`U` shaped wire is placed in front of a concave mirror of radius of curvature `20cm` as shown. The total length of the image of the wire `ABCD` is nearly

Options

A. `2.5cm`
B. `6cm`
C. `12.5cm`
D. `15cm`

Answer

B

Explanation

Focal length of concave mirror `f=(R )/(2)=-10cm`
for the left arm (`AB`) of the `U`-wire, `u=-40cm`
`(1)/(f)=(1)/(v)+(1)/(u)implies v_(1)=(-40)/(3)cm`
for the right arm (`CD`), `u_(2)=-30cm`
`-(1)/(10)=(1)/(v_(2))+(1)/(-30)implies v_(2)=-15cm`
Magnification of `AB=(v_(1))/(u_(1))=((40)/(3))/(40)=(1)/(3)`
`implies A^(1)B^(1)=(AB)/(3)=(5)/(3)cm`
Magnification of `CD=(v_(2))/(u_(2))=(15)/(30)=(1)/(2)cm`
`impliesC_(1)D_(1)=(CD)/(2)=(5)/(2)cm`
Total length of the image of the wire
`=A^(1)B^(1)+B^(1)C^(1)+C^(1)D^(1)`, `=(5)/(3)+(5)/(3)+(5)/(2)~~6cm`