Problem 16 - University Physics with Modern Physics

Question

Coherent light that contains two wavelengths, 660 \(\mathrm{nm}\) (red) and 470 nm (blue), passes through two narrow slits separated by \(0.300 \mathrm{mm},\) and the interference pattern is observed on a screen 5.00 \(\mathrm{m}\) from the slits. What is the distance on the screen between the first-order bright fringes for the two wavelengths?

Step-by-Step Solution

Answer

The distance is 3.17 mm between the first-order bright fringes for the red and blue wavelengths.

Step 1: Identify Key Variables and Formula

First, note the given wavelengths: 660 nm (red) and 470 nm (blue), as well as the slit separation \(d = 0.300\, \text{mm} = 0.0003\, \text{m}\) and the distance to the screen \(L = 5.00\, \text{m}\). The formula for the position \(y\) of the m-th order bright fringe is \(y = \frac{m \lambda L}{d}\), where \(m\) is the order (1 in this case).

Step 2: Calculate Position for Red Fringe (660 nm)

Substitute the red wavelength into the formula: \(\lambda = 660 \times 10^{-9}\, \text{m}\), \(m = 1\), \(L = 5.00\, \text{m}\), \(d = 0.0003\, \text{m}\). So, \(y_\text{red} = \frac{660 \times 10^{-9} \times 5.00}{0.0003}\).

Step 3: Solve for Position of Red Fringe

Calculate \(y_\text{red}\):\[y_\text{red} = \frac{660 \times 10^{-9} \times 5.00}{0.0003} = 0.011\, \text{m} = 11.0\, \text{mm}.\]

Step 4: Calculate Position for Blue Fringe (470 nm)

Use \(\lambda = 470 \times 10^{-9}\, \text{m}\). Substitute into the formula: \(y_\text{blue} = \frac{470 \times 10^{-9} \times 5.00}{0.0003}\).

Step 5: Solve for Position of Blue Fringe

Calculate \(y_\text{blue}\):\[y_\text{blue} = \frac{470 \times 10^{-9} \times 5.00}{0.0003} = 0.00783\, \text{m} = 7.83\, \text{mm}.\]

Step 6: Find Distance Between First Order Fringes

The distance between the first-order bright fringes for the two wavelengths is the difference: \(\Delta y = y_\text{red} - y_\text{blue}\).

Step 7: Calculate Delta y

Calculate the distance: \[\Delta y = 11.0\, \text{mm} - 7.83\, \text{mm} = 3.17\, \text{mm}.\]

Key Concepts

Coherent Light

Coherent light is a type of light where the waves have a constant phase difference, the same frequency, and, often, the same amplitude. These properties make coherent light essential for producing clear and stable interference patterns, such as those seen in physics experiments involving light and slits. In most cases, coherent light is achieved using lasers, due to their precise control over these properties.

• Constant Phase Difference: This means that at any given point, the phase difference between two light waves remains unchanged over time.
• Same Frequency: Coherent waves also have the same frequency, meaning the number of oscillations or vibrations per second is identical.
• Same Amplitude: While not always necessary, having the same amplitude helps in achieving maximum interference effects.

Coherent light is crucial for experiments like the one described, where precise measurements of interference patterns are required.

Wavelength

Wavelength is the distance between consecutive peaks of a wave, often represented by the symbol \(\lambda\). It is a fundamental characteristic of all waves, including light, and determines the color of visible light. In the exercise, two different wavelengths are used: 660 nm (red light) and 470 nm (blue light).

• Red Light (660 nm): Longer wavelength, and is less refracted through different mediums, which often makes red light appear more spread out in an interference pattern.
• Blue Light (470 nm): Shorter wavelength, which results in higher frequencies and often more pronounced bending and scattering.

The difference in these wavelengths is what allows us to see distinct positions of their corresponding bright fringes on the screen.

Slit Separation

Slit separation, denoted as \(d\), is the distance between the two slits through which the light passes in a double-slit experiment. This distance is a critical factor in determining the nature and spacing of the interference pattern created. In this exercise, the slit separation is 0.300 mm.The role of slit separation is vital because:

• The smaller the slit separation, the wider the fringes will appear on the screen.
• Conversely, increasing the slit separation will cause the fringes to be closer together.

By adjusting the slit separation, one can control the layout of the interference pattern, a principle commonly applied in various optical devices and tests.

Bright Fringes

Bright fringes, which appear in interference patterns, are areas of constructive interference where two or more light waves superimpose each other and the amplitudes add up to create a brighter effect. These fringes are the result of coherent light waves meeting in phase. Their precise positions are determined by multiple factors:

• Wavelength: Different wavelengths will produce bright fringes at different positions due to varying distances between wave peaks.
• Order of Interference: In the given problem, we consider the first-order (\(m=1\)) bright fringe, which is the closest to the central maximum.
• Slit Separation: As mentioned, the wider the slits are apart, the closer the bright fringes will be.

In practical terms, bright fringes are where you observe the most noticeable and vivid light bands, crucial for analyzing wave behaviors.