91Ó°ÊÓ

Wavelength of light$\mathrm{λ}=589 \text{nm}$

The reactive index of water $n=\text{1.33}$

Angularly separation of fringenear the centre of the pattern$\mathrm{Δ}\mathrm{θ}=0.30°$

Interference fringe, a bright or dark band caused by beams of light that are in phase or out of phase with one another.

In double slit experiment the expression for the angular separation $\mathrm{Δ}\mathrm{θ}$between the fringes and wavelength$\mathrm{λ}$ of light used is

$\mathrm{Δ}\mathrm{θ}=\frac{\mathrm{λ}}{d}$

Here d is the distance between the two slits.

$\left(n=1.33\right)$

Now, we have to find the wavelength of light in the water refractive index

In the double sheet experiment wavelength of light $\text{'}\mathrm{λ}\text{'}=589\text{mm}$

For the bright fringe,

The angle $\mathrm{Δ}\mathrm{θ}=0.30°$

The reactive index of water $\left(n=\text{1.33}\right)$

The angular separation between the fringes when we observed the interference pattern in air medium

The angular fringe separation

$\mathrm{Δ}\mathrm{θ}=\frac{\mathrm{λ}}{d}$ ...(i)

The angular separation between the fringes when we observed the interference pattern in water medium

$\mathrm{Δ}{\mathrm{θ}}_w=\frac{{\mathrm{λ}}_w}{d}$ ...(ii)

Here ${\mathrm{λ}}_w$ is wave length of light in the water medium

${\mathrm{λ}}_w=\frac{\mathrm{λ}}{n}$ ...(iii)

Substitution equation (iii) in equation (ii) we get

$\mathrm{Δ}{\mathrm{θ}}_w=\frac{\mathrm{λ}}{nd}$ ...(iv)

Dividing equation (iv) and (i), we get

$\frac{\mathrm{Δ}{\mathrm{θ}}_w}{\mathrm{Δ}\mathrm{θ}}=\frac{1}{n}$

And rearranging the above equation and substitute all values, we get angular separation between the fringes as

$\begin{array}{rcl}\mathrm{Δ}{\mathrm{θ}}_w& =& \frac{1}{n}\left(\mathrm{Δ}\mathrm{θ}\right)\\ & =& \frac{1}{1.33}\left(0.30°\right)\\ & =& 0.23°\end{array}$

Hence, the wavelength of light in the water refractive index is $0.23°$