(a) What is the width of a single slit that produces its first minimum at \(60.0^{\circ}\) for 600 -nm light? (b) Find the wavelength of light that has its first minimum at \(62.0^{\circ}\)

Short Answer

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(a) The width of the single slit is approximately 692.8 nm. (b) The wavelength of light that has its first minimum at \(62.0^{\circ}\) is approximately 611.8 nm.

Step by step solution

01

Part (a) - Finding the width of the slit

To find the width of the single slit, we will use the following formula for the first minimum in the interference pattern: \[\sin\theta = \dfrac{\lambda}{a}\] where \(\theta\) is the angle of the first minimum, \(\lambda\) is the wavelength of the light, and \(a\) is the width of the slit. We are given that the angle of the first minimum is \(60.0^{\circ}\), and the wavelength of the light is 600 nm. Let's rearrange the formula to find the width of the slit: \[a = \dfrac{\lambda}{\sin\theta}\] Now, plug in the given values and use the equation to find the width of the slit a.
02

Part (a) - Calculation

Now, let's plug in the given values: \[a = \dfrac{600\,\text{nm}} {\sin{60.0^{\circ}}}\] \[a \approx \dfrac{600\,\text{nm}}{0.866}\] \[a \approx 692.8\,\text{nm}\] So, the width of the single slit is approximately 692.8 nm.
03

Part (b) - Finding the wavelength of the light

To find the wavelength of light that has its first minimum at \(62.0^{\circ}\), we will use the same formula as before but rearrange it to solve for \(\lambda\): \[\lambda = a \cdot \sin\theta\] We are given that the angle of the first minimum is \(62.0^{\circ}\), and the width of the slit from part (a) is 692.8 nm.
04

Part (b) - Calculation

Now, let's plug in the given values: \[\lambda = 692.8\,\text{nm} \cdot \sin{62.0^{\circ}}\] \[a \approx 692.8\,\text{nm} \cdot 0.882\] \[a \approx 611.8\,\text{nm}\] So, the wavelength of light that has its first minimum at \(62.0^{\circ}\) is approximately 611.8 nm.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Diffraction Pattern
When a wave encounters an obstacle or a slit that is comparable in size to its wavelength, it bends around it, creating a phenomenon known as diffraction. When light diffuses through a single slit, it produces a diffraction pattern characterized by a central bright band with several darker and lighter bands to each side.

This occurs because different parts of the wavefront interfere with each other. In a single slit diffraction pattern, the central maximum is flanked by symmetrically spaced minima and maxima, which become progressively fainter and more narrowly spaced as one moves away from the center.

Understanding the placement and intensity of these bands requires an appreciation of the relationship between the slit width, the wavelength of the light, and the interference that occurs as the light waves spread out from the slit. The first minimum occurs at a specific angle where the light waves from one end of the slit interfere destructively with waves from the midpoint, canceling out the light.
Wavelength of Light
Wavelength is one of the fundamental properties of waves, defined as the distance between two consecutive points that are in phase. For light, which is an electromagnetic wave, the wavelength determines its color when it is within the visible spectrum, ranging from around 400 nm (violet) to 700 nm (red).

When dealing with diffraction, we speak of a light wave's wavelength in terms of how it interacts with objects in its path. In the exercise's context, the 600-nm wavelength falls within the visible spectrum as a deep red hue. This specific wavelength is essential when calculating the width of a slit required to create a particular diffraction pattern. By modifying this slit width, or conversely the wavelength, one can manipulate the resulting pattern on the observation screen, which has practical applications in optics and photonics.
Interference in Physics
Interference is the process where two or more waves superimpose to form a resultant wave of greater, lower, or the same amplitude. There are two types of interference in physics: constructive and destructive. Constructive interference happens when the peaks of two waves align, and their amplitudes add together, while destructive interference occurs when the peak of one wave aligns with the trough of another, reducing or canceling out their amplitudes.

In the context of single slit diffraction, destructive interference is responsible for the dark bands (minima) observed in the diffraction pattern. It happens at specific angles where the path difference between two waves corresponds to a half-integer multiple of the light's wavelength, causing those waves to cancel each other out. The formula \( \sin\theta = \dfrac{\lambda}{a} \) is used to predict these angles of destructive interference, linking the experimentally observed phenomena back to the fundamental properties of wave behavior.

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