Final answer:
To find the minimum difference in rate between the waves from the slits, we can use the formula for calculating the intensity at a fringe in an interference pattern. Setting the intensity at the fringe to 35% of the intensity of the central maximum, we can solve for the minimum difference in rate by finding the value of the angle of the fringe that satisfies the equation. The minimum difference in rate occurs when the phase difference between the waves is π radians (180 degrees), which corresponds to an angle of 30 degrees. Substituting the given wavelength, we can solve for the distance between the slits, which is the minimum difference in rate.
Step-by-step explanation:
In order to calculate the minimum difference in rate between the waves from the slits, we need to use the concept of interference. When light passes through two slits, it creates an interference pattern on a screen. The pattern consists of bright and dark fringes, with the central maximum being the brightest. The intensity of the light at a particular fringe depends on the phase difference between the waves from the two slits.
The formula for calculating the intensity at a fringe is given by:
I = I0 * cos2(πd sinθ/λ)
Where:
- I is the intensity at the fringe
- I0 is the intensity of the central maximum
- d is the distance between the slits
- θ is the angle of the fringe
- λ is the wavelength of the light
In this case, we are given the intensity at the fringe, which is 35% of the intensity of the central maximum. We can set up the equation:
0.35I0 = I0 * cos2(πd sinθ/λ)
Simplifying, we get:
cos2(πd sinθ/λ) = 0.35
Now, we need to find the minimum difference in rate between the waves from the slits. The minimum difference in rate occurs when the phase difference between the waves is π radians (180 degrees). This means that the argument of the cosine function, πd sinθ/λ, must be equal to ±π/2 (±90 degrees). We can set up the equation:
πd sinθ/λ = ±π/2
Simplifying, we get:
sinθ = (±1/2) * (λ/d)
To find the minimum difference in rate, we need to consider the case where sinθ = 1/2. This occurs when θ = 30 degrees. Substituting this value into the equation, we get:
(1/2) * (λ/d) = 1/2
Simplifying, we find:
λ/d = 1
Finally, substituting the given wavelength of 604 nm (or 0.604 µm), we can solve for d:
d = λ/0.604 = 0.604 µm
Therefore, the minimum difference in rate between the waves from the slits must be 0.604 µm.