Question

The origin of a coordinate system is placed at the center of curvature of a spherical mirror with radius of curvature \(R=59.3 \mathrm{~cm}\) (see the figure). An object is placed at \(x_{\mathrm{o}}=39.5 \mathrm{~cm} .\) What is the mirror's magnification?

Short answer

Answer: The magnification of the mirror is -0.454, meaning the image is reduced and inverted compared to the original object.

Step-by-step solution

Write down the given information

The given radius of curvature, \(R = 59.3 cm\). The object distance, \(x_o = 39.5 cm\).

Recognize the type of mirror

Since the object is placed in front of the center of curvature, we are dealing with a concave mirror.

Apply the mirror equation

The mirror equation is given by: \(\frac{1}{f} = \frac{1}{x_o} + \frac{1}{x_i}\) For a concave mirror, the focal length (\(f\)) is half the radius of curvature: \(f = \frac{R}{2} = \frac{59.3}{2} = 29.65 cm\) Plug in the given values of \(x_o\) and \(f\) into the mirror equation and solve for the image distance \(x_i\): \(\frac{1}{29.65} = \frac{1}{39.5} + \frac{1}{x_i}\)

Solve for the image distance

To solve for \(x_i\), first we will calculate a common denominator for the fractions: \( \frac{39.5x_i + 29.65 x_i}{(29.65)(39.5)} = 0\) Now, combine the fractions and solve for \(x_i\): \(x_i = \frac{(29.65)(39.5)}{29.65+39.5} = 17.95 cm\)

Apply the magnification formula

The magnification formula is given by: \(M = -\frac{x_i}{x_o}\) Plug in the values of \(x_i\) and \(x_o\) to find the magnification: \(M = -\frac{17.95}{39.5} = -0.454\)

Interpret the result

The magnification is -0.454, which means the image is reduced and inverted (due to the negative sign) with respect to the originally placed object.