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Question
A convex mirror has a focal length of 18 cm. The image of an object kept in front of the mirror is half the height of the object. What is the distance of the object from the mirror?
Solution
Given:
Focal length (f) = 18 cm
Magnifiction (M) = `("Height of the image" ("h"_2))/("Height of the object" ("h"_1))`
= `1/2`
Find: Object distance (u)
Formula:
- M = `("h"_2)/("h"_1) = -"v"/"u"`
- `1/"f" = 1/"v" + 1/"u"`
Calculations:
According to formula (i),
`1/2 = -"v"/"u"`
∴ v = `-"u"/2`
According to formula (ii),
`1/"f" = 1/"v" + 1/"u" = ("u" + "v")/("uv")`
∴ f = `("uv")/ ("u" + "v")`
Now, v = `-"u"/2`
∴ f = `("u"((-"u")/2))/("u" + ((-"u")/2))`
∴ `18 = ((-"u"^2)/2)/((2"u" - "u")/2) = (-"u"^2)/"u"`
∴ u = −18 cm
A negative sign indicates that the object is placed to the left of the mirror.
Hence, the distance of the object from the mirror is 18 cm.
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