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Question
A concave mirror of curvature 40 cm, used for shaving purposes produces image of double size as that of the object. Object distance must be ______.
Options
10 cm only
20 cm only
30 cm only
10 cm or 30 cm
Solution
A concave mirror of curvature 40 cm, used for shaving purposes produces image of double size as that of the object. Object distance must be 10 cm or 30 cm.
Explanation:
The radius of curvature (R) of the concave mirror is 40 cm
The focal length (f) is given by:
`f = R/2 = 40/2 = 20` cm
The image is double the size of the object (m = −2m = −2m= −2) since concave mirrors produce inverted magnified images for real images.
The magnification m is related to object distance (u) and image distance (v) by:
`m = (-v)/u`
For magnification m = −2
`-2 = (-v)/u`
`-2 = (-v)/u or v = 2u`
Substitute f = 20 and v = 2u
`1/20 = 1/(2u) + 1/u`
`1/20 = (1 + 2)/(2u)`
= `3/(2u)`
u = `(3 xx 20)/2`
30 cm
Now, consider magnification of +2 (virtual image for shaving purpose): For virtual, upright image, m = 2, so;
`2 = (-v)/u ⇒ v = -2u`
Substituting v = −2u in the mirror formula:
`1/20 1/(-2u) + 1/u`
`1/20 =( -1 + 2)/(2u)`
= `1/(2u)`
u = 10 cm
Hence possible u = 10 or 30 cm
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