Find the radius of a star image formed on the retina of the eye if the aperture (pupil) diameter at night is 0.70 cm and the length of the eye is 3.1 cm. Assume the wavelength of the starlight in the eye is 500 nm. (1nm=10^9 m)
To find the radius of the star image on the retina of the eye, we can use the formula for the angular resolution of a telescope:
θ = 1.22 (λ / D)
Where:
θ is the angular resolution (in radians)
λ is the wavelength of light (in meters)
D is the diameter of the aperture (pupil) of the eye (in meters)
First, let's convert the aperture diameter to meters:
D = 0.70 cm = 0.70 * 10^(-2) m
Next, let's convert the wavelength of the starlight to meters:
λ = 500 nm = 500 * 10^(-9) m
Now we can plug these values into the formula to find the angular resolution:
θ = 1.22 * (500 * 10^(-9) / 0.70 * 10^(-2))
Simplifying the expression:
θ = 1.22 * (5 * 10^(-7) / 7 * 10^(-4))
θ = 1.22 * (5/7) * (10^(-7) / 10^(-4))
θ = 1.22 * (5/7) * 10^(-3)
Now, we can find the radius of the star image on the retina by multiplying the angular resolution by the length of the eye. The length of the eye is given as 3.1 cm, which we will convert to meters:
Length of the eye = 3.1 cm = 3.1 * 10^(-2) m
Finally, we can calculate the radius of the star image on the retina:
Radius = θ * Length of the eye
Radius = (1.22 * (5/7) * 10^(-3)) * 3.1 * 10^(-2)
Simplifying the expression:
Radius = (1.22 * 5/7) * 10^(-3) * 3.1 * 10^(-2)
Radius = 1.22 * 5 * 3.1 * 10^(-3 - 2)
Radius = 1.22 * 5 * 3.1 * 10^(-5)
Radius ≈ 0.189 mm
Therefore, the radius of the star image formed on the retina of the eye is approximately 0.189 mm.