If this telescope is used to view a 100 m tall tower 3 km away, what is the height of the image of the tower formed by the objective lens?

To determine the height of the image of a 100 m tall tower viewed from a distance of 3 km using a telescope's objective lens, we need to employ some basic optical principles. Let's break this down step by step to make it clearer.

Understanding the Basics of Lenses

The objective lens of a telescope is designed to gather light and form an image of distant objects, such as our 100 m tall tower. The height of the image formed by a lens can be found using the magnification formula:

Magnification Formula

The magnification (M) of a lens is defined as the ratio of the height of the image (h') to the height of the object (h):

• M = h' / h

Where:

• h' = height of the image
• h = height of the object (in this case, the tower)

Calculating the Magnification

To find the height of the image, we first need to determine the magnification produced by the telescope. The magnification is also related to the focal lengths of the objective lens (f_o) and the eyepiece (f_e):

• M = f_o / f_e

However, since we do not have the focal lengths provided, let’s assume a typical magnification value for a powerful telescope, say M = 100. This means that the image will be 100 times the size of the object.

Finding the Height of the Image

Now that we have our magnification, we can calculate the height of the image of the tower. Given that the height of the tower (h) is 100 m:

• h' = M * h
• h' = 100 * 100 m
• h' = 10,000 m

Conclusion

The height of the image of the tower formed by the objective lens, using a magnification of 100, would be 10,000 m. This example highlights how telescopes can effectively magnify distant objects, allowing us to see them in greater detail.

Real-World Implications

In practice, while this calculation provides a theoretical height, actual observations may vary due to atmospheric conditions, lens imperfections, and other factors. Nonetheless, understanding these basic principles allows astronomers and enthusiasts alike to appreciate the power and utility of telescopes in observing our world and beyond.