Problem 46 - University Physics with Modern Physics

Question

Coherent monochromatic light of wavelength l passes through a narrow slit of width \(a\), and a diffraction pattern is observed on a screen that is a distance \(x\) from the slit. On the screen, the width \(w\) of the central diffraction maximum is twice the distance \(x\). What is the ratio \(a/ \lambda\) of the width of the slit to the wavelength of the light?

Step-by-Step Solution

Answer

The ratio \( a/\lambda \) is 1.

Step 1: Identify the Condition for Central Maximum

The condition for the central maximum in a single-slit diffraction pattern is that the first minima occur where \( a \sin \theta = m \lambda \), with \( m \) as an integer. For the first minima adjacent to the central maximum, \( m = \pm 1 \).

Step 2: Relate Width of Maximum to Angle

The width \( w \) of the central maximum is the angular spread between the first minima on either side. The angle \( \theta \) for the first minima can be given as \( \sin \theta = \lambda / a \).

Step 3: Calculate the Width of the Central Maximum

Since the width \( w \) on the screen is given as twice the distance \( x \), we have \( w = 2x \). This translates to path lengths where \( \theta \) angles sum up as \( 2 \sin \theta \approx \sin(\theta_1) + \sin(\theta_2) \approx \theta_1 + \theta_2 = 2\theta \approx 2\lambda/a \).

Step 4: Substitute and Rearrange

Given \( w = 2x \), substitute the expression \( 2x = 2x \lambda / a \). Simplifying, we get \( 2 \lambda x / a = 2x \), thus \( a = \lambda \), hence \( a/\lambda = 1 \).

Step 5: Conclude the Ratio

Through simplifying the above steps, the ratio of the slit width \( a \) to the wavelength \( \lambda \) results in \( a / \lambda = 1 \).

Key Concepts

Diffraction Pattern

When light passes through a narrow slit, it does not just go straight through in parallel lines. Instead, it spreads out in a pattern that we refer to as a diffraction pattern. This occurs due to the bending of waves around the edges of the slit. The result is a series of dark and light bands on a screen. The light bands, or maxima, are where light waves constructively interfere, and the dark bands, or minima, are where the waves destructively interfere. For a single-slit diffraction, the pattern forms because light waves emanating from different parts of the slit reach the screen at slightly different angles and interfere with each other.

• The central maximum is the brightest and widest part of the pattern.
• First-order minima flank the central maximum and are found where the path difference is \(m \lambda\) for \(m = \pm 1\).

Monochromatic Light

Monochromatic light refers to light consisting of a single wavelength. Think of it like a single note played on an instrument rather than a chord of several notes. When this type of light passes through a slit, it forms a distinct diffraction pattern. This is because light of only one wavelength interacts in a consistent way, leading to clear and predictable patterns on a screen.

• Having a single wavelength allows precise calculations and predictions of the pattern's features.
• Common sources of monochromatic light in experiments include lasers.

Utilizing monochromatic light simplifies the analysis because there are no overlapping patterns from different wavelengths.

Wavelength

Wavelength is the distance between successive peaks of a wave. In terms of light, it's a measure of the color; shorter wavelengths are towards the violet end of the spectrum, and longer wavelengths are toward the red. In the context of diffraction, the wavelength of the light determines how much the light waves will spread out after passing through a slit.

• Shorter wavelengths bend less, causing less spreading in the diffraction pattern.
• Longer wavelengths bend more, resulting in wider patterns.

The relationship between the slit width and the wavelength is crucial: if the slit width is comparable to the wavelength, a noticeable diffraction pattern will emerge.

Central Maximum

The central maximum is the principal bright band in a diffraction pattern. It is the most intense part of the pattern, sitting directly opposite the slit. This intensity is due to constructive interference from light waves that have similar path lengths as they travel through the slit and strike the screen. The width of the central maximum, denoted as \(w\), is determined by the geometry of the setup and how the light diffracts.

• In a single-slit experiment, the width \(w\) of the central maximum can be observed to be related to the distance to the screen \(x\).
• The angular spread of this maximum depends inversely on the slit width and directly on the light wavelength \(\frac{\lambda}{a}\).

Understanding the central maximum's formation helps in deducing properties like the wavelength when other parameters are known.