(III) An object is placed a distance r in front of a wall, where r exactly equals the radius of curvature of a certain concave mirror. At what distance from the wall should this mirror be placed so that a real image of the object is formed on the wall? What is the lateral magnification of the image?

Verified step by step guidance

1. Identify the given values: the object distance from the wall is \( r \), and the radius of curvature of the mirror is also \( r \). The radius of curvature \( R \) is related to the focal length \( f \) of the mirror by \( f = R/2 \).
2. Calculate the focal length of the mirror using \( f = r/2 \) since \( R = r \).
3. Set up the mirror equation \( \frac{1}{f} = \frac{1}{d_o} + \frac{1}{d_i} \) where \( d_o \) is the object distance from the mirror and \( d_i \) is the image distance from the mirror. We need \( d_i \) to be equal to \( r - d_o \) because the total distance from the object to the wall is \( r \).
4. Solve the mirror equation for \( d_o \) by substituting \( d_i = r - d_o \) and \( f = r/2 \) into the equation, and solve for \( d_o \). Then find \( d_m \), the distance from the wall to the mirror, by subtracting \( d_o \) from \( r \).
5. Calculate the lateral magnification using the formula \( M = -\frac{d_i}{d_o} \). Substitute \( d_i \) and \( d_o \) to find the magnification, which indicates the ratio of the image size to the object size and the orientation of the image.