Question

A dentist uses a small mirror with a radius of 36 mm to locate a cavity in a patient's tooth. If the mirror is concave and is held 15 mm from the tooth, what is the magnification of the image?

Answer 1

A dentist uses a small mirror with a radius of $36\mathrm{mm}$ to locate a cavity in a patient's tooth. If the mirror is concave and is held $15\mathrm{mm}$ from the tooth, what is the magnification of the image? Please explain your answer.

Solution

The Gaussian mirror equation relates the object distance $\mathrm{d}_0$ and image distance $\mathrm{d}_i$ to the focal length $f$:

$$\frac {1}{f} = \frac {1}{d _ {0}} + \frac {1}{d _ {i}}$$

Magnification is defined as:

$$m = - \frac {d _ {i}}{d _ {0}}$$

http://en.wikipedia.org/wiki/Curved_mirror#Analysis

For the concave mirror focal length related with radius as:

$$f = \frac {R}{2}$$

We are given

$$R = 36 \mathrm{mm}$$ $$d _ {0} = 15 \mathrm{mm}$$ $$\frac {2}{R} = \frac {1}{d _ {0}} + \frac {1}{d _ {i}}$$

So

$$d _ {i} = \frac {1}{\frac {2}{R} - \frac {1}{d _ {0}}}$$

Thus magnification is:

$$m = - \frac {\frac {1}{2} - \frac {1}{R} - \frac {1}{d _ {0}}}{d _ {0}} = - \frac {1}{\frac {2 d _ {0}}{R} - 1} = \frac {1}{1 - \frac {2 d _ {0}}{R}}$$

Calculation

$$m = \frac {1}{1 - \frac {2 * 15}{36}} = 6$$

Answer:

$$m = 6$$