Question

The radius of curvature of a convex mirror is \(-25.0 \mathrm{~cm}\). What is its focal length?

Short answer

Answer: The focal length of the convex mirror is -12.5 cm.

Step-by-step solution

Write down the mirror equation

The mirror equation is given by: $$ \frac{1}{f} = \frac{1}{d_o} + \frac{1}{d_i} $$ Where \(f\) is the focal length, \(d_o\) is the object distance, and \(d_i\) is the image distance. In this exercise, we only need to find the focal length of the convex mirror while knowing its radius of curvature.

Recall the relationship between the focal length and the radius of curvature

For a convex mirror, the focal length is half the radius of curvature and has a positive value. The relationship is given by: $$ f = \frac{R}{2} $$ Where \(f\) is the focal length and \(R\) is the radius of curvature.

Substitute the given value of the radius of curvature

The radius of curvature of the convex mirror is given as \(-25.0 \mathrm{~cm}\). The negative sign indicates that it is a convex mirror. To find the focal length, plug this value into the equation we obtained in step 2: $$ f = \frac{-25.0 \mathrm{~cm}}{2} $$

Calculate the focal length of the convex mirror

Divide the radius of curvature by 2 to find the focal length: $$ f = -12.5 \mathrm{~cm} $$ The negative sign indicates that it is a convex mirror. Thus, the focal length of the convex mirror is \(-12.5 \mathrm{~cm}\).