91Ó°ÊÓ

Magnification (\( m \)) for mirrors is defined as the ratio of the image height (\( h' \)) to the object height (\( h \)). It is also expressed in terms of the image distance (\( v \)) and object distance (\( u \)) as:\[ m = -\frac{v}{u}\]For the concave mirror, we know that the magnification is given as +2.0. This indicates the image is upright and virtual, as positive magnification for mirrors indicates these characteristics.

Since the object distance (\( u \)) remains the same for both the concave and convex side uses, we can use the same object distance (let's call it \( u \)) for the convex mirror scenario as well. We have:\[m_{concave} = -\frac{v_{concave}}{u} = +2.0\]Thus, \( v_{concave} = -2.0u \). This information will help us determine the focal length for the concave side.

For a concave mirror, the mirror equation is:\[\frac{1}{f} = \frac{1}{u} + \frac{1}{v}\]Substitute \( v_{concave} = -2.0u \) into the equation:\[\frac{1}{f} = \frac{1}{u} - \frac{1}{2.0u} = \frac{2.0u - u}{2.0u^2} = \frac{1}{2.0u}\]Thus, the focal length for the concave side is \( f = 2.0u \).

Since the surfacing of the mirror can be assumed as the same material, the radius of curvature remains consistent, implying the focal lengths of the concave and convex sides are the same magnitude but opposite in sign. Hence, the focal length for the convex side is \( f = -2.0u \).For the convex mirror, we use the same mirror equation:\[\frac{1}{f} = \frac{1}{u} + \frac{1}{v}\]where \( f = -2.0u \). Solving for this:\[\frac{1}{-2.0u} = \frac{1}{u} + \frac{1}{v_{convex}}\]Rearranging terms:\[\frac{1}{v_{convex}} = \frac{1}{-2.0u} - \frac{1}{u}\]\[\frac{1}{v_{convex}} = \frac{-1 - 2.0}{2.0u} = \frac{-3}{2.0u}\]Thus, \( v_{convex} = -\frac{2.0u}{3} \). This gives the magnification for the convex side as \( m_{convex} = -\frac{-\frac{2.0u}{3}}{u} = \frac{2.0}{3} \).