Two slits are 1 mm apart and the screen is located 1 m away from the slits. A light wavelength \( 500 \, \text{nm} \) is used. The width of each slit to obtain 10 maxima of the double slit pattern within the central maximum of the single slit pattern is \(\dots\) \(\times \, 10^{-4} \, \text{m}\).
Correct Answer: 2
Approach Solution - 1
To find the width of each slit that results in 10 maxima of the double-slit pattern within the central maximum of the single-slit pattern, we begin by analyzing the interference and diffraction conditions.
For a double-slit, the condition for maxima is:
\(d \sin \theta = m\lambda\),
where \(d = 1 \, \text{mm} = 1 \times 10^{-3} \, \text{m}\) (distance between slits), \(\lambda = 500 \, \text{nm} = 500 \times 10^{-9} \, \text{m}\), and \(m\) is the order of the maximum.
The angular width of the central maximum of single-slit diffraction is given by:
\(\sin \theta = \frac{\lambda}{a}\),
where \(a\) is the slit width.
Given that 10 maxima of the double-slit pattern fit within the central maximum of the single-slit pattern, the angle for the 5th order (half of 10) double-slit maximum should equal the angle for the first minimum of the single-slit pattern:
\(\frac{5\lambda}{d} = \frac{\lambda}{a}\).
Solving for \(a\):
\(a = \frac{d}{5} = \frac{1 \times 10^{-3}}{5} = 0.2 \times 10^{-3} \, \text{m} = 2 \times 10^{-4} \, \text{m}\).
This calculated width of the slit \(a = 2 \times 10^{-4} \, \text{m}\) falls within the specified range. Therefore, the width of each slit is \(2 \times 10^{-4} \, \text{m}\).
Approach Solution -2
Given:
\[d = 1 \, \text{mm} = 10^{-3} \, \text{m}, \quad D = 1 \, \text{m}, \quad \lambda = 500 \, \text{nm} = 5 \times 10^{-7} \, \text{m}.\]
For the central maximum of the single-slit pattern to contain 10 maxima of the double-slit pattern:
\[10 \times \frac{\lambda D}{d} = \frac{2\lambda D}{a},\]
where $a$ is the slit width.
Rearranging for $a$:
\[a = \frac{d}{5}.\]
Substitute $d = 10^{-3} \, \text{m}$:
\[a = \frac{10^{-3}}{5} = 2 \times 10^{-4} \, \text{m}.\]
Thus, the slit width is:
\[a = 2 \times 10^{-4} \, \text{m}.\]
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