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*34. A concave mirror has a focal length of 30.0 cm. The distance be- tween an object and its image is 45.0 cm. Find the object and image dis- tances, assuming that (a) the object lies beyond the center of curvature and (b) the object lies between the focal point and the mirror.

*34. A concave mirror has a focal length of 30.0 cm. The distance be- tween an object and its image is 45.0 cm. Find the object and image dis- tances, assuming that (a) the object lies beyond the center of curvature and (b) the object lies between the focal point and the mirror.

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### Problem 34: Concave Mirror and Image Formation A concave mirror has a focal length of 30.0 cm. The distance between an object and its image is 45.0 cm. Find the object and image distances, assuming that: 1. **(a)** The object lies beyond the center of curvature. 2. **(b)** The object lies between the focal point and the mirror. ### Solution Approach Start by recalling the mirror equation and magnification formula: \[ \frac{1}{f} = \frac{1}{d_o} + \frac{1}{d_i} \] \[ m = -\frac{d_i}{d_o} \] where: - \( f \) = focal length of the mirror (30.0 cm in this case) - \( d_o \) = object distance from the mirror - \( d_i \) = image distance from the mirror - \( m \) = magnification of the image ### Case (a): Object Beyond Center of Curvature When the object lies beyond the center of curvature (C), the object distance (\( d_o \)) is greater than the radius of curvature (R), which is twice the focal length (i.e., \( R = 2f = 60 \) cm). ### Case (b): Object Between Focal Point and Mirror In this case, the object lies within the focal length, \( d_o < f \). #### Detailed Steps and Diagrams 1. **Determine possible locations for \( d_o \) and \( d_i \) based on given total distance**: - Total distance \( d_o + d_i = 45.0 \) cm 2. **Use the mirror equation to find the exact distances**: - Solve \( \frac{1}{f} = \frac{1}{d_o} + \frac{1}{d_i} \) 3. **Graphical Representation**: - A ray diagram can be used to illustrate both cases by drawing the principal axis, focal point (F), center of curvature (C), object (O), and image formation (I). If there are specific values, steps, or graphical representations the educational content should include, please provide that information or request further elaboration on particular points.
Transcribed Image Text:### Problem 34: Concave Mirror and Image Formation A concave mirror has a focal length of 30.0 cm. The distance between an object and its image is 45.0 cm. Find the object and image distances, assuming that: 1. **(a)** The object lies beyond the center of curvature. 2. **(b)** The object lies between the focal point and the mirror. ### Solution Approach Start by recalling the mirror equation and magnification formula: \[ \frac{1}{f} = \frac{1}{d_o} + \frac{1}{d_i} \] \[ m = -\frac{d_i}{d_o} \] where: - \( f \) = focal length of the mirror (30.0 cm in this case) - \( d_o \) = object distance from the mirror - \( d_i \) = image distance from the mirror - \( m \) = magnification of the image ### Case (a): Object Beyond Center of Curvature When the object lies beyond the center of curvature (C), the object distance (\( d_o \)) is greater than the radius of curvature (R), which is twice the focal length (i.e., \( R = 2f = 60 \) cm). ### Case (b): Object Between Focal Point and Mirror In this case, the object lies within the focal length, \( d_o < f \). #### Detailed Steps and Diagrams 1. **Determine possible locations for \( d_o \) and \( d_i \) based on given total distance**: - Total distance \( d_o + d_i = 45.0 \) cm 2. **Use the mirror equation to find the exact distances**: - Solve \( \frac{1}{f} = \frac{1}{d_o} + \frac{1}{d_i} \) 3. **Graphical Representation**: - A ray diagram can be used to illustrate both cases by drawing the principal axis, focal point (F), center of curvature (C), object (O), and image formation (I). If there are specific values, steps, or graphical representations the educational content should include, please provide that information or request further elaboration on particular points.
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