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In u-v method for finding the focal leng...

In `u-v` method for finding the focal length of a concave mirror. The mirror is fixed at position `A` marked `20.0cm` on an optical bench and an object needle is placed at position `B` marked `45.0cm` on an optical banch. For no parallax between object needle and image needle the image needle at position `C 57.5cm` on optical bench. Then percentage error in the measurement of focal length of the mirror is

Text Solution

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Distance of the object, `u=45.0-20.0`
`=25.0 cm`
Distance of the image, `v=57.5-20.0=37.5 cm`
Using mirror formula,
`f=(uv)/(u+v)=(25xx37.5)/(25+37.5)`
`=(25xx37.5)/(62.5)=15.0 cm`
The maximum permissible error in focal length is given by
`(Delta f)/f=(Deltau)/u+(Deltav)/v+(Delta(u+v))/((u+v))`
`(Delta f)/f=(Delta u)/u+(Delta v)/v+(Deltau)/((u+v))+(Deltav)/((u+v))`
`=0.2/25.0+0.2/37.5+0.2/62.5+0.2/62.5`
`=0.008+0.005+0.003+0.003=0.019`
`:. Deltaf=0.019xxf=0.019xx15.0=0.285 cm cong 0.3 cm`
`:.` The focal length of concave mirror is `(15.0 pm 0.3) cm`.
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