}{L} = \lambda m$

$Y = \frac{m\lambda L}{d}$

The distance between slits d can be expressed also as $d= \frac{L}{N}$ Where N is the number of the fringes, then

$Y_n = mN\lambda L$

Similarly when there is added a new Fringe we have the change of the distance would be :

$Y_{n+1} = (m+1)N\lambda L$

Linear distance between fringes is

$\Delta Y = \Delta Y_{m+1}-Y_m$

$\Delta Y = (m+1)N\lambda L - mN\lambda L$

Therefore the answer is

$\Delta Y = N\lambda L$