Experiment 8: Single-slit Diffraction and Double slit diffraction/interference Analysis Single Slit Diffraction: Do the various orders give the same slit width within a reasonable error ( $$ \pm 5 \% $$ )? If not, discuss the reasons for the discrepancy. 0 Single-slit diffraction (2) Does the distance between diffraction minima increase or decrease when the slit width is increased? Explain with reference to your data (observation) and theory. The distance between liffraction minima decreases when the slit width incrases. Double-slit interference. Double Slit Interference: (1) Does the distance between interference maxima increase, decrease, or stay the same when the slit separation is increased? Explain with reference to your data (observation) and theory. The distance between inferferenu maxima stays the same, but the fringes may beome less visible as the slit width increases. (2) Does the distance between interference maxima increase, decrease, or stay the same when the slit width is increased? Explain with reference to your data (observation) and your understanding of the theory. The distance between interference maxima stays the same, but the Fringes may become less visible as the slit width increases 113

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1. **Single-Slit Diffraction**

- **Comparing Slit-Width Values from Various Orders:**  
     When using the diffraction minima to calculate the slit width, we use the condition  
     $$
     a\sin\theta = m\lambda,
     $$  
     where $$a$$ is the slit width, $$m$$ is the order of the minimum ($$m = \pm1, \pm2, \dots$$), and $$\lambda$$ is the wavelength. In an ideal experiment, using different orders $$m$$ should yield the same value for $$a$$ within experimental uncertainty (typically within $$\pm 5\%$$).  
     
     **Discrepancies** may arise due to several reasons:  
     
     - **Measurement Errors:** Small errors in measuring the position of the minima or the angle $$\theta$$ can lead to variations in the calculated $$a$$.  
     - **Alignment Issues:** If the setup is not perfectly aligned (e.g., the laser beam not perpendicular to the slit), the apparent positions of the minima may shift.  
     - **Finite Slit Effects:** The theoretical model assumes an ideal slit, but real slits have imperfections and edge effects that can cause deviations.  
     - **Approximations in Small-Angle Assumption:** For larger orders, the small-angle approximation ($$\sin\theta \approx \theta$$ in radians) may break down, leading to further error.

- **Dependence of the Distance Between Diffraction Minima on Slit Width:**  
     The position of the minima is given by  
     $$
     a\sin\theta = m\lambda.
     $$  
     For small angles, $$\sin\theta \approx \theta$$ (with $$\theta$$ expressed in radians), and the linear distance $$y$$ on a screen at distance $$L$$ can be approximated as  
     $$
     y \approx L\theta \approx \frac{m\lambda L}{a}.
     $$  
     From this relation, if the slit width $$a$$ is increased while keeping all other parameters constant, the value of $$\theta$$ (and hence $$y$$) decreases. Therefore, **the distance between adjacent diffraction minima decreases as the slit width increases.**

2. **Double-Slit Interference**

- **Effect of Increasing Slit Separation on the Distance Between Interference Maxima:**  
     The condition for constructive interference (maxima) in a double-slit experiment is  
     $$
     d\sin\theta = m\lambda,
     $$  
     where $$d$$ is the slit separation and $$m$$ is an integer representing the order of the maximum.  
     
     For small angles, using $$\sin\theta \approx \theta$$ and the position on the screen given by $$y \approx L\theta$$, we have  
     $$
     y \approx \frac{m\lambda L}{d}.
     $$  
     From this equation, if the slit separation $$d$$ is increased (with constant $$\lambda$$ and $$L$$), the spacing $$\Delta y$$ between adjacent maxima, given by  
     $$
     \Delta y = \frac{\lambda L}{d},
     $$  
     will **decrease**.  
     
     *Observation:* In some experimental setups, increasing $$d$$ might not make a dramatic change in observable fringe spacing if other factors (such as the overall envelope intensity or resolution of the detection method) are limiting. However, the theoretical prediction is that the fringe spacing decreases.

- **Effect of Increasing Slit Width on the Distance Between Interference Maxima:**  
     Although the interference condition is governed by the slit separation $$d$$, each slit has a finite width $$a$$. The single-slit diffraction pattern from each slit forms an envelope for the double-slit interference fringes.  
     
     The positions of the interference maxima remain determined by  
     $$
     d\sin\theta = m\lambda,
     $$  
     which means the spacing $$\Delta y = \frac{\lambda L}{d}$$ is independent of $$a$$.  
     
     However, as the slit width $$a$$ increases, the envelope of the diffraction pattern narrows because the minima for a single slit occur at  
     $$
     a\sin\theta = m\lambda.
     $$  
     A narrower envelope means that although the positions of the interference fringes remain the same, **their visibility can be reduced** because fewer interference maxima lie within the central bright region of the envelope. In summary, increasing $$a$$ does not change the spacing between interference maxima but does affect the overall intensity distribution.

**Single-Slit Diffraction:** - **Slit Width Comparison:** Using different orders of diffraction minima should give the same slit width within a 5% error. Discrepancies may arise from measurement errors, alignment issues, and real-world slit imperfections. - **Distance Between Minima:** As the slit width increases, the distance between diffraction minima decreases. **Double-Slit Interference:** - **Effect of Slit Separation:** Increasing the slit separation decreases the distance between interference maxima. - **Effect of Slit Width:** Increasing the slit width does not change the distance between interference maxima but may reduce the visibility of the fringes.

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