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Ch. 34 - The Wave Nature of Light: Interference and Polarization
Giancoli Douglas - Physics for Scientists and Engineers 5th edition
Giancoli Douglas5th editionPhysics for Scientists and EngineersISBN: 9780137488179Non è quello che usi tu?Cambia libro di testo
Tutti i libri di testoGiancoli Douglas 5th editionCh. 34 - The Wave Nature of Light: Interference and PolarizationProblema 23
Capitolo 33, Problema 23

Suppose that one slit of a double-slit apparatus is wider than the other so that the intensity of light passing through it is twice as great. Determine the intensity I as a function of position (θ) on the screen for coherent light.

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Start by recalling the formula for the intensity of light in a double-slit interference pattern: \( I(\theta) = I_0 \cos^2(\phi/2) \), where \( \phi \) is the phase difference between the two waves arriving at a point on the screen.
Since one slit is wider and allows twice the intensity of light compared to the other, the amplitudes of the waves from the two slits will differ. Let the amplitude of the wave from the narrower slit be \( A \), and the amplitude from the wider slit will be \( \sqrt{2}A \) (since intensity is proportional to the square of the amplitude).
The total amplitude at a point on the screen is the vector sum of the two amplitudes. Using trigonometry, the resultant amplitude \( A_{\text{total}} \) is given by: \( A_{\text{total}} = \sqrt{A^2 + (\sqrt{2}A)^2 + 2A(\sqrt{2}A)\cos(\phi)} \).
Simplify the expression for \( A_{\text{total}} \): \( A_{\text{total}} = A\sqrt{1 + 2 + 2\sqrt{2}\cos(\phi)} = A\sqrt{3 + 2\sqrt{2}\cos(\phi)} \).
Finally, the intensity \( I(\theta) \) is proportional to the square of the total amplitude: \( I(\theta) = I_0 \left( 3 + 2\sqrt{2}\cos(\phi) \right) \), where \( \phi = \frac{2\pi d \sin(\theta)}{\lambda} \), \( d \) is the slit separation, and \( \lambda \) is the wavelength of the light.

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Double-Slit Experiment

The double-slit experiment demonstrates the wave nature of light through interference patterns created when coherent light passes through two closely spaced slits. When light waves from the slits overlap, they can constructively or destructively interfere, resulting in bright and dark fringes on a screen. Understanding this setup is crucial for analyzing how varying slit widths affect intensity distribution.
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Young's Double Slit Experiment

Intensity of Light

Intensity refers to the power per unit area carried by a wave, which in the context of light, is related to the amplitude of the light wave. In the double-slit experiment, the intensity at a point on the screen is influenced by the contributions from both slits, with wider slits allowing more light to pass through, thus affecting the overall intensity pattern observed.
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Wave Intensity

Interference Pattern

An interference pattern is formed when two or more coherent light waves overlap, leading to regions of constructive interference (bright spots) and destructive interference (dark spots). The intensity at any point on the screen can be calculated using the principle of superposition, taking into account the amplitude contributions from each slit, which is particularly important when one slit is wider than the other.
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Wave Interference & Superposition
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