Question

Light shines through a single 8.60 x 10^–4 m slit. A diffraction pattern forms on a screen 5.00 m away. The separation between the middle of the central maximum and the first dark fringe is 4.5 mm. Calculate the wavelength of the light.

Answer 1

Given:

The separation between the slits is,

$d=8.60\times10^{-4}\text{ m}$

The distance between the slit and the screen is,

$D=5.00\text{ m}$

The separation between the central maximum and the first dark fringe is,

$\begin{gathered} y=4.5\text{ mm} \\ =4.5\times10^{-3}\text{ m} \end{gathered}$

To find:

The wavelength of the light

Explanation:

The diagram of the arrangement is shown below:

The separation between the central fringe and the mth bright fringe is,

$dsin\theta=m\lambda$

Here,

$\begin{gathered} m=1 \\ tan\theta=\frac{y}{D} \\ tan\theta=\frac{4.5\times10^{-3}}{5.00} \\ \theta=tan^{-1}(9\times10^{-4}) \\ \theta=0.051\degree \end{gathered}$

Now we can write,

$\begin{gathered} 8.60\times10^{-4}\times sin(0.051\degree)=1\times\lambda \\ \lambda=7.65\times10^{-7}\text{ m} \end{gathered}$

Hence, the wavelength is

$7.65\times10^{-7}\text{ m}$