Q3DQ - University Physics with Modern Physics

Question

You use a lens of diameter  and light of wavelength  and frequency  to form an image of two closely spaced and distant objects. Which of the following will increase the resolving power? (a) Use a lens with a smaller diameter; (b) use light of higher frequency; (c) use light of longer wavelength. In each case justify your answer.

Step-by-Step Solution

Answer

The correct option is (b) use light of higher frequency.

Step 1: Define the resolving power of the lens and Rayleigh’s criteria of resolution.

Resolving power is defined as the reciprocal of the smallest angle suspended in a lens by two-point objects, which can just be distinguished as separate.

According to Rayleigh’s criterion, two point objects are just barely resolved when their angular separation    is given by,

$\mathbf{sin\theta }\mathbf{=}\mathbf{1}\mathbf{.}\mathbf{22}\frac{\mathbf{\lambda }}{\mathbf{D}}$ 

Where $\mathbf{\lambda }$ is the wavelength of light and   is the diameter of the lens (circular aperture).

Resolving power is reciprocal of the angular separation.

Step 2: Determine factors which affect the resolving power of the lens

As Resolving power is reciprocal of the angular separation, the narrow angular separation, better the resolving power. 

(a) The diameter of lens is inversely proportional to resolving power. Thus, reducing the diameter D will increase the angular separation which leads to decrease in resolving power.

(b) Increasing the frequency of light will result in decreasing the wavelength. the wavelength of light is directly proportional to the angular separation. Thus, decreasing the wavelength will decrease the angular separation and hance, increase the resolving power.

(c) Increasing the wavelength will increase the angular separation and decrease the resolving power. 

Thus, the only factor which leads to an increase in the resolving power of the lens is the use of light having a higher frequency.