Question

Wavelength of light used in an optical instrument are ${\lambda }_1=4000\stackrel{\circ }{\mathrm{A}}$ and ${\lambda }_2=5000\stackrel{\circ }{\mathrm{A}}$, then ratio of their respective resolving powers (corresponding to ${\lambda }_1$ and ${\lambda }_2$) is:

• A. $16:25$
• B. $9:1$
• C. $4:5$
• D. $5:4$

Answer 1

The correct option is D

$5:4$

Step 1. Given:

${\lambda }_1=4000\stackrel{\circ }{\mathrm{A}}$

${\lambda }_2=5000\stackrel{\circ }{\mathrm{A}}$

Step 2. Relation between resolving power and wavelength:

Resolving power is the power to create different images of two nearby objects.

$R=\frac{2\mu \sin \theta }{\lambda }$

Here, $\mu$ is the refractive index of objects

$\theta$ is the angle subtended by the radius of objective lens.

$\lambda$ is the wavelength

Clearly notice that, for constant medium, $R\propto \frac{1}{\lambda }$

Therefore,

$\begin{array}{rcl}\frac{\mathrm{Resolving}\mathrm{power}\mathrm{at}{\lambda }_1}{\mathrm{Resolving}\mathrm{power}\mathrm{at}{\lambda }_2}& =& \frac{{\lambda }_2}{{\lambda }_1}\\ & =& \frac{5000}{4000}\\ & =& \frac{5}{4}\end{array}$

Therefore, the required ratio is $5:4$

Hence, the correct option is (D).