Q52P - University Physics with Modern Physics

Question

An underwater camera has a lens with focal length in air of 35.0 mm and a maximum aperture of f/2.80. The film it uses has an emulsion that is sensitive to light of frequency $\mathbf{6}\mathbf{.}\mathbf{00}\mathbf{\times }{\mathbf{10}}^{\mathbf{14}}\mathbf{Hz}$. If the photographer takes a picture of an object 2.75 m in front of the camera with the lens wide open, what is the width of the smallest resolvable detail on the subject if the object is (a) a fish underwater with the camera in the water and (b) a person on the beach with the camera out of the water?

Step-by-Step Solution

Answer

1. The width of the finest detail on the subject, if the object is a fish underwater with the camera in the water, is 0.101mm
2. The width of the finest detail on the subject if the object is a person on the beach with the camera out of the water is 0.134mm

Step 1: Rayleigh Criterion

According to the Rayleigh criterion for the diffraction limit to resolution, two images are only resolvable when the center of one's diffraction pattern is directly over the first minimum of the other's diffraction pattern.

Step 2: The width of the finest detail on the subject if the object is a fish underwater with the camera in the water

The problem makes use of resolving power and f-number (a lens's f-number is its focal length f divided by the aperture diameter D), as well as the Rayleigh criterion to determine D and the angular separation of the barely resolvable detail on the object.

1. Aperture diameter is;

       

the angular separation of the object's finest detail in a medium with a refractive index n is;

As 

In water, n=1.33

Hence, the width of the finest detail on the subject if the object is a fish underwater with the camera in the water is 0.101mm

Step 3: The width of the finest detail on the subject if the object is a person on the beach with the camera out of the water

      1. In the air, n=1

         

Hence, the width of the finest detail on the subject if the object is a person on the beach with the camera out of the water is 0.134mm