Two slits spaced 0.450 mm apart are placed 75.0 cm from a screen. What is the distance between the second and third dark lines of the interference pattern on the screen when the slits are illuminated with coherent light with a wavelength of 500 nm?

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Step 1: Understand the problem. This is a double-slit interference problem where we need to calculate the distance between the second and third dark lines (minima) on the screen. The formula for the position of dark fringes is given by: \( y_m = \frac{(m + 0.5) \lambda L}{d} \), where \( m \) is the order of the dark fringe, \( \lambda \) is the wavelength of light, \( L \) is the distance to the screen, and \( d \) is the slit separation.

Step 2: Identify the given values. From the problem, \( d = 0.450 \; \text{mm} = 0.450 \times 10^{-3} \; \text{m} \), \( L = 75.0 \; \text{cm} = 0.75 \; \text{m} \), and \( \lambda = 500 \; \text{nm} = 500 \times 10^{-9} \; \text{m} \).

Step 3: Calculate the position of the second dark fringe (\( m = 2 \)). Substitute \( m = 2 \) into the formula \( y_m = \frac{(m + 0.5) \lambda L}{d} \): \( y_2 = \frac{(2 + 0.5) \cdot 500 \times 10^{-9} \cdot 0.75}{0.450 \times 10^{-3}} \).

Step 4: Calculate the position of the third dark fringe (\( m = 3 \)). Substitute \( m = 3 \) into the formula \( y_m = \frac{(m + 0.5) \lambda L}{d} \): \( y_3 = \frac{(3 + 0.5) \cdot 500 \times 10^{-9} \cdot 0.75}{0.450 \times 10^{-3}} \).

Step 5: Find the distance between the second and third dark lines. Subtract the position of the second dark fringe from the position of the third dark fringe: \( \Delta y = y_3 - y_2 \). This gives the required distance between the second and third dark lines.

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주요 개념

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Young's Double Slit Experiment

This experiment demonstrates the wave nature of light through the interference pattern created when coherent light passes through two closely spaced slits. The resulting pattern consists of alternating bright and dark fringes on a screen, which can be analyzed to determine the positions of these fringes based on the wavelength of the light and the geometry of the setup.

Interference Pattern

An interference pattern arises from the superposition of waves from the two slits, leading to constructive interference (bright fringes) and destructive interference (dark fringes). The positions of these fringes depend on the path difference between the waves from the two slits, which is influenced by the wavelength of the light and the distance to the screen.

Path Difference and Fringe Spacing

The path difference between the light waves from the two slits determines whether they interfere constructively or destructively. For dark fringes, the path difference must equal an odd multiple of half the wavelength. The distance between adjacent dark fringes can be calculated using the formula for fringe spacing, which incorporates the slit separation, distance to the screen, and wavelength of the light.