Question

An object is 30.0cmfrom a spherical mirror, along the mirror’s central axis. The mirror produces an inverted image with a lateral magnification of absolute value 0.500. What is the focal length of the mirror?

Short answer

Focal length of the mirror is 10.0cm.

Step-by-step solution

Determine the given quantities

Object distance p =30cm.

Lateral magnification m=0.500

The image is inverted.

Determine the concept and formula for magnification

Use the equation of magnification to calculate the image distance. Then we can use the mirror equation to find the focal length of the mirror.

Consider the formula for the magnification as:

$\frac{1}{p}+\frac{1}{i}=\frac{1}{f}$

$m=-\frac{i}{p}$

Step 3: Determine the focal length of the mirror

The mirror equation relates an object distance p, mirror’s focal length f and the image distance i as follows:

$\frac{1}{p}+\frac{1}{i}=\frac{1}{f}$ (1)

The lateral magnification is

$m=-\frac{i}{p}$

But we have the image as inverted, and the absolute value of magnification is 0.500.

So, m= -0.500.

The minus sign shows that the image is inverted.

$$\begin{gathered} -\frac{1}{2}=-\frac{i}{p} \\ i=\frac{p}{2} \end{gathered}$$

Substituting i in (1), and solve as:

$$\begin{gathered} \frac{1}{p}+\frac{2}{p}=\frac{1}{f} \\ \frac{3}{p}=\frac{1}{f} \\ f=\frac{p}{3} \\ =\frac{30.0cm}{3} \\ =10.0cm \end{gathered}$$

Focal length of the mirror is 10.0cm