Chapter 17.2, Problem 19PP

To determine

The lightbulb’s image position and its diameter.

Answer to Problem 19PP

  $10.68\text{cm}$ behind the mirror and $1.07\text{cm}$ , respectively.

Explanation of Solution

Given:

The focal length of the convex mirror, $f=-13.0\text{cm}$

The diameter of the bulb, $h_o=6.0\text{cm}$

The object distance from the mirror, $d_o=60.0\text{cm}$

Formula used:

Mirror equation,

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

Where,

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

And,

Magnification, $\frac{h_i}{h_o}=-\frac{d_i}{d_o}$

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,

  $\begin{array}{c}\frac{1}{f}=\frac{1}{d_i}+\frac{1}{d_o}\\ \frac{1}{-13}=\frac{1}{d_i}+\frac{1}{60}\\ d_i=-\left(\frac{60\times 13}{60+13}\right)\\ =-10.68\text{cm}\end{array}$

Negative sign denotes that the image is formed behind the mirror.

And, the diameter of the image will be,

  $\begin{array}{c}\frac{h_i}{h_o}=-\frac{d_i}{d_o}\\ \frac{h_i}{6.0}=-\frac{\left(-10.68\right)}{60.0}\\ h_i=1.07\text{cm}\end{array}$

Conclusion:

Hence, the lightbulb’s image distance and the diameter of the bulb is $10.68\text{cm}$ behind the mirror and $1.07\text{cm}$ , respectively.