Question

What is the focal length of a makeup mirror that produces a magnification of \(1.50\)when a person’s face is \(12.0\;cm\) away? Explicitly show how you follow the steps in the Problem-Solving Strategy for Mirrors.

Short answer

The value of the focal length of the given makeup mirror is \(0.36\;m\).

Step-by-step solution

Definition of focal length

The focal length of an optical system is the inverse of the system's optical power; it measures how strongly the system converges or diverges light. A system with a positive focal length converges light, while a system with a negative focal length diverges light.

Information Provided

• The magnification value:\(1.50\).
• Distance between lens and person’s face: \(12.0\;cm\).

Calculation for the focal length of the mirror

Use the magnification equation:

\(m = - \dfrac{{{d_i}}}{{{d_o}}}\)

where \(m\) denotes the magnification, \({d_o}\) denotes the distance between the object and the lens, and\({d_i}\) denotes the distance from the lens to the projected image that is in focus.

To find the image distance\({d_i}\)and then use it to solve the equation of the focal length via\(\dfrac{1}{f} = \dfrac{1}{{{d_i}}} + \dfrac{1}{{{d_o}}}\).

Where \(f\) denotes the lens's focal length, \({d_o}\) denotes the distance between the object and the lens, and \({d_i}\) denotes the distance from the lens to the projected image that is in focus.

\(\begin{aligned}f &= {\left( {\dfrac{{ - 1}}{{m{d_o}}} + \dfrac{1}{{{d_o}}}} \right)^{ - 1}}\\f &= {\left( {\dfrac{{ - 1}}{{1.50 \times 0.12\;m}} + \dfrac{1}{{0.12\;m}}} \right)^{ - 1}}\\f &= 0.36\;m\end{aligned}\)

Therefore, the value of the focal length is \(0.36\;m\).