Chapter 25, Problem 74GP

To determine

The location of red and yellow images.

Answer to Problem 74GP

The location of where red image is formed: $\text{79}\text{.4}\text{cm}$

The location of where yellow image is formed: $\text{75}\text{.5}\text{cm}$

Explanation of Solution

Given:

A Lucite Planoconvex lens has one flat surface and other has $R=\text{18}\text{.4}\text{cm}$ .

Distance of object is $\text{66}\text{.0}\text{cm}$

The index of refraction of the glass is $\text{1}\text{.5106}$ for red light and $\text{1}\text{.5226}$ for yellow light.

Formula used:

It is known that

  $\frac{1}{d_o}+\frac{1}{d_i}=\frac{1}{f}$

Where,

  $f$ is a focal length

  $d_o$ is distance from the source to lens 

  $d_i$ is distance from the screen to the lens

Calculation:

First calculate the focal lengths of lens for the two colors:

  $\frac{1}{f_{red}}=\left(n_{red}-1\right)\left(\frac{1}{R_1}+\frac{1}{R_2}\right)$

  $\text{=}\left(\text{1}\text{.5106}-\text{1}\right)\left(\left(\frac{\text{1}}{\text{18}\text{.4}\text{cm}}\right)\text{+}\left(\frac{\text{1}}{\infty }\right)\right)$

  $f_{red}\text{=}\text{36}\text{.04}\text{cm}$

  $\frac{1}{f_{yellow}}=\left(n_{yellow}-1\right)\left(\frac{1}{R_1}+\frac{1}{R_2}\right)$

  $\text{=}\left(\text{1}\text{.5226-1}\right)\left(\left(\frac{\text{1}}{\text{18}\text{.4}\text{cm}}\right)\text{+}\left(\frac{\text{1}}{\infty }\right)\right)$

  $f_{yellow}\text{=}\text{35}\text{.21}\text{cm}$

Now calculate the image distances by using the formula:

  $\frac{1}{d_o}+\frac{1}{d_{i\left(red\right)}}=\frac{1}{f_{red}}$

  $\frac{\text{1}}{\text{66}\text{.0}\text{cm}}\text{ + }\frac{1}{d_{i\left(red\right)}}\text{ = }\frac{\text{1}}{\text{36}\text{.04}\text{cm}}$

  $d_{i\left(red\right)}\text{=}\text{79}\text{.4}\text{cm}$

  $\frac{\text{1}}{\text{66}\text{.0}\text{cm}}+\frac{1}{d_{i\left(yellow\right)}}\text{ = }\frac{\text{1}}{\text{35}\text{.21}\text{cm}}$

  $d_{i\left(yellow\right)}\text{=}\text{75}\text{.5}\text{cm}$

Conclusion:

The location where red and yellow images formed are $\text{79}\text{.4}\text{cm}$ and $\text{75}\text{.5}\text{cm}$ .