Light of 600 nm wavelength passes through a single slit and creates a 2.0-cm-wide central maximum on a screen behind the slit. What wavelength of light will create a 3.0-cm-wide central maximum on a screen twice as far away?

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Understand the problem: The central maximum width in a single-slit diffraction pattern is determined by the wavelength of light, the slit width, and the distance to the screen. The relationship is given by the formula for the angular width of the central maximum: \( \Delta \theta = \frac{2 \lambda}{a} \), where \( \lambda \) is the wavelength, and \( a \) is the slit width. The linear width on the screen is related to \( \Delta \theta \) and the screen distance \( L \).

Express the linear width of the central maximum as \( w = 2L \tan(\Delta \theta / 2) \). For small angles, \( \tan(\Delta \theta / 2) \approx \Delta \theta / 2 \), so \( w \approx \frac{2 \lambda L}{a} \). This shows that the width \( w \) is proportional to both the wavelength \( \lambda \) and the screen distance \( L \).

Set up the ratio of the two cases: Let \( \lambda_1 = 600 \ \text{nm} \), \( w_1 = 2.0 \ \text{cm} \), \( w_2 = 3.0 \ \text{cm} \), and \( L_2 = 2L_1 \). Using the proportionality \( w \propto \lambda L \), write \( \frac{w_2}{w_1} = \frac{\lambda_2 L_2}{\lambda_1 L_1} \).

Substitute \( L_2 = 2L_1 \) into the ratio: \( \frac{w_2}{w_1} = \frac{\lambda_2 (2L_1)}{\lambda_1 L_1} \). Simplify to get \( \frac{w_2}{w_1} = 2 \cdot \frac{\lambda_2}{\lambda_1} \).

Solve for \( \lambda_2 \): Rearrange the equation to find \( \lambda_2 = \frac{w_2}{2w_1} \cdot \lambda_1 \). Substitute the known values of \( w_1 \), \( w_2 \), and \( \lambda_1 \) to calculate \( \lambda_2 \).

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주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

Single Slit Diffraction

Single slit diffraction occurs when light passes through a narrow opening, causing it to spread out and create a pattern of light and dark bands on a screen. The width of the central maximum is influenced by the wavelength of the light and the width of the slit. This phenomenon is described by the diffraction formula, which relates the angle of the minima to the wavelength and slit width.

Wavelength and Central Maximum Width

The width of the central maximum in a diffraction pattern is directly related to the wavelength of the light used. A longer wavelength results in a wider central maximum, while a shorter wavelength produces a narrower one. This relationship is crucial for predicting how changes in wavelength will affect the observed diffraction pattern.

Geometric Relationships in Diffraction

In diffraction experiments, the distance from the slit to the screen affects the size of the diffraction pattern. When the screen is moved further away, the central maximum expands. The relationship between the width of the central maximum, the distance to the screen, and the wavelength can be expressed mathematically, allowing for calculations of how changes in these parameters influence the observed pattern.