A plano-covex lens is made of a material of refractive index n. When a small object is placed 30cm away in front of the curved surface of the lens, an image of double the size of the object is produced. Due to reflection from the convex surface of the lens, another faint image is observed at a distance of 10 cm away from the lens. Which of the following statement (s) is (are) true?

Options

A. The refractive index of the lens is 2.5
B. The radius of curvature of the convex surface is 45 cm
C. The faint image is erect and real
D. The focal length of the lens is 20 cm

Answer

A, D

Explanation

Distance of object from convex lens is 30 cm.
`u =-30 cm`.
It is convex lens and if we assume that given image is real then
`rArr" "m=-2=(v)/(u)`
`v=-2u=-2xx-30=60cm`
Now we can use lens formula to find focal length of the lens.
`(1)/(f)=(1)/(v)-(1)/(u)`
`rArr" "(1)/(f)=(1)/(60)-(1)/(-30)`
`rArr" "(1)/(f)=(1)/(60)+(1)/(30)=(1+2)/(60)`
`rArr" "f=20cm`
Hence, option (d) is correct.
Now the faint image that we are getting is due to reflection on convex surface and hence due to convex mirror. Image formed by convex mirror is always virtual and erect, hence, option (c) is wrong. But we can use mirror formula to get radius of curvature for convex surface.
`(1)/(f)=(1)/(v)-(1)/(u)rArr (2)/(R )=(1)/(+10)+(1)/(-30)`
`rArr" "(2)/(R )=(1)/(10)-(1)/(30)=(2)/(30)`
`rArr" "R=30cm`
Hence, option (b) is wrong.
Now after knowing the radius of curvature of convex surface of plano - convex lens we can apply lens maker.s formula.
`(1)/(f)=(mu-1)((1)/(R_(1))-(1)/(R_(2)))`
`rArr" "(1)/(20)=(mu-1)((1)/(+30)-(1)/(oo))`
`rArr" "(1)/(20)=((mu-1))/(30)`
`rArr" "3=2mu-2`
`rArr" "mu=2.5`
Hence, option (a) is correct.
Finally only options (a) and (d) are correct.