A small candle, 2.5 cm in size is placed at 27 cm in front of a concave mirror of radius of curvature 36 cm. At what distance from the mirror should a screen be placed in order to obtain a sharp image? Describe the nature and size of the image. If the candle is moved closer to the mirror, how would the screen have to be moved?
The screen should be placed at a distance of 54 cm from the mirror. The image is real, inverted, and magnified with a size of 5 cm. If the candle is moved closer to the mirror, the screen would have to be moved farther away from the mirror.
Explanation
This problem applies the mirror formula and magnification formula from the mirror equation. The context shows similar solved examples (Example 9.3) where the mirror formula 1/v + 1/u = 1/f is used to find image position, and magnification m = -v/u determines image characteristics. The object is placed beyond the focal point (27 cm > 18 cm), so a real image forms on the same side as the object.
Solution Steps
Step 1: Calculate focal length: f = R/2 = 36/2 = 18 cm. For concave mirror, f = -18 cm (by sign convention).
Step 2: Apply mirror formula: 1/v + 1/u = 1/f. Given u = -27 cm, f = -18 cm.
Step 3: Substitute values: 1/v + 1/(-27) = 1/(-18). Solving: 1/v = -1/18 + 1/27 = (-3+2)/54 = -1/54.
Step 4: Calculate image distance: v = -54 cm. The negative sign indicates image is formed on the same side as the object, 54 cm from the mirror.
Step 5: Calculate magnification: m = -v/u = -(-54)/(-27) = -2. Negative magnification means image is inverted.
Step 6: Calculate image size: Image height = m × object height = - = -5 cm. The image is 5 cm tall and inverted.
Step 7: For object moved closer: Since object is beyond focus, moving it closer moves the image farther. Screen must be moved away from mirror.