when the distance between object and screen is more than 4f, we can obtain the image of the object on the screen for the two positions of the lens.in one case image is magnified.if the I₁ and I₂ be the sizes of two images,then the size of the object is?

To tackle your question about the relationship between object distance, image distance, and magnification in lens systems, let's break it down step by step. When dealing with lenses, particularly convex lenses, we often refer to the focal length (f) and the distances involved in forming images. The scenario you described involves an object placed at a distance greater than 4f from the lens, which indeed allows for the formation of two distinct images on the screen.

Understanding Image Formation with Lenses

When an object is positioned beyond 4f from a convex lens, two images can be formed due to the lens's ability to refract light in different ways depending on the object's position relative to the focal points. The two positions of the lens create two different images, which we can denote as I1 and I2.

Magnification and Image Size

The magnification (M) produced by a lens is defined as the ratio of the height of the image (h) to the height of the object (H). This can be expressed mathematically as:

• M = h / H

For a convex lens, the magnification can also be related to the object distance (u) and the image distance (v) using the formula:

• M = v / u

In your case, since you have two images (I1 and I2), each will have its own magnification:

• M1 = v1 / u
• M2 = v2 / u

Relating Image Sizes to Object Size

Given that I1 and I2 are the sizes of the two images formed, we can express them in terms of the object size (H) and their respective magnifications:

• I1 = M1 * H
• I2 = M2 * H

From these equations, we can derive the size of the object (H) if we know the sizes of the images (I1 and I2) and their respective magnifications:

• H = I1 / M1
• H = I2 / M2

Example Calculation

Let’s say you have the following values:

• I1 = 10 cm (size of the first image)
• M1 = 2 (magnification for the first image)
• I2 = 5 cm (size of the second image)
• M2 = 1 (magnification for the second image)

Using the magnification formulas, we can find the object size:

• H = I1 / M1 = 10 cm / 2 = 5 cm
• H = I2 / M2 = 5 cm / 1 = 5 cm

Both calculations give us the same object size of 5 cm, confirming the consistency of the lens equations.

Final Thoughts

In summary, when the object is placed beyond 4f, two images can be formed, and their sizes can be related back to the object size using the magnification formulas. This relationship is crucial in optics and helps us understand how lenses manipulate light to form images. If you have any further questions or need clarification on any point, feel free to ask!