Question

A telescope has been properly focused on the Sun. You want to observe the Sun visually, but to protect your sight you don't want to look through the eyepiece; rather, you want to project an image of the Sun on a screen \(1.5 \mathrm{~m}\) behind (the original position of ) the eyepiece, and observe that. If the focal length of the eyepiece is \(8.0 \mathrm{~cm}\), how must you move the eyepiece?

Short answer

Answer: The eyepiece should be moved approximately 9 cm closer to the objective lens.

Step-by-step solution

Write down the lens formula

We need to use the lens formula to relate the object distance (u), image distance (v), and the focal length (f). The formula is as follows: (1/v) - (1/u) = (1/f)

Find the object distance (u) and focal length (f)

In this problem, we are given the focal length of the eyepiece as f = 8.0 cm. Also, when looking at the sun, the object distance can be considered to be infinity.

Find the initial image distance (v1)

Since the telescope was properly focused to look at the Sun initially, substitute the object distance as infinity into the lens formula: (1/v1) - (1/∞) = (1/8.0) (1/v1) = (1/8.0) So, the initial image distance v1 = 8.0 cm

Find the final image distance (v2) and final object distance (u2)

The screen is placed 1.5 meters (150 cm) behind the initial position of the eyepiece. Therefore, the total distance between the image and the eyepiece is v2 = v1 + 150 = 8.0 + 150 = 158 cm Now, to find the new object distance (u2), use the lens formula again: (1/v2) - (1/u2) = (1/f) (1/158) - (1/u2) = (1/8.0)

Find the new object distance (u2)

Solve the equation for u2: (1/u2) = (1/158) - (1/8) (1/u2) ≈ -0.111 Therefore, u2 ≈ -9 cm

Determine how far to move the eyepiece

The change in object distance is the difference between the initial object distance (infinity) and the new object distance (u2): Δu = u2 - infinity Since the original distance is infinity, the telescope needs to be moved approximately 9 cm closer to the objective lens to project the image 1.5 meters behind its current position. Keep in mind that in practice, distances and movement might vary due to necessary adjustments for practical purposes such as clear image focus.