A spy satellite orbiting at 160 km above Earth's surface has a lens with a focal length of 3.6 m and can resolve objects on the ground as small as 30 cm. For example, it can easily measure the size of an aircraft's air intake port. What is the effective diameter of the lens as determined by diffraction consideration alone? Assume wavelength=550 nm.
To find the effective diameter of the lens as determined by diffraction considerations alone, we can use the formula for the minimum resolvable angle:
θ = 1.22 * (λ / D)
Where:
θ is the minimum resolvable angle (in radians)
λ is the wavelength of light (in meters)
D is the effective diameter of the lens (in meters)
We can rearrange the formula to solve for D:
D = 1.22 * (λ / θ)
Given values:
λ = 550 nm = 550 * 10^(-9) meters
θ = 30 cm = 30 * 10^(-2) meters
Now, let's substitute these values into the formula to find the effective diameter:
D = 1.22 * (550 * 10^(-9) meters / (30 * 10^(-2) meters))
Calculating this expression, we get:
D = 1.22 * (550 * 10^(-9) meters / 0.3 meters)
= 1.22 * (1.833333 * 10^(-6))
≈ 2.2385 * 10^(-6) meters
≈ 2.24 micrometers
Therefore, the effective diameter of the lens as determined by diffraction considerations alone is approximately 2.24 micrometers.