A transparent sphere of radius and refractive index is kept in the air. At what distance from the surface of the sphere should a point object be placed on the principal axis so as to form a real image at the same distance from the second surface of the sphere?

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answer is C.

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Detailed Solution

Concept- By maintaining the point object at a specific distance from the sphere's first surface in order to acquire its image at infinity (since the image is real) from the second surface, we may make a ray diagram.
Refraction will occur as a result of a change in the medium, thus we can use the equation for refraction through a spherical surface to calculate the distance at which the item must be placed. Air's refractive index is always assumed to be 1.
Light will be refracted as it travels from the air to the sphere because the sphere is a denser medium than the air. The following is the ray diagram for an item and its reflection via a spherical surface:
The item is preserved at O, u distance from the sphere's first surface, while its image is created at I, v distance from the sphere's second surface. The picture must be at an endless distance from the sphere for it to be genuine. According to sign conventions, the distances on the left are assumed to be negative, and as a result, the distance of the object (u) in the ray diagram is also taken to be negative.
The object's distance from the sphere may be determined using the following equation for the refraction through a sphere:

Here,
Refractive index of air from where incident light start travel

Refractive index of sphere where incident light strikes

(given)
Distance of object from the sphere

(to be calculated; to the left side of the sphere, hence negative)
Distance of object from the sphere

(for the image to be real)
Radius of sphere

(given)
Substituting these values, we get:

Therefore, the distance of a point object from the surface of the sphere to form a real image at the same distance from the second surface of the sphere is

.
Hence, the correct option is 3) .

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