Let's break down the problem **step-by-step**: ### Step 1: **Minima Condition for Single Slit** The minima (dark spots) for a single slit occur at: \[ \sin\theta_{\text{min}} = m\frac{\lambda}{D} \] where \( m = \pm1, \pm2, \pm3, \ldots \) (but not 0). ### Step 2: **Where Do Maxima Occur?** The central maximum is at \( m = 0 \). The **first maximum (not the central one)** will be between the first two minima. - The **first minimum** is at \( \sin\theta = \frac{\lambda}{D} \). - The **second minimum** is at \( \sin\theta = \frac{2\lambda}{D} \). The **first (secondary) maximum** will be somewhere between these two minima. ### Step 3: **Location of the First Secondary Maximum** From the intensity formula for a single slit: \[ I(\theta) = I_0 \left[\frac{\sin(\beta)}{\beta}\right]^2 \] where \( \beta = \frac{\pi D}{\lambda} \sin\theta \). The maxima (other than the central one) occur approximately halfway between the minima. More precisely, the position of the first secondary maximum is found by solving: \[ \tan\beta = \beta \] for \( 0 < \beta < \pi \). The first solution (other than \(\beta=0\)) is approximately \(\beta \approx 1.43\) radians. - At the first minimum: \(\beta = \pi\) - At the first maximum: \(\beta \approx 1.43\) But for **multiple-choice tests**, the standard approximation is that the first maximum is **midway** between the minima: \[ \sin\theta_{\text{max}} \approx \frac{3\lambda}{2D} \] ### Step 4: **Compare With the Options** - (A) \( \sin\theta_{\text{max}} = \lambda/D \) ⟶ **First minimum** (not maximum) - (B) \( \sin\theta_{\text{max}} = 3\lambda/2D \) ⟶ **Between first and second minima** - (C) \( \sin\theta_{\text{max}} = 2\lambda/D \) ⟶ **Second minimum** ### **Final Answer** \[ \boxed{(B)\ \sin\theta_{\text{max}} = \frac{3\lambda}{2D}} \] **Explanation:** The first secondary maximum occurs approximately halfway between the first two minima, which is at \(\sin\theta = \frac{3\lambda}{2D}\).

Let's analyze each option carefully, considering the physics of single-slit diffraction and the positions of minima and maxima. --- ### **Background: Single-Slit Diffraction Pattern** - **Minima (dark fringes):** Occur where the slit causes destructive interference. \[ \boxed{\sin \theta_\text{min} = m \frac{\lambda}{D}, \quad m = \pm 1, \pm 2, \pm 3, \ldots} \] Here: - \(\lambda\) = wavelength of light - \(D\) = width of the slit - \(\theta\) = diffraction angle - **Maxima (bright fringes apart from the central maximum):** These are not at the minima, but occur approximately **between** the minima. The **central maximum** is at \(m=0\). --- ### **Step-by-Step Explanation of Each Option** #### **Option (A): \(\sin \theta = \lambda / D\)** - This corresponds **exactly** to the position of the **first minimum** (since \(m=1\)): \[ \sin \theta_{\text{min},1} = \frac{\lambda}{D} \] - **Interpretation:** This is **not** the position of a maximum; it's where the first minima occur. The maxima **lie between** minima, so this option suggests the maximum occurs **at the same position** as the first minimum, which is **incorrect**. --- #### **Option (B): \(\sin \theta = \frac{3\lambda}{2D}\)** - This value is **approximately halfway** between the first two minima: \[ \text{First minima at } m=1: \sin \theta = \frac{\lambda}{D} \] \[ \text{Second minima at } m=2: \sin \theta = \frac{2\lambda}{D} \] - The **first secondary maximum** (the first bright spot after the central maximum) occurs **roughly halfway** between these two minima. - **Approximate location:** \[ \sin \theta \approx \frac{3\lambda}{2D} \] - **Conclusion:** This option accurately indicates the **position of the first maximum** (excluding the central maximum). --- #### **Option (C): \(\sin \theta = \frac{2\lambda}{D}\)** - This coincides with the **second minimum** (since \(m=2\)): \[ \sin \theta_{\text{min},2} = \frac{2\lambda}{D} \] - The maxima **do not** occur at minima. They occur **between** minima, so this is **not** the position of the first maximum. --- ### **Summary of the Positions** | Feature | Approximate \(\sin \theta\) | Location | | -------- | --------------------------- | --------- | | First minimum | \(\lambda / D\) | First dark fringe beyond center | | First maximum (not central) | \(\approx 3\lambda / 2D\) | Between first and second minima | | Second minimum | \(2\lambda / D\) | Second dark fringe | --- ### **Final Answer:** \[ \boxed{\textbf{(B)}\ \sin \theta = \frac{3\lambda}{2D}} \] This corresponds to the **first maximum** (after the central maximum), which occurs roughly halfway between the first and second minima. --- ### **Summary:** - **Option (A):** At the first minimum — **not** a maximum. - **Option (B):** Between the first and second minima — **correct** location for the first secondary maximum. - **Option (C):** At the second minimum — **not** where maxima occur. **Therefore, the correct choice is (B).**