Question

Suppose the distance between the slits in a double-slit experiment is \(2.00 \cdot 10^{-5} \mathrm{~m}\). A beam of light with a wavelength of \(750 \mathrm{nm}\) is shone on the slits. What is the angular separation between the central maximum and an adjacent maximum? a) \(5.00 \cdot 10^{-2} \mathrm{rad}\) c) \(3.75 \cdot 10^{-2} \mathrm{rad}\) b) \(4.50 \cdot 10^{-2} \mathrm{rad}\) d) \(2.50 \cdot 10^{-2} \mathrm{rad}\)

Short answer

Answer: The angular separation between the central maximum and an adjacent maximum is approximately \(2.50 \cdot 10^{-2} \mathrm{rad}\).

Step-by-step solution

Write down the given information

We are given the distance between the slits, \(d = 2.00 \cdot 10^{-5} \mathrm{~m}\), and the wavelength of the light, \(\lambda = 750 \mathrm{nm} = 750 \cdot 10^{-9} \mathrm{~m}\).

Write down the double-slit interference formula

The double-slit interference formula is given by: \(\sin{\theta} = \frac{m\lambda}{d}\), where \(\theta\) is the angular separation between the central maximum and an adjacent maximum, \(m\) is the order of the maximum (an integer), \(d\) is the distance between the slits, and \(\lambda\) is the wavelength of the light.

Solve for the angular separation

In this problem, we want to find the angular separation between the central maximum and an adjacent maximum, which corresponds to \(m=1\). Plugging the values into the formula, we get: \(\sin{\theta} = \frac{1 \cdot (750 \cdot 10^{-9} \mathrm{~m})}{2.00 \cdot 10^{-5} \mathrm{~m}}.\)

Evaluate the expression and find the angle

Evaluating the expression, we get: \(\sin{\theta} = 0.0375.\) To find the angle \(\theta\), we take the inverse sine (also known as arcsine) on both sides: \(\theta = \arcsin{(0.0375)} \approx 2.15 \cdot 10^{-2} \mathrm{rad}\).

Compare with the given options and choose the closest answer

Comparing our calculated value of \(\theta\) to the given options: a) \(5.00 \cdot 10^{-2} \mathrm{rad}\) c) \(3.75 \cdot 10^{-2} \mathrm{rad}\) b) \(4.50 \cdot 10^{-2} \mathrm{rad}\) d) \(2.50 \cdot 10^{-2} \mathrm{rad}\) We see that our calculated value of \(\theta \approx 2.15 \cdot 10^{-2} \mathrm{rad}\) is closest to option d) \(2.50 \cdot 10^{-2} \mathrm{rad}\). Therefore, we choose option d) as our answer.

Key concepts

Interference Pattern

The phenomenon called the interference pattern is a hallmark of wave behavior, which occurs when two or more waves overlap and combine. In the context of the double-slit experiment, each slit acts as a source of light waves. As these waves emerge from the slits, they interact with each other—sometimes the waves reinforce one another (constructive interference), leading to bright bands on a screen, and sometimes they cancel each other out (destructive interference), resulting in dark bands.

The pattern of alternating bright and dark bands is the interference pattern, and it serves as strong evidence of the wave nature of light. The pattern is highly predictable, based on the path difference that the waves travel from the slits to a point on the screen. This is a central concept in understanding wave optics and underpins much of the behavior seen in various optical phenomena.

Angular Separation

Angular separation refers to the angle formed when lines are drawn from each of the slits in the double-slit experiment to a particular point on the interference pattern. It can be seen as the angular distance between successive bright (or dark) bands in the pattern. This measurement is key to calculating the positions of the fringes on the screen.

Mathematically, the angular separation in the double-slit experiment can be found using the formula \(\sin{\theta} = \frac{m\lambda}{d}\), where \(\theta\) is the angular separation, \(d\) is the distance between slits, \(\lambda\) is the wavelength of the light, and \(m\) is an integer that represents the order of the maximum. The formula relates the physical setup (slit separation and wavelength) to the observed interference pattern.

Wavelength

Wavelength, symbolized by \(\lambda\), is a fundamental property of waves that represents the distance between consecutive points of similar phase, such as peaks or troughs. In the context of light, wavelength is directly associated with color—each color in the visible spectrum corresponds to a different wavelength.

In the double-slit experiment, wavelength determines the spacing of the interference pattern fringes on the observing screen; shorter wavelengths produce more closely spaced fringes, whereas longer wavelengths result in fringes that are farther apart. The wavelength is crucial in the calculation for determining the angular separation and, consequently, the specific positions of the light and dark bands in the experiment.

Diffraction

Diffraction is a wave phenomenon where waves bend around obstacles or spread out as they pass through narrow openings. In the double-slit experiment, diffraction occurs as light waves pass through the two small slits. It is the diffraction of the waves that allows them to emerge from the slits and subsequently interfere with one another to create the interference pattern.

Diffraction plays a crucial role in the formation of the fringe pattern seen on a screen. Without the ability of light to diffract, or spread out, the distinctive bands of light would not form, and the fundamental nature of light as a wave would be far less apparent. Therefore, diffraction is essential for observing wave interference and helps demystify the wave-particle duality of light.