Question

A biconvex thin lens is prepared from glass of refractive index 3/2​. The two bounding surfaces have equal radii of 25 cm each. One of the surfaces is silvered from outside to make it reflecting. Where should an object be placed before this lens so that the image is formed at the same position as the object?

a. At a distance less than 25 cm from the lens
b. At a distance of 25 cm from the lens
c. At a distance between 25 cm and 50 cm from the lens
d. At a distance greater than 50 cm from the lens

Answer 1

To form an image at the same position as the object using a biconvex thin lens with one side silvered, the object must be placed at twice the focal length of the lens, which in this case is 50 cm away from the lens.

Step-by-step explanation:

To determine where an object should be placed before a lens that has one surface silvered (functioning as a mirror) so that its image is formed at the same position as the object, we can apply the principles of both lens and mirror optics. First, we calculate the focal length of the biconvex lens using the Lens Maker's Equation and the given refractive index and radius of curvature.

The thin-lens equation is given by 1/f = (n-1)(1/R1 - 1/R2), where n is the refractive index and R1 and R2 are the radii of curvature for the two lens surfaces. For a biconvex lens with equal and opposite radii, this simplifies to 1/f = 2(n-1)/R.

Using the refractive index (n=3/2) and radius of curvature (R=25 cm), the focal length (f) can be calculated. Once the focal length is known, we need to double it to account for the mirror. An object must be placed at twice the focal length from the lens in order for the object and its image to coincide after reflection from the silvered surface. Therefore, the object should be placed at a distance of 50 cm from the lens. This is because at two times the focal length, the image formed by the lens would be at the focal point of the mirrored surface, which would then reflect back to the original position of the object.