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Two light sources can be adjusted to emit monochromatic light of any visible wavelength. The two sources are coherent, 2.04 \(\mu\)m apart, and in line with an observer, so that one source is 2.04 \(\mu\)m farther from the observer than the other. (a) For what visible wavelengths (380 to 750 nm) will the observer see the brightest light, owing to constructive interference? (b) How would your answers to part (a) be affected if the two sources were not in line with the observer, but were still arranged so that one source is 2.04 \(\mu\)m farther away from the observer than the other? (c) For what visible wavelengths will there be \(destructive\) interference at the location of the observer?

Short Answer

Expert verified
(a) For constructive interference: 680 nm, 510 nm, etc. (b) Same results apply if not inline but equidistant changes. (c) For destructive interference: 582 nm, etc.

Step by step solution

01

Understanding Constructive Interference

For constructive interference to occur, the path difference between the light from the two sources must be an integer multiple of the wavelength. This condition can be expressed with the equation: \[ \Delta d = m \lambda \] where \( \Delta d \) is the path difference, \( m \) is an integer (0, 1, 2, ...), and \( \lambda \) is the wavelength. Given that \( \Delta d = 2.04 \mu m \), we can find \( \lambda \) for different values of \( m \).
02

Calculating Wavelengths for Constructive Interference

We rearrange the equation to find the wavelength: \[ \lambda = \frac{\Delta d}{m} = \frac{2.04 \mu m}{m} \] Convert \( 2.04 \mu m \) to nm: \( 2.04 \mu m = 2040 \) nm. We seek wavelengths in the range of 380 nm to 750 nm. Solve for \( m \) by checking the integer values that keep \( \lambda \) within this range: \( m = 3, 4, 5, \) and so on until those conditions are met.
03

Solving for Constructive Interference Wavelengths

Calculate \( \lambda \) for each integer \( m \): - For \( m = 3 \), \( \lambda = \frac{2040}{3} = 680 \) nm (valid)- For \( m = 4 \), \( \lambda = \frac{2040}{4} = 510 \) nm (valid)- Check larger values to see if they fit within the visible range. These calculated wavelengths ensure constructive interference occurs.
04

Analyzing Effect if Sources Are Not Inline

If the sources are not inline but maintain the distance such that one source is 2.04 \( \mu m \) farther away, the condition for constructive interference doesn't change. Thus, the wavelengths calculated remain the same.
05

Understanding Destructive Interference

For destructive interference, the path difference is an odd half multiple of the wavelength. The condition for this is given by: \[ \Delta d = \left( m + \frac{1}{2} \right) \lambda \]. Using similar conversion and path difference \( 2040 \) nm, solve for \( \lambda \) within the visible range.
06

Calculating Wavelengths for Destructive Interference

The wavelengths can be calculated using \[ \lambda = \frac{2040}{m + \frac{1}{2}} \] ensuring \( \lambda \) is between 380 nm to 750 nm. Solve for integer \( m \) with values such as \( m = 2, 3, 4, \) etc., similar to the constructive interference.
07

Solving for Destructive Interference Wavelengths

With the formula, calculate for different \( m \):- For \( m = 2 \), \( \lambda = \frac{2040}{2.5} = 816 \) nm (not visible)- For \( m = 3 \), \( \lambda = \frac{2040}{3.5} = 582 \) nm (valid)- Find all \( \lambda \) within those bounds for visible light. It reveals the wavelengths where destructive interference occurs.

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Key Concepts

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

Constructive Interference
When we talk about light and interference, constructive interference refers to the situation where two light waves meet and combine to produce a wave of greater amplitude. This leads to brighter light observations. For constructive interference to occur, the path difference between waves from two sources must be an integer multiple of their wavelength.

To calculate the specific wavelengths at which constructive interference happens, use the formula:
\[ \Delta d = m \lambda \]Here, \( \Delta d \) is the path difference, \( m \) is an integer, and \( \lambda \) is the wavelength of the light.

By substituting the given path difference of 2040 nm, you can determine valid wavelengths of light by checking for integer values of \( m \) which provide wavelengths within the visible range of 380 nm to 750 nm. Some valid wavelengths include 680 nm for \( m = 3 \) and 510 nm for \( m = 4 \). What changes if the arrangement of light sources is altered but maintains the path difference is that the conditions for constructive interference still apply. Thus, the calculated wavelengths remain valid regardless of the positioning of sources relative to the observer.
Destructive Interference
Contrasting with constructive interference, destructive interference is when two waves meet and cancel each other out, resulting in a wave of reduced or no amplitude. This leads to dimmer light or darkness. Destructive interference occurs when the path difference is an odd multiple of half the wavelength:

\[\Delta d = \left( m + \frac{1}{2} \right) \lambda\]This equation means that waves are out of phase, creating cancellation.

To find visible wavelengths that lead to destructive interference, solve the equation by substituting the path difference of 2040 nm and varying \( m \). The condition will only be satisfied for certain wavelengths that fall within the visible light spectrum of 380 nm to 750 nm. For instance, a wavelength of 582 nm occurs for \( m = 3 \), demonstrating how specific values of \( \lambda \) result in destructive interference.
Keep in mind that while calculating, the wavelengths must fall within the visible spectrum to observe them. The math reveals how and why certain wavelengths become 'invisible' to an observer due to destructive interference.
Visible Wavelengths
Visible wavelengths are the portion of the electromagnetic spectrum that can be detected by the human eye. These wavelengths range from about 380 nm (violet) to 750 nm (red). The colors we perceive depend on the wavelength of light. Shorter wavelengths are closer to violet and blue, while longer wavelengths shift towards red.

When discussing interference, the visibility of light can dramatically shift due to effects like constructive and destructive interference. Constructive interference can make specific wavelengths stand out more brightly, while destructive interference can cause them to 'disappear'.
To calculate which wavelengths are affected, it's critical to understand that the interaction between light waves and their relative phase plays a pivotal role in what the observer sees. For instance, when the wavelength corresponds to constructive interference conditions, that wavelength will appear more vibrant, whereas destructive interference can completely negate the visibility of a particular wavelength.
Understanding visible wavelengths is key to predicting and explaining the patterns and colors observed when light waves interact, whether through diffraction or interference. This knowledge helps elucidate how natural phenomena like rainbows or soap bubbles get their vivid colors from specific interference patterns.

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Most popular questions from this chapter

Laser light of wavelength 510 nm is traveling in air and shines at normal incidence onto the flat end of a transparent plastic rod that has \(n\) = 1.30. The end of the rod has a thin coating of a transparent material that has refractive index 1.65. What is the minimum (nonzero) thickness of the coating (a) for which there is maximum transmission of the light into the rod; (b) for which transmission into the rod is minimized?

Coherent light with wavelength 450 nm falls on a pair of slits. On a screen 1.80 m away, the distance between dark fringes is 3.90 mm. What is the slit separation?

Two flat plates of glass with parallel faces are on a table, one plate on the other. Each plate is 11.0 cm long and has a refractive index of 1.55. A very thin sheet of metal foil is inserted under the end of the upper plate to raise it slightly at that end, in a manner similar to that discussed in Example 35.4. When you view the glass plates from above with reflected white light, you observe that, at 1.15 mm from the line where the sheets are in contact, the violet light of wavelength 400.0 nm is enhanced in this reflected light, but no visible light is enhanced closer to the line of contact. (a) How far from the line of contact will green light (of wavelength 550.0 nm) and orange light (of wavelength 600.0 nm) first be enhanced? (b) How far from the line of contact will the violet, green, and orange light again be enhanced in the reflected light? (c) How thick is the metal foil holding the ends of the plates apart?

What is the thinnest soap film (excluding the case of zero thickness) that appears black when illuminated with light with wavelength 480 nm? The index of refraction of the film is 1.33, and there is air on both sides of the film.

In your research lab, a very thin, flat piece of glass with refractive index 1.40 and uniform thickness covers the opening of a chamber that holds a gas sample. The refractive indexes of the gases on either side of the glass are very close to unity. To determine the thickness of the glass, you shine coherent light of wavelength \(\lambda_0\) in vacuum at normal incidence onto the surface of the glass. When \(\lambda_0\) = 496 nm, constructive interference occurs for light that is reflected at the two surfaces of the glass. You find that the next shorter wavelength in vacuum for which there is constructive interference is 386 nm. (a) Use these measurements to calculate the thickness of the glass. (b) What is the longest wavelength in vacuum for which there is constructive interference for the reflected light?

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