A reflecting telescope is used to observe two distant point sources that are 3.50 m apart with light of wavelength 600 nm. The telescope's mirror has a radius of 5.0 cm .
What is the maximum distance in meters at which the two sources may be distinguished?
To calculate the maximum distance at which the two sources can be distinguished, we need to determine the minimum angular separation between the sources that the telescope can resolve.
The angular resolution of a telescope is determined by the formula:
θ = 1.22 * (λ / D)
where θ is the angular resolution, λ is the wavelength of light being observed, and D is the diameter of the telescope's mirror.
In this case, the angular resolution is the minimum angle between the two point sources that the telescope can distinguish. We can rearrange the formula to solve for D:
D = 1.22 * (λ / θ)
Using the given values:
λ = 600 nm = 600 * 10^-9 m
D = 5.0 cm = 5.0 * 10^-2 m
Now we can calculate the minimum angle between the two sources:
θ = 1.22 * (600 * 10^-9 m / 5.0 * 10^-2 m)
Simplifying the equation:
θ = 1.22 * (600 * 10^-9 m / 5.0 * 10^-2 m) = 0.01464 radians
To determine the maximum distance at which the two sources can be distinguished, we can use the formula:
Distance = 2 * D * tan(θ/2)
Plugging in the values:
Distance = 2 * (3.50 m) * tan(0.01464 radians / 2)
Calculating the distance:
Distance = 2 * (3.50 m) * tan(0.00732 radians) = 0.051 m
Therefore, the maximum distance at which the two sources can be distinguished is 0.051 meters or 51 centimeters.