Question

Laser light of wavelength \(500.0 \mathrm{nm}\) illuminates two identical slits, producing an interference pattern on a screen \(90.0 \mathrm{~cm}\) from the slits. The bright bands are \(1.00 \mathrm{~cm}\) apart, and the third bright bands on either side of the central maximum are missing in the pattern. Find the width and the separation of the two slits.

Short answer

Slit width and separation are both approximately 45 μm (micrometers).

Step-by-step solution

Understand the Problem

We have laser light with a wavelength of \(500.0 \mathrm{nm}\), and it creates an interference pattern on a screen 90.0 cm away. The bright bands are 1.00 cm apart. The third-order bright band is missing. We need to find the slit width \(a\) and separation \(d\).

Use Young's Double-Slit Formula

The position \(y\) of the m-th bright fringe is given by: \[ y_m = \frac{m \cdot \lambda \cdot L}{d} \]where \(m\) is the order number, \(\lambda = 500 \times 10^{-9} \mathrm{m}\), and \(L = 0.9 \mathrm{m}\). The distance between consecutive bright fringes (fringe separation) is given by: \[ \Delta y = \frac{\lambda \cdot L}{d} \]We know \(\Delta y = 0.01 \mathrm{m} \).

Calculate the Slit Separation

From the fringe separation formula: \[ d = \frac{\lambda \cdot L}{\Delta y} \]Plug in the values: \[ d = \frac{500 \times 10^{-9} \cdot 0.9}{0.01} \approx 4.5 \times 10^{-5} \mathrm{m} \] This is the separation between the slits.

Understand Missing Order and its Implication

The third bright fringe being missing suggests destructive interference due to the single-slit diffraction condition: \[ a \cdot \sin \theta = m' \cdot \lambda \]For the third order, \(m'=3\), we test with small angles where \(\sin \theta \approx \tan \theta \approx \frac{y}{L}\).

Relate Missing Order to Slit Width

The position of the third interference fringe is: \[ y_3 = \frac{3 \cdot \lambda \cdot L}{d} \]From single-slit diffraction, the third order is at: \[ a \cdot \frac{3\cdot 0.01}{0.9} = 3 \cdot 500 \times 10^{-9} \]Solve for slit width \(a\): \[ a = \frac{0.9 \cdot 1500 \times 10^{-9}}{3\cdot 0.01} = 4.5 \times 10^{-5} \mathrm{m} \]

Conclusion

Both slit width \(a\) and separation \(d\) are approximately \(4.5 \times 10^{-5} \mathrm{m}\). They are equal due to the condition of the missing third-order fringe.

Key concepts

Wavelength

In the context of light and interference patterns, the wavelength is a fundamental concept. It refers to the distance between two corresponding points on consecutive cycles of a wave, such as from crest to crest. For visible light, wavelengths range from about 400 nm to 700 nm, with specific colors associated with particular wavelengths.

In the given problem, the laser light has a wavelength of 500 nm. This wavelength interacts with the slits to produce an interference pattern. Knowing the wavelength is essential, as it determines the spacing of the interference fringes on the detection screen. This pattern results from the wave nature of light experiencing constructive and destructive interference.

• The wavelength helps determine fringe position in experiments like Young's double-slit.
• Shorter wavelengths like ultraviolet light create narrower separation between fringes.
• Wavelength is crucial for calculating optical properties and understanding wave behaviors.

Fringe Separation

Fringe separation refers to the distance between adjacent bright or dark bands in an interference pattern. It is a critical measurement in Young's double-slit experiment, helping to determine properties like slit separation.

In the problem, the fringe separation is 1.00 cm. This is the distance between successive bright fringes that appear on a screen due to interference of light waves passing through the double slits. The separation provides insight into actual physical properties of the slits.

• The formula \( \Delta y = \frac{\lambda \cdot L}{d} \) describes fringe separation, where \( \Delta y \) is the fringe spacing.
• Increasing screen distance \( L \) increases fringe separation.
• Larger fringe separation indicates a larger distance between slits \( d \).

Destructive Interference

Destructive interference is a phenomenon where two waves combine to form a reduced amplitude. In the case of light waves, this results in dark bands on the interference pattern. It occurs when waves are out of phase by half of a wavelength, leading to cancellation.

In the exercise, the absence of the third bright fringe indicates destructive interference. The condition is imposed by single-slit effects interacting with the double-slit pattern, causing the third fringe to "disappear." This helps us find the slit dimensions.

• Destructive interference results in missing fringes or dark areas in patterns.
• Occurs at angles where the path difference between two waves is an odd multiple of half-wavelengths.
• Can also indicate slit width characteristics when specific bright fringes are absent.

Young's Double-Slit Experiment

Young's double-slit experiment is a classic demonstration of the wave nature of light. It shows how light waves passing through two slits produce an interference pattern on a screen.

The experiment provides a way to measure wavelengths and explore interference effects. In the problem, it is utilized to calculate slit dimensions based on the resulting interference pattern. The appearance of bright and dark fringes is directly related to wave interference.

• The experiment helps visualize wave phenomena like interference and diffraction.
• It validates the principle of superposition of waves.
• Can be adjusted to evaluate both light and particle behaviors with modern quantum experiments.