For refraction at a concave spherical surface (separating glass-air medium), the distance of the object should be greater than 'z' times the radius of curvature of the refracting surface for the image to be real, the value of 'z' is:

Question

Accepted Answer

To determine why Option 4 is the correct answer, we need to understand the principles of refraction at a concave spherical surface, particularly when transitioning from a denser medium (glass) to a rarer medium (air).

### Explanation for Option 4:

1. **Refraction at a Concave Surface**: When light travels from a denser medium (glass) to a rarer medium (air) through a concave spherical surface, the refraction is governed by the lens maker's formula and the geometry of the surface.

2. **Lens Maker's Formula**: The formula for refraction at a spherical surface is given by:
   $$
   \frac{n_2}{v} - \frac{n_1}{u} = \frac{n_2 - n_1}{R}
   $$
   where $ n_1 $ and $ n_2 $ are the refractive indices of glass and air, respectively, $ u $ is the object distance, $ v $ is the image distance, and $ R $ is the radius of curvature.

3. **Real Image Formation**: For the image to be real, the image distance $ v $ must be positive. This condition is satisfied when the object distance $ u $ is greater than a certain multiple of the radius of curvature $ R $.

4. **Derivation for 'z'**: By manipulating the formula and considering the refractive indices (where $ n_1 > n_2 $), we find that the object distance $ u $ must be greater than 3 times the radius of curvature $ R $ for the image to be real. This is because the light rays need to converge on the opposite side of the surface to form a real image.

5. **Conclusion**: Therefore, the value of 'z' is 3, making Option 4 the correct answer.

### Explanation for Why Other Options are Incorrect:

- **Option 1 (z = 2)**: If $ z $ were 2, the object distance would not be sufficient to ensure that the refracted rays converge to form a real image on the opposite side. The rays would still diverge or converge insufficiently, leading to a virtual image.

- **Option 2 (z = 3/2)**: A value of $ z = 3/2 $ is too small to allow the rays to converge properly. The object would be too close to the surface, resulting in a virtual image rather than a real one.

- **Option 3 (z = 2/3)**: This value is even smaller and would certainly not allow for the formation of a real image. The object would be much too close to the surface, causing the rays to diverge after refraction.

In summary, the correct value of 'z' is 3, as stated in Option 4, ensuring that the object distance is sufficient for the formation of a real image when light refracts through a concave spherical surface from glass to air.