Chapter 17, Problem 55A

To determine

The image position and image height.

Answer to Problem 55A

The image position is 30 cm.

The height of the image is 1.8 cm.

Explanation of Solution

Given:

The focal length of the concave mirror, $f=15.0\text{cm}$

The height of the object, $h_o=1.8\text{cm}$

The object distance from the mirror, $d_o=30.0\text{cm}$

Formula used:

Mirror equation,

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

Where,

  $\begin{array}{l}f=\text{focal length}\\ d_i=\text{ image distance}\\ d_o=\text{object distance}\end{array}$

Also, the magnification is

  $\frac{h_i}{h_o}=-\frac{d_i}{d_o}$

Where,

  $\begin{array}{l}{h}_i=\text{image height}\\ h_o=\text{object height}\end{array}$

Calculation:

Using the given values in mirror equation, the image distance will be,

  $\begin{array}{c}\frac{1}{f}=\frac{1}{d_i}+\frac{1}{d_o}\\ \frac{1}{15}=\frac{1}{d_i}+\frac{1}{30}\\ d_i=\frac{30\times 15}{30-15}\\ =30\text{cm}\end{array}$

Then, the image height will be,

  $\begin{array}{c}\frac{h_i}{h_o}=-\frac{d_i}{d_o}\\ \frac{h_i}{1.8}=-\frac{30.0}{30.0}\\ h_i=-1.8\text{cm}\end{array}$

Negative sign denotes inverted image.

Conclusion:

Hence, the image position is $30.0\text{cm}$ and image height is $1.8\text{cm}$ .