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The convex surface of a thin concavo– convex lens of glass of refractive index 1.5 has a radius of curvature 20 cm. The concave surface has a radius of curvature of 60 cm. The convex side is coated with silver and placed at a horizontal surface as shown in the figure. 

(a) Where a pin should be placed on the optical axis such that its image is formed at the same place? 

(b) If the concave part is filled with water of refractive index 4/3, find the distance through which the pin should be moved, so that the image of the pin again coincides with pin.

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When the convex side of the concavo–convex lens is coated with silver, the combination becomes a mirror. This combination consist two lenses and one mirror placed close to each other. The powers of all the three will be added. When the water is filled on concave side, we get plano-convex water lens whose focal length is found by lens makers formula. This combination consists of four lenses (two lenses of glass and two lenses of water) and one mirror. 

(a) The refraction takes place from the first surface, reflection from the lower surface and finally refraction from the first surface of focal lengths  fg , fm and fg , respectively. The combined focal length F is given by

For the image to be informed at the same point as the object

u = 2F = 2 x 7.5 = 15 cm.

The object should be placed at a distance of 15 cm from the lens on the optical axis.

(b) If fw is the focal length of lens in water, the focal length F’ of this combination is given by

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