In a two-slit interference pattern, the intensity at the peak of the central maximum is I0. (a) At a point in the pattern where the phase difference between the waves from the two slits is 60.0°, what is the intensity?

Verified step by step guidance

1. Identify the phase difference given in the problem, which is 60.0°. Convert this angle to radians for calculations, using the conversion factor \\( \pi \\text{ radians} = 180° \\).
2. Use the formula for the intensity at a point in a two-slit interference pattern, which is given by \\( I = I_0 \cos^2(\frac{\Delta \phi}{2}) \\), where \\( I_0 \\) is the intensity at the central maximum and \\( \Delta \phi \\) is the phase difference between the two waves.
3. Substitute the phase difference in radians into the formula. Calculate \\( \frac{\Delta \phi}{2} \\) first, as it is used in the cosine function.
4. Compute \\( \cos^2(\frac{\Delta \phi}{2}) \\) using a calculator or cosine table to find the square of the cosine of the half phase difference.
5. Finally, multiply \\( I_0 \\) by \\( \cos^2(\frac{\Delta \phi}{2}) \\) to find the intensity \\( I \\) at the point where the phase difference is 60.0°.

Key Concepts

Interference of Waves

Interference occurs when two or more waves overlap, resulting in a new wave pattern. In the context of the two-slit experiment, constructive interference happens when waves from the slits are in phase, leading to increased intensity, while destructive interference occurs when they are out of phase, reducing intensity. The resulting pattern is characterized by alternating bright and dark fringes.

Phase Difference

Phase difference refers to the difference in the phase of two waves at a given point in time. It is measured in degrees or radians and is crucial in determining the type of interference that occurs. For example, a phase difference of 0° results in constructive interference, while a phase difference of 180° leads to destructive interference. In this question, a phase difference of 60° will affect the intensity of the resulting wave.

Intensity of Light

The intensity of light is a measure of the power per unit area carried by a wave. In interference patterns, the intensity at any point can be calculated using the formula I = I0 * cos²(Δφ/2), where Δφ is the phase difference. This relationship shows how the intensity varies with phase difference, allowing us to determine the intensity at points in the interference pattern based on the initial intensity I0.