A short object of length `L` is placed along the principal axis of a concave mirror away from focus. The object distance is `u`. If the mirror has a focal length `f`, what will be the length of the image ? You may take `L lt lt | u - f |`.

Explanation

Since , the object distances ins u. Let us aconsider the two ends of the object be at distance `u_(1)=u-L//2 and u_(2)=u+L//2`, respectively so that `|u_(1)-u_(2)|` =L . Let the image of the two ends be formed at `v_(1) and v_(2)`, respectively so that the image length would be
`" " L'=|v_(1)-v_(2)|`
Applying mirror formula, we have
`" " (1)/(u)+(1)/(v)=(1)/(f)or v=(fu)/(u-f)`
On solving, the positions of two images are given by
`" " v_(1)=(f(u-L//2))/(u-f-L//2),v_(2)=(f(u+L//2))/(u-f+L//2)`
For length , substituting the value in (i), we have
`" " L'=|v_(1)-v_(2)|=(f^(2)L)/((u-f)^(2)xxL^(2)//4)`
Since, the object is short and kept away from focus, we have
`" " L^(2)//4 lt lt (u-f)^(2)`
Hencce, finally `" " L'=(f^(2))/((u-f)^(2))L`
Thus is the required expression of length of image.