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Question
A U-shaped wire is placed before a concave mirror having radius of curvature 20 cm as shown in figure. Find the total length of the image.
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Solution
Given,
Radius of curvature of concave mirror, R = 20 cm
So its focal length will be f = \[\frac{R}{2}\] = −10 cm
For part AB of the U shaped wire, PB = 30 + 10 = 40 cm
Therefore, u = −40 cm
Using mirror formula,
`1\v = -1/10 - 1/-40 =-3/40`
⇒ `V = -40/3 = 13. 3 cm`
So , PB' = 13.3 cm
`m = (A' B')/( AB ) = -( v )/ u`
\[= \frac{- ( - 13 . 3)}{- 40} = - \frac{1}{3}\]
\[ \Rightarrow A'B' = - \frac{10}{3} = - 3 . 33 \text{ cm }\]
For part CD of the U shaped wire, PC = 30 cm
Therefore, u = −30 cm
Again, using mirror equation:
`1/V = -1/10 - 1/-30`
`1/V = -1/10 + 1/-30 = -1/15`
⇒ v = −15 cm = PC'
\[m = \frac{C'D'}{CD} = - \frac{v}{u}\]
`= -( -15 )/ -30 = -1/2`
⇒ C'D' = 5 cm
⇒ B'C' = PC' − PB'
= 15 − 13.3 = 1.7 cm
Hence, the total length of the U- shaped wire is A'B' + B'C' + C'D'
= (3.3) + (1.7) + 5 = 10 cm
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