Two converging lenses, one with ƒ = 4.0 cm and the other with ƒ = 41 cm , are made into a telescope. (a) Determine the telescope’s length and its magnification. Which lens should be the eyepiece? (b) Assume these lenses are now combined to make a microscope; if the magnification needs to be 125 x , how long would the microscope be? Explain why the result in (b) is so impractical.

Verified step by step guidance

1. Step 1: Understand the problem. The problem involves two converging lenses being used in two different configurations: as a telescope and as a microscope. For the telescope, we need to calculate its length and magnification, and determine which lens should be the eyepiece. For the microscope, we need to calculate its length for a given magnification and explain why the result is impractical.
2. Step 2: For the telescope, the length of the telescope is the sum of the focal lengths of the two lenses. Use the formula: $L\; =\; f\_1\; +\; f\_2$, where $f\_1$ is the focal length of the objective lens and $f\_2$ is the focal length of the eyepiece. The magnification of the telescope is given by: $M\; =\; -f\_1\; /\; f\_2$. The lens with the shorter focal length should be the eyepiece because it provides higher magnification.
3. Step 3: For the microscope, the total magnification is the product of the magnifications of the objective and the eyepiece. Use the formula: $M\_\{total\}\; =\; M\_\{objective\}\; imes\; M\_\{eyepiece\}$. The magnification of each lens is related to its focal length and the distance between the lenses. The length of the microscope is approximately the sum of the focal lengths of the two lenses plus the tube length required for focusing.
4. Step 4: To calculate the length of the microscope for a magnification of 125x, rearrange the magnification formula to solve for the required tube length. Use the relationship: $M\_\{objective\}\; =\; L\; /\; f\_\{objective\}$ and $M\_\{eyepiece\}\; =\; 25\; /\; f\_\{eyepiece\}$, where 25 cm is the near point of the human eye. Combine these equations to find the total length of the microscope.
5. Step 5: Explain why the result for the microscope is impractical. The impracticality arises because the focal lengths of the lenses are very different, leading to a very long microscope. This is not feasible for practical use, as microscopes are typically designed to be compact and ergonomic.

Key Concepts

Lens Formula

The lens formula relates the focal length (f), object distance (u), and image distance (v) of a lens, expressed as 1/f = 1/v - 1/u. This formula is essential for understanding how lenses form images and is crucial for calculating the effective focal lengths when combining lenses in optical devices like telescopes and microscopes.

Magnification

Magnification is the ratio of the height of the image to the height of the object, and it can also be defined in terms of the distances involved in lens systems. For telescopes, the total magnification is the product of the magnifications of the objective and eyepiece lenses, which helps determine how much larger an object appears when viewed through the device.

Telescope and Microscope Design

The design of telescopes and microscopes involves specific arrangements of lenses to achieve desired focal lengths and magnifications. In telescopes, the eyepiece lens typically has a shorter focal length to provide higher magnification, while in microscopes, the combination of lenses must be optimized to achieve high magnification without excessive length, which can lead to impractical designs.