Two 0.010-mm-wide slits are 0.030 mm apart (center to center). Determine (a) the spacing between interference fringes for 520-nm light on a screen 1.0 m away and (b) the distance between the two diffraction minima on either side of the central maximum of the envelope.

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Step 1: Identify the given values for part (a). The slit separation (d) is 0.030 mm = 3.0 × 10⁻⁵ m, the wavelength of light (λ) is 520 nm = 5.2 × 10⁻⁷ m, and the distance to the screen (L) is 1.0 m. The formula for fringe spacing (Δy) in a double-slit interference pattern is Δy = (λ × L) / d.

Step 2: Substitute the given values into the formula for fringe spacing. Use Δy = (λ × L) / d, where λ = 5.2 × 10⁻⁷ m, L = 1.0 m, and d = 3.0 × 10⁻⁵ m. Simplify the expression to find Δy.

Step 3: For part (b), identify the given values for the diffraction envelope. The slit width (a) is 0.010 mm = 1.0 × 10⁻⁵ m, and the wavelength of light (λ) is 520 nm = 5.2 × 10⁻⁷ m. The formula for the angular position of the first diffraction minimum is sin(θ) = mλ / a, where m = ±1 for the first minima.

Step 4: Solve for the angular position (θ) of the first diffraction minima using sin(θ) = λ / a. Substitute λ = 5.2 × 10⁻⁷ m and a = 1.0 × 10⁻⁵ m into the equation. Then, calculate θ for both m = +1 and m = -1.

Step 5: To find the distance between the two diffraction minima on the screen, use the small-angle approximation, where y = L × tan(θ) ≈ L × sin(θ). Calculate the positions of the two minima (y₁ and y₂) on the screen and find the distance between them as |y₁ - y₂|.

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주요 개념

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

Double-Slit Interference

Double-slit interference occurs when light waves pass through two closely spaced slits, creating an interference pattern on a screen. The pattern consists of alternating bright and dark fringes due to constructive and destructive interference of the light waves. The spacing between these fringes can be calculated using the formula: Δy = λL/d, where Δy is the fringe spacing, λ is the wavelength of light, L is the distance to the screen, and d is the distance between the slits.

Diffraction

Diffraction is the bending of waves around obstacles and the spreading of waves when they pass through narrow openings. In the context of the double-slit experiment, diffraction causes the light waves to spread out after passing through the slits, leading to a broader envelope of intensity that contains the interference pattern. The positions of the minima in the diffraction pattern can be determined using the formula: a sin(θ) = mλ, where a is the slit width, θ is the angle of the minima, m is the order of the minimum, and λ is the wavelength.

Interference and Diffraction Minima

In the context of the double-slit experiment, interference minima occur at points where the path difference between light from the two slits leads to destructive interference. Similarly, diffraction minima are points where the intensity of the light is significantly reduced due to the wave nature of light interacting with the slit width. The distance between these minima can be calculated based on the geometry of the setup and the wavelength of the light used.