Problem 62 - Physics

Question

Concept Questions (a) In a single-slit diffraction pattern the width of the central bright fringe is defined by the location of the first dark fringe that lies on either side of it. For a given slit width, does the width of the central bright fringe increase, decrease, or remain the same as the wavelength of the light increases? (b) For a given wavelength, does the width of the central bright fringe increase, decrease, or remain the same as the slit width increases? (c) When both the wavelength and the slit width change, it is possible for the width of the central bright fringe to remain the same. What condition must be satisfied for this to happen? In each case, give your reasoning. Problem A slit has a width of \(W_{1}=2.3 \times 10^{-6} \mathrm{~m}\). When light with a wavelength of \(\lambda_{1}=510\) tim passes through this slit, the width of the central bright fringe on a flat observation screen has a certain value. With the screen kept in the same place, this slit is replaced with a second slit (width \(W_{2}\) ) and a wavelength of \(\lambda_{2}=740 \mathrm{nm}\) is used. The width of the central bright fringe on the screen is observed to be unchanged. Find \(W_{2}\).

Step-by-Step Solution

Answer

(a) Increases. (b) Decreases. (c) \(\frac{\lambda}{a}\) constant. \(W_2 = 3.33 \times 10^{-6} \text{ m}.\)

Step 1: Understand Single-Slit Diffraction

The single-slit diffraction pattern consists of a central bright fringe and a series of diminishing bright and dark fringes on either side. The central bright fringe width is defined by the position of the first minimum (dark fringe). This occurs when the path difference between light from the edges of the slit leads to destructive interference.

Step 2: Determine the Condition for the Central Bright Fringe

The position of the first dark fringe in a diffraction pattern occurs at angles where the condition \(a\sin\theta = m\lambda\) is satisfied, where \(m = 1\) for the first dark fringe, \(a\) is the slit width, and \(\lambda\) is the wavelength of light. The angular width of the central fringe is thus determined by the angle \(\theta\) where \(\sin\theta = \frac{\lambda}{a}\).

Step 3: Relate Fringe Width to Slit Width and Wavelength

The width of the central bright fringe on the screen, \(W_{\text{fringe}}\), can be approximated as \(W_{\text{fringe}} = 2L\tan\theta\), where \(L\) is the distance from the slit to the screen. For small angles, \(\tan\theta \approx \sin\theta\), giving \(W_{\text{fringe}} \approx 2L\frac{\lambda}{a}\).

Step 4: Evaluate Changes with Wavelength

(a) As the wavelength \(\lambda\) increases, the width of the central bright fringe, \(W_{\text{fringe}}\), increases since \(W_{\text{fringe}} \propto \lambda\).

Step 5: Evaluate Changes with Slit Width

(b) As the slit width \(a\) increases, the width of the central bright fringe, \(W_{\text{fringe}}\), decreases since \(W_{\text{fringe}} \propto \frac{1}{a}\).

Step 6: Condition for Keeping Central Fringe Width Unchanged

(c) The width of the central bright fringe will remain unchanged if the ratio \(\frac{\lambda}{a}\) remains constant. Thus, if the wavelength \(\lambda\) increases by a factor, the slit width \(a\) should also increase by the same factor.

Step 7: Solve the Problem Using Given Values

Given the first peripheral conditions: \(\lambda_{1} = 510\) nm and \(W_{1} = 2.3 \times 10^{-6}\) m, and the second set of conditions: \(\lambda_{2} = 740\) nm and \(W_{2}\) unknown, remain unchanged. From \(\frac{\lambda_{1}}{W_{1}} = \frac{\lambda_{2}}{W_{2}}\), solve for \(W_{2}\):\[ W_{2} = W_{1}\times \frac{\lambda_{2}}{\lambda_{1}} = (2.3 \times 10^{-6} \text{ m}) \times \frac{740 \text{ nm}}{510 \text{ nm}} \approx 3.33 \times 10^{-6} \text{ m}. \]

Step 8: Final Answer

The new slit width \(W_{2}\) that keeps the width of the central bright fringe unchanged with the specified changes in wavelength is \(3.33 \times 10^{-6} \text{ m}\).

Key Concepts

Central Bright Fringe

In a single-slit diffraction pattern, the central bright fringe is the most prominent feature seen on the screen. This region of light appears at the very center of the diffraction pattern and is wider and much brighter than other fringes. It is bordered by two dark regions, called the first dark fringes, which occur on either side. The central bright fringe results from constructive interference, where the light waves pass through the slit and arrive in phase at the screen. The exact width of this central fringe is crucial, as it helps us understand how different factors such as slit width and wavelength affect the diffraction pattern. Its boundaries are defined by the position of the first dark fringe determined by the formula: \[ a \sin\theta = m\lambda \]Here, \(a\) represents the slit width, \(\theta\) is the angle at the diffraction minimum, \(m\) is the order number (for the first dark fringe, \(m = 1\)), and \(\lambda\) is the wavelength of the light used.

Wavelength Influence on Diffraction

The wavelength of light plays a significant role in diffraction patterns. When light waves encounter an obstacle or slit, they bend around it. The extent of this bending depends on the wavelength. A longer wavelength results in more noticeable diffraction and vice versa.

• As the wavelength increases, the light rays spread out more, causing the fringes to become wider and further apart.
• This is why when the wavelength of the incident light increases, the width of the central bright fringe also increases. This relationship can be described by the equation \(W_{\text{fringe}} \propto \lambda\).

Unlike shorter wavelengths, which create sharper and narrower fringes, longer wavelengths cause more pronounced interference effects. Thus, knowing the wavelength is essential in predicting how the diffraction pattern will appear.

Slit Width Effects

The width of the slit, through which light passes, significantly impacts the characteristics of the diffraction pattern. Analyzing how changes in slit width affect the pattern can help understand core diffraction concepts better.

• When the slit width increases, the width of the central bright fringe decreases. This is because wider slits allow less diffraction, causing the light waves to focus more tightly to the central region.
• Conversely, a narrower slit results in broader central fringes, since the light diffracts more extensively.

This behavior is derived from the relationship \(W_{\text{fringe}} \propto \frac{1}{a}\), where \(a\) represents the slit width. Understanding how slit width affects diffraction patterns is crucial for applications and experiments involving light interference and diffraction. Therefore, manipulating slit dimensions can help control how detailed and spread out the diffraction effects appear on a screen.