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In a single slit diffraction set up, sec...

In a single slit diffraction set up, second minima is observed at an angle of `60^(@)`. The expected position of first minima is

A

`25^(@)`

B

`20^(@)`

C

`30^(@)`

D

`45^(@)`

Text Solution

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To solve the problem of finding the expected position of the first minima in a single slit diffraction setup, given that the second minima is observed at an angle of \(60^\circ\), we can follow these steps: ### Step-by-Step Solution: 1. **Understanding the Condition for Minima**: In a single slit diffraction pattern, the condition for the minima is given by the formula: \[ d \sin \theta = n \lambda \] where \(d\) is the width of the slit, \(\theta\) is the angle at which the minima occurs, \(n\) is the order of the minima (1 for the first minima, 2 for the second minima, etc.), and \(\lambda\) is the wavelength of the light used. 2. **Applying the Given Information**: We know that the second minima occurs at an angle of \(60^\circ\). Thus, for the second minima (\(n = 2\)): \[ d \sin(60^\circ) = 2 \lambda \] 3. **Calculating \(\sin(60^\circ)\)**: The value of \(\sin(60^\circ)\) is: \[ \sin(60^\circ) = \frac{\sqrt{3}}{2} \] Substituting this into the equation gives: \[ d \cdot \frac{\sqrt{3}}{2} = 2 \lambda \] 4. **Rearranging to Find \(\lambda\)**: Rearranging the equation to solve for \(\lambda\): \[ \lambda = \frac{d \sqrt{3}}{4} \] 5. **Finding the First Minima**: Now we need to find the angle for the first minima (\(n = 1\)): \[ d \sin \theta_1 = 1 \lambda \] Substituting \(\lambda\) from the previous step: \[ d \sin \theta_1 = \frac{d \sqrt{3}}{4} \] 6. **Dividing Both Sides by \(d\)**: Assuming \(d \neq 0\), we can divide both sides by \(d\): \[ \sin \theta_1 = \frac{\sqrt{3}}{4} \] 7. **Calculating \(\theta_1\)**: To find \(\theta_1\), we take the inverse sine: \[ \theta_1 = \sin^{-1}\left(\frac{\sqrt{3}}{4}\right) \] 8. **Finding the Angle**: Using a calculator or sine table, we find: \[ \theta_1 \approx 25^\circ \] ### Final Answer: The expected position of the first minima is at an angle of approximately \(25^\circ\).
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