Double Slit Fringe Shift Explained

In a double-slit experiment, introducing a transparent sheet in front of one slit changes the optical path length, causing the interference pattern to shift. The shift occurs because the light travels slower through the sheet, increasing the path length.

Calculating Path Difference

The path difference introduced by a transparent sheet of thickness $t$ and refractive index $n$ is given by the formula:

$ \Delta = (n-1)t $

Where:

• $n$ is the refractive index of the sheet ($n = 1.56$).
• $t$ is the thickness of the sheet ($t = \alpha \times 10^{-9} \text{ m}$).

Fringe Shift Condition

When the central fringe (originally at the center, order $m=0$) shifts to the position of the $m^{th}$ bright fringe, the path difference introduced by the sheet is equivalent to $m$ wavelengths of the light source.

In this problem:

• The central fringe shifts to the position of the $7^{th}$ bright fringe, so $m = 7$.
• The wavelength of light is $\lambda = 450 \text{ nm} = 450 \times 10^{-9} \text{ m}$.

Therefore, the condition becomes:

$ (n-1)t = m\lambda $

Determining Mica Sheet Thickness

Substitute the known values into the equation:

$ (1.56 - 1) \times (\alpha \times 10^{-9} \text{ m}) = 7 \times (450 \times 10^{-9} \text{ m}) $

Simplify the equation:

$ 0.56 \times \alpha \times 10^{-9} = 3150 \times 10^{-9} $

Cancel out the $10^{-9}$ term from both sides:

$ 0.56 \alpha = 3150 $

Solve for $\alpha$:

$ \alpha = \frac{3150}{0.56} $

$ \alpha = 5625 $

The thickness of the mica sheet is $5625 \times 10^{-9} \text{ m}$.