Problem 104 - Physics for Scientists and Engineers with Modern Physics

Question

Two converging lenses, one with \(f=4.0 \mathrm{cm}\) and the other with \(f=44 \mathrm{cm},\) are made into a telescope. \((a)\) What are the length and magnification? Which lens should be the eyepiece? \((b)\) Assume these lenses are now combined to make a microscope; if the magnification needs to be \(25 \times\) , how long would the microscope be?

Step-by-Step Solution

Answer

Telescope: 48 cm, magnification 11, 4 cm eyepiece. Microscope: 44 cm, 25x magnification.

Step 1: Understanding the Telescope Setup

In a telescope setup, the objective lens has a longer focal length than the eyepiece. So, the lens with a focal length of 44 cm should be the objective, and the lens with a focal length of 4.0 cm should be the eyepiece.

Step 2: Calculating the Telescope Length

The length of the telescope is the sum of the focal lengths of the objective and the eyepiece: \( L = f_o + f_e = 44\, \text{cm} + 4\, \text{cm} = 48\, \text{cm} \).

Step 3: Calculating the Telescope Magnification

The magnification of a telescope is given by the ratio of the focal lengths: \( M = \frac{f_o}{f_e} = \frac{44}{4} = 11 \).

Step 4: Understanding the Microscope Setup

In a microscope, the lens with a shorter focal length should be used as the objective, and the one with a longer focal length as the eyepiece. Therefore, for the microscope, the 4.0 cm lens will be the objective, and the 44 cm lens will be the eyepiece.

Step 5: Calculating Microscope Length for Given Magnification

The total magnification \( M \) of a microscope is given by \( M = \frac{L}{f_o} \cdot \frac{D}{f_e} \), where \( L \) is the length of the microscope, \( D \) is the near point distance (usually 25 cm), \( f_o = 4\, \text{cm}, \ f_e = 44\, \text{cm} \). Rearrange to find \( L \):\[ M = \frac{L}{f_o} \cdot \frac{D}{f_e} = \frac{L \cdot 25}{4 \cdot 44} \]\[ L = \frac{M \cdot f_o \cdot f_e}{D} = \frac{25 \cdot 4 \cdot 44}{25} = 44\, \text{cm} \].

Step 6: Summarizing the Results

The telescope is 48 cm long with a magnification of 11, and the eyepiece should be the 4.0 cm lens. The microscope needs to be 44 cm long for a magnification of 25x, with the 4.0 cm lens as the objective.

Key Concepts

Telescope

A telescope is an amazing optical instrument that allows us to see distant objects like stars and planets with more clarity. It works by collecting light through a large lens, known as the objective, and then brings that light to a focus where it can be viewed through a smaller lens called the eyepiece.
In an astronomical telescope, the lens with the longer focal length is the objective. This is because it needs to gather light from faraway objects. Meanwhile, the eyepiece lens with a shorter focal length helps magnify the image of the object formed by the objective lens.
To compute the overall length of a telescope, you simply add the focal lengths of both lenses. This gives us the distance needed between them to focus the image properly for viewing.

Microscope

A microscope allows us to observe very tiny objects that are not visible to the naked eye, such as cells and microorganisms. Unlike telescopes, microscopes use their lenses in the opposite order.
In a microscope, the lens with a shorter focal length is used as the objective. This is to form an enlarged image of the small object. The eyepiece, which has a longer focal length, further magnifies this image so that we can see details. This system of lenses allows for a very high total magnification.
Setting up a microscope requires adjusting the distance between the objective and eyepiece to achieve the desired magnification. By doing this properly, we can ensure clear and detailed viewing of tiny specimens.

Lens Focal Length

When dealing with optical instruments like telescopes and microscopes, the focal length of a lens is crucial. The focal length is the distance from the lens where light rays converge to a point. It essentially determines how much the lens can bend light rays.

• A longer focal length means the lens gathers light over a wider area and focuses it at a point further away.
• A shorter focal length bends the light more quickly, focusing it closer to the lens.

In optical devices, choosing the right combination of long and short focal lengths is key to determining how much you can magnify an object or how wide a field you can capture. This impacts everything from the clarity of the image to the overall size of the device.

Telescope Magnification

Magnification in telescopes is a measure of how much larger an object appears when viewed through the instrument. The magnification depends on the relationship between the focal lengths of the objective lens and the eyepiece.

• The formula for magnification is: \( M = \frac{f_o}{f_e} \), where \(f_o\) is the focal length of the objective lens and \(f_e\) is the focal length of the eyepiece lens.
• A higher magnification means you can see more detail but might reduce the brightness and field of view.
• Choosing the right magnification is about balancing detail and clarity, especially when observing faint and distant objects.

By adjusting the lenses in your telescope, you can explore everything from a sweeping view of the night sky to close-up details of faraway celestial bodies.