In a two-slit interference experiment, the path length to a certain point P on the screen differs for one slit in comparison with the other by 1.25λ.
(a) What is the phase difference between the two waves arriving at point P?
(b) Determine the intensity at P, expressed as a fraction of the maximum intensity Iₒ on the screen.

Verified step by step guidance

1. To determine the phase difference between the two waves arriving at point P, recall that the phase difference Δϕ is related to the path difference Δx by the formula: $\Delta \varphi =2\pi \Delta x/\lambda$. Here, the path difference Δx is given as 1.25λ. Substitute this value into the formula to calculate Δϕ.
2. Simplify the expression for Δϕ: $\Delta \varphi =2\pi (1.25)$. This will give the phase difference in radians.
3. To determine the intensity at point P, use the formula for intensity in a two-slit interference pattern: $I=I$_o(cos²(Δϕ/2)), where $I$_o is the maximum intensity and $\Delta \varphi$ is the phase difference calculated earlier.
4. Substitute the value of $\Delta \varphi$ from step 2 into the intensity formula. Simplify the expression to find the intensity at point P as a fraction of the maximum intensity $I$_o.
5. The final result will express the intensity at point P as a fraction of the maximum intensity. Ensure that the cosine function is evaluated correctly, and the fraction is simplified if possible.

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 the two-slit experiment, constructive interference happens when waves from both slits arrive in phase, while destructive interference occurs when they arrive out of phase. The resulting intensity pattern on the screen is a direct consequence of these interference effects.

Phase Difference

Phase difference refers to the difference in the phase of two waves at a given point in time. It is typically measured in radians or wavelengths (λ). In the two-slit experiment, a path length difference of 1.25λ corresponds to a phase difference of 2.5π radians, which indicates how much one wave is ahead or behind the other when they reach point P.

Intensity of Light

The intensity of light at a point in an interference pattern is related to the amplitude of the resultant wave at that point. The maximum intensity (Iₒ) occurs at points of constructive interference, while the intensity at point P can be calculated using the formula I = Iₒ * (cos²(Δφ/2)), where Δφ is the phase difference. This relationship allows us to express the intensity at P as a fraction of the maximum intensity.