Two stars are 8 light years from earth (a light year is the distance light travels in one year). The 550 nm light from the two stars can be barely distinguished in the diffraction pattern of 90 cm wide single slit. How far apart are the stars?
To determine the distance between the two stars, we can use the concept of diffraction and the formula for the angular separation of fringes in a single-slit diffraction pattern.
1. First, let's convert the wavelength of light into meters. The given wavelength is 550 nm, so we have to divide it by 10^9 to convert it to meters. Thus, the wavelength is 550 nm * (1 m / 10^9 nm) = 550 × 10^-9 m.
2. Next, we need to find the angular separation of the fringes in the diffraction pattern. The angular separation can be given by the formula:
θ = λ / (a * n)
Where:
θ is the angular separation,
λ is the wavelength of light,
a is the width of the single slit, and
n is the order of the fringe (in this case, n = 1 since we are looking at the first fringe).
Here, we are given the width of the single slit, which is 90 cm. We need to convert it to meters by dividing it by 100. So, a = 90 cm * (1 m / 100 cm) = 0.9 m.
Plugging in the values, we get:
θ = (550 × 10^-9 m) / (0.9 m * 1) = 6.11 × 10^-7 radians
3. Now, we can calculate the distance between the two stars. The angular separation of the fringes can be related to the distance between the two stars using the formula:
θ = d / D
Where:
θ is the angular separation of the fringes,
d is the distance between the stars, and
D is the distance from Earth to the stars (which is equal to 8 light years).
Rearranging the formula, we get:
d = θ * D = (6.11 × 10^-7 radians) * 8 light years
Since we want the distance in meters, we need to convert light years to meters. There are approximately 9.461 × 10^15 meters in a light year, so:
d = (6.11 × 10^-7 radians) * 8 * (9.461 × 10^15 meters / 1 light year)
Evaluating the expression, we get:
d ≈ 3.672 × 10^9 meters
Therefore, the two stars are approximately 3.672 billion meters apart.