A spy camera is said to be able to read the numbers on a car's license plate. If the numbers on the plate are 6.0 cm apart, and the spy satellite is at an altitude of 160 km, what must be the diameter of the camera's aperture? (Assume light with a wavelength of 600 nm.)
To be able to resolve 6cm the lens would be ~3.905m (Diffraction limited ground sample distance= 2.44 * wavelength * altitude / aperture (diameter) of lens.
To determine the diameter of the camera's aperture, we need to use the concept of diffraction limit. Diffraction occurs when light waves passing through an aperture or around an obstacle spread out and interfere with each other.
We can use the Rayleigh criterion, also known as the angular resolution limit, to calculate the minimum resolvable distance. The formula for the angular resolution (θ) is:
θ = 1.22 * (λ / D)
Where:
θ = angular resolution (in radians)
λ = wavelength of light (in meters)
D = diameter of the aperture (in meters)
First, let's convert the wavelength of light from nanometers to meters:
λ = 600 nm = 600 * 10^(-9) m
Next, we need to determine the distance (L) from the spy camera to the car's license plate. Since the spy satellite is at an altitude of 160 km, we can use basic trigonometry to calculate L. Considering the altitude as the hypotenuse of a right triangle, we have:
L = hypotenuse = √(altitude^2 + base^2)
L = √(160,000 m^2 + (6.0 cm / 100 m/cm)^2)
Now, we can calculate the angular resolution using the determined values of λ and L:
θ = 1.22 * (λ / D)
To find the minimum resolvable distance, we set θ equal to the separation between the numbers on the license plate:
θ = separation distance / L
Finally, by rearranging the formula, we can solve for D:
D = λ / (1.22 * separation distance / L)
Substituting the known values, we can calculate the diameter of the camera's aperture.