In a two-slit interference pattern, the intensity at the peak of the central maximum is I₀. 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. Step 1: Understand the relationship between intensity and phase difference in a two-slit interference pattern. The intensity at a given point is determined by the superposition of the electric fields from the two slits. The formula for intensity is: \( I = I_0 \cos^2(\Delta \phi / 2) \), where \( \Delta \phi \) is the phase difference and \( I_0 \) is the maximum intensity.
2. Step 2: Identify the given values in the problem. The maximum intensity \( I_0 \) is provided, and the phase difference \( \Delta \phi \) is given as 60.0°.
3. Step 3: Convert the phase difference from degrees to radians if necessary, since trigonometric functions in physics often use radians. Use the conversion formula: \( \Delta \phi_{\text{radians}} = \Delta \phi_{\text{degrees}} \times \frac{\pi}{180} \).
4. Step 4: Substitute the phase difference \( \Delta \phi \) into the formula \( I = I_0 \cos^2(\Delta \phi / 2) \). Divide the phase difference by 2 to calculate \( \Delta \phi / 2 \), and then compute \( \cos(\Delta \phi / 2) \).
5. Step 5: Square the result of \( \cos(\Delta \phi / 2) \) to find \( \cos^2(\Delta \phi / 2) \), and multiply this value by \( I_0 \) to determine the intensity \( I \) at the given phase difference.

Key Concepts

Interference of Waves

Interference occurs when two or more waves overlap and combine to form a new wave pattern. In the context of light waves, constructive interference happens when waves are in phase, leading to increased intensity, while destructive interference occurs when waves are out of phase, resulting in reduced intensity. The two-slit experiment exemplifies this phenomenon, demonstrating how light can create patterns of 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° or 360° 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 resulting intensity at that point in the interference pattern.

Intensity of Light

The intensity of light is a measure of the power per unit area carried by a wave and is proportional to the square of the amplitude of the wave. In interference patterns, the intensity at any point can be calculated using the formula I = I0 * (1 + cos(Δφ)), where I0 is the maximum intensity and Δφ is the phase difference. This relationship allows us to determine the intensity at points in the pattern based on the phase differences between the interfering waves.