Question:
When light of wavelength $\lambda$ incidents on a single slit of width 'a', then the angular width between the third order diffraction maxima on either side of the central maximum is:
When light of wavelength $\lambda$ incidents on a single slit of width 'a', then the angular width between the third order diffraction maxima on either side of the central maximum is:
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The angular width between any symmetric $n^{\text{th}}$ secondary maxima on either side of the central peak is always:
$\Delta \theta = (2n + 1)\frac{\lambda}{a}$.
For $n = 3$, substituting gives $\Delta \theta = (2(3) + 1)\frac{\lambda}{a} = \frac{7\lambda}{a}$.
$\Delta \theta = (2n + 1)\frac{\lambda}{a}$.
For $n = 3$, substituting gives $\Delta \theta = (2(3) + 1)\frac{\lambda}{a} = \frac{7\lambda}{a}$.
Updated On: Jul 22, 2026
- $\frac{9\lambda}{a}$
- $\frac{7\lambda}{a}$
- $\frac{5\lambda}{a}$
- $\frac{3\lambda}{a}$
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The Correct Option is B
Solution and Explanation
Step 1: Understanding the Question:
The question asks for the angular separation between the third-order secondary diffraction maxima located on opposite sides of the central maximum in a single-slit diffraction pattern.
Step 2: Key Formula and Approach:
In single-slit diffraction, the positions of secondary maxima are given by:
\[ a \sin\theta \approx \left(n + \frac{1}{2}\right)\lambda \] For small angles, $\sin\theta \approx \theta$, so:
\[ \theta_n = \left(n + \frac{1}{2}\right)\frac{\lambda}{a} \] We will find the position of the third-order maximum ($n=3$) and calculate the total angular width between the maxima on both sides.
Step 3: Detailed Explanation:
• Position of the $n^{\text{th}}$ secondary maximum:
The central maximum lies at $\theta = 0$.
The secondary maxima are located symmetrically on both sides of the center at angles:
\[ \theta_n = \pm \left(n + \frac{1}{2}\right)\frac{\lambda}{a} \]
• For $3^{\text{rd}}$ order secondary maxima ($n = 3$):
The angular position is:
\[ \theta_3 = \left(3 + \frac{1}{2}\right)\frac{\lambda}{a} = \frac{7}{2}\frac{\lambda}{a} \]
• Angular width between them on either side ($\Delta\theta$):
The separation between $+\theta_3$ and $-\theta_3$ is:
\[ \Delta\theta = 2 \times \theta_3 \] \[ \Delta\theta = 2 \times \left(\frac{7\lambda}{2a}\right) = \frac{7\lambda}{a} \]
Step 4: Final Answer:
The angular width is $\frac{7\lambda}{a}$, which corresponds to Option (B).
The question asks for the angular separation between the third-order secondary diffraction maxima located on opposite sides of the central maximum in a single-slit diffraction pattern.
Step 2: Key Formula and Approach:
In single-slit diffraction, the positions of secondary maxima are given by:
\[ a \sin\theta \approx \left(n + \frac{1}{2}\right)\lambda \] For small angles, $\sin\theta \approx \theta$, so:
\[ \theta_n = \left(n + \frac{1}{2}\right)\frac{\lambda}{a} \] We will find the position of the third-order maximum ($n=3$) and calculate the total angular width between the maxima on both sides.
Step 3: Detailed Explanation:
• Position of the $n^{\text{th}}$ secondary maximum:
The central maximum lies at $\theta = 0$.
The secondary maxima are located symmetrically on both sides of the center at angles:
\[ \theta_n = \pm \left(n + \frac{1}{2}\right)\frac{\lambda}{a} \]
• For $3^{\text{rd}}$ order secondary maxima ($n = 3$):
The angular position is:
\[ \theta_3 = \left(3 + \frac{1}{2}\right)\frac{\lambda}{a} = \frac{7}{2}\frac{\lambda}{a} \]
• Angular width between them on either side ($\Delta\theta$):
The separation between $+\theta_3$ and $-\theta_3$ is:
\[ \Delta\theta = 2 \times \theta_3 \] \[ \Delta\theta = 2 \times \left(\frac{7\lambda}{2a}\right) = \frac{7\lambda}{a} \]
Step 4: Final Answer:
The angular width is $\frac{7\lambda}{a}$, which corresponds to Option (B).
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