A diffraction grating with 1000 slits/cm is oriented parallel to the y-axis and its center is located on the x-axis at x = -5.00 m. A source of light of wavelength 650 nm is located on the x-axis at x = -125 m. At how many locations does constructive interference occur on the y-axis? Use the symbol N for the number of locations.
For this problem, I used y = m(lambda)L / d. I have d as 10^-5 m, lambda as 650 nm, or 6.5 * 10^-7 m, L = 125 m, and y = 5 m. The answer comes out to 0.615 m, which becomes 1.
Is this correct?
To solve this problem, you correctly used the formula for the position of constructive interference on the y-axis for a single slit in a diffraction grating:
y = m(lambda)L / d,
where y is the position of the interference pattern on the y-axis, m is the order of the interference pattern, lambda is the wavelength of light, L is the distance between the light source and the grating, and d is the distance between the slits in the grating.
In this problem, you are given:
- d = 1 / (1000 cm) = 1 * 10^(-4) m (as there are 1000 slits in 1 cm)
- lambda = 650 nm = 650 * 10^(-9) m
- L = 125 m
You want to find the number of locations (N) where constructive interference occurs on the y-axis.
First, calculate the maximum order (m_max) of the interference pattern for which the interference is constructive. To do this, rearrange the formula as:
m_max = (y_max * d) / (lambda * L),
where y_max is the maximum y position that yields constructive interference. In this case, y_max = 5 m.
Substituting the given values, we have:
m_max = (5 * 10^0 m * 1 * 10^(-4) m) / (650 * 10^(-9) m * 125 m)
≈ 0.615
Now, since the number of locations is rounded to the nearest whole number, N = 1.
So, the answer is correct. There is only one location on the y-axis where constructive interference occurs.