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Step-by-Step Solution
Step 1: Express All Given Quantities in SI Units
• Wavelengths:
$ \lambda_1 = 400\,\text{nm} = 400 \times 10^{-9}\,\text{m} \,,\quad \lambda_2 = 560\,\text{nm} = 560 \times 10^{-9}\,\text{m} .$
• Slit separation:
$ d = 0.1\,\text{mm} = 0.1 \times 10^{-3}\,\text{m} .$
• Distance to screen:
$ D = 1\,\text{m} .$
Step 2: General Expression for Position of Dark Fringes
The distance $y$ of the nth dark fringe (from the central maximum) for a wavelength $ \lambda $ is given by
$ y = \frac{(2n - 1)\,\lambda\,D}{2\,d}\,.$
Here, $n$ is an integer (1, 2, 3, …), $d$ is the slit separation, $D$ is the distance to the screen, and $ \lambda $ is the wavelength of light.
Step 3: Condition for Overlapping Dark Fringes
We have two wavelengths $ \lambda_1 $ and $ \lambda_2 $. For their dark fringes to overlap at the same point on the screen:
$ \frac{(2n - 1)\,\lambda_1\,D}{2\,d} \;=\; \frac{(2m - 1)\,\lambda_2\,D}{2\,d} \,, $
where $n$ and $m$ are the order numbers (positive integers) for the dark fringes of $ \lambda_1 $ and $ \lambda_2 $, respectively. By canceling the common factors $ \frac{D}{2d} $ from both sides, we get:
$ \frac{2n - 1}{2m - 1} = \frac{\lambda_2}{\lambda_1} = \frac{560}{400} = \frac{7}{5}\,.$
Step 4: Finding the Smallest Positive Integer Solution
We need integers $ n $ and $ m $ that satisfy
$ (2n - 1) : (2m - 1) = 7 : 5 .$
The smallest solution is:
$ 2n - 1 = 7 \;\Rightarrow\; n = 4,\quad 2m - 1 = 5 \;\Rightarrow\; m = 3 .$
Step 5: Find Positions of These Overlapping Dark Fringes
If $ n = 4 $ for $ \lambda_1 $, the position on the screen is:
$ y_1 = \frac{(2 \times 4 - 1)\,\lambda_1\,D}{2\,d}
= \frac{7 \times (400 \times 10^{-9}\,\text{m}) \times 1\,\text{m}}{2 \times 0.1 \times 10^{-3}\,\text{m}} \,. $
Converting to millimeters:
$ y_1 = 14\,\text{mm}\,.
The next overlapping set happens at a larger fringe order, for instance $ n = 11 $ (ensuring the ratio remains $7:5$). Then:
$ y_2 = \frac{(2 \times 11 - 1)\,\lambda_1\,D}{2\,d}
= \frac{21 \times (400 \times 10^{-9}\,\text{m}) \times 1\,\text{m}}{2 \times 0.1 \times 10^{-3}\,\text{m}} \,=\, 42\,\text{mm}\,.
Step 6: Calculate the Minimum Distance Between Two Successive Overlapping Dark Fringes
The difference between these positions is:
$ y_2 - y_1 = 42\,\text{mm} - 14\,\text{mm} = 28\,\text{mm}\,.
Hence, the minimum distance between two successive regions of complete darkness is 28 mm.
