Question

The separation between successive fringes in a double slit arrangement is $x$. If the whole arrangement is dipped under water what will be the new fringe separation? [The wavelenght of light being used is 5000Å]

• A. 1.5x
• B. x
• C. 0.75x
• D. 2x

Answer 1

Answer: (C)

0.75x

Answer 2

The fringe separation in a double-slit experiment is directly proportional to the wavelength of light. When the setup is dipped in water, the wavelength of light decreases by a factor equal to the refractive index of water ($\mu_w$). Since the fringe separation is proportional to the wavelength, the new fringe separation in water will be the original fringe separation divided by the refractive index of water.

Initial fringe separation $x = \frac{\lambda_{air} D}{d}$.

New fringe separation $\beta_{water} = \frac{\lambda_{water} D}{d}$.

Wavelength in water $\lambda_{water} = \frac{\lambda_{air}}{\mu_w}$.

So, $\beta_{water} = \frac{(\lambda_{air}/\mu_w) D}{d} = \frac{1}{\mu_w} \left(\frac{\lambda_{air} D}{d}\right) = \frac{x}{\mu_w}$.

Assuming the refractive index of water $\mu_w = 4/3$, the new fringe separation is $\beta_{water} = \frac{x}{4/3} = \frac{3}{4}x = 0.75x$.