Problem 80 - Physics for Scientists and Engineers with Modern Physics

Question

(III) Given two 12 -cm-focal-length lenses, you attempt to make a crude microscope using them. While holding these lenses a distance \(55 \mathrm{~cm}\) apart, you position your microscope so that its objective lens is distance \(d_{0}\) from a small object. Assume your eye's near point \(N=25 \mathrm{~cm}\). (a) For your microscope to function properly, what should \(d_{\mathrm{o}}\) be? (b) Assuming your eye is relaxed when using it, what magnification \(M\) does your microscope achieve? (c) Since the length of your microscope is not much greater than the focal lengths of its lenses, the approximation \(M \approx N \ell / f_{\mathrm{c}} f_{\mathrm{o}}\) is not valid. If you apply this approximation to your microscope, what \% error do you make in your microscope's true magnification?

Step-by-Step Solution

Answer

(a) 16.6 cm, (b) 5.39, (c) 77.14% error.

Step 1: Determine image distance for the objective lens

For the microscope to work, its final image should be at the near point, 25 cm from your eye. Assuming a relaxed eye, the image created by the eyepiece should be at infinity. Thus, the intermediate image created by the objective lens should fall at the focal point of the eyepiece to ensure this. This means the distance between the lenses and the image distance for the objective are related by:\[(d_{o} + f_{c}) = 55 \text{ cm}\]Where \(f_{c}\) is the focal length of the eyepiece (12 cm). Therefore, the image distance \(d_{i}\) from the objective lens is:\[d_{i} = 55 \text{ cm} - 12 \text{ cm} = 43 \text{ cm}\]

Step 2: Use lens formula for the objective lens

The lens formula is given by: \[\frac{1}{f_{o}} = \frac{1}{d_{o}} + \frac{1}{d_{i}}\]Given \(f_{o} = 12 \text{ cm}\) (focal length of objective lens) and \(d_{i} = 43 \text{ cm}\), substitute these values into the lens equation:\[\frac{1}{12} = \frac{1}{d_{o}} + \frac{1}{43}\]

Step 3: Solve for object distance \(d_{o}\)

Rearrange the lens equation to solve for \(\frac{1}{d_{o}}\):\[\frac{1}{d_{o}} = \frac{1}{12} - \frac{1}{43}\]Calculate the right hand side:\[\frac{1}{d_{o}} = \frac{43 - 12}{516}\]\[\frac{1}{d_{o}} = \frac{31}{516}\]Thus, \[d_{o} \approx 16.6 \text{ cm}\]

Step 4: Compute magnification \(M\) in the relaxed case

The total magnification \(M\) is the product of the magnifications from both the objective and eyepiece. With the eyepiece forming an image at infinity, its magnification \(M_{c}\) is:\[M_{c} = \frac{N}{f_{c}} = \frac{25}{12} \approx 2.08\]The magnification by the objective \(M_{o}\) is given by:\[M_{o} = \frac{d_{i}}{d_{o}} = \frac{43}{16.6} \approx 2.59\]So the total magnification is:\[M = M_{o} \times M_{c} \approx 2.59 \times 2.08 \approx 5.39\]

Step 5: Calculate magnification using approximation and find error

Using the approximation \(M \approx \frac{N \ell}{f_{c} f_{o}}\):\[M \approx \frac{25 \times 55}{12 \times 12} = \frac{1375}{144} \approx 9.55\]The actual magnification calculated was 5.39. The percent error is calculated as:\[\text{Percent Error} = \frac{|9.55 - 5.39|}{5.39} \times 100 \approx 77.14\%\]

Key Concepts

Lens Formula

The lens formula is a fundamental equation used to relate the focal length, object distance, and image distance of a lens. It is expressed as:
\[\frac{1}{f} = \frac{1}{d_o} + \frac{1}{d_i}\]
where:

• \(f\) is the focal length of the lens.
• \(d_o\) is the distance from the object to the lens (object distance).
• \(d_i\) is the distance from the image to the lens (image distance).

The lens formula is derived from geometrical optics, and it is quite useful for calculating unknown distances when you have two of these values. This equation is crucial in understanding the behavior of light as it passes through convex or concave lenses, commonly found in devices like cameras, glasses, and microscopes.
The formula works for both converging and diverging lenses, but remember to consider the sign convention used in optics:

• For a converging lens, the focal length \(f\) is positive.
• For a diverging lens, the focal length \(f\) is negative.
• Distances are positive if they are on the same side as the incoming light (real side).

In our microscope example, using this formula helped determine how far the object should be placed from the lens to achieve the desired magnification.

Image Distance

The image distance \(d_i\) in lens systems is the space between the lens and the image formed by the lens. Understanding this concept is important for applications ranging from photography to microscopy.
In optics, the actual position where the image forms depends on the object distance \(d_o\) and the focal length \(f\) of the lens, which we can calculate using the lens formula. In the context of a microscope, the choice of image distance affects the clarity and magnification of the viewed object.
**Factors Influencing Image Distance**- **Lens Type**: Convex lenses converge light and can form real images; concave lenses diverge light and generally form virtual images.- **Object Position**: When an object is placed closer to a convex lens than its focal length, a virtual, upright, and magnified image is formed on the same side as the object.
When designing optical instruments like microscopes, choosing the correct image distance is crucial for creating a viewable image, especially when an observer’s eye needs to focus on it comfortably. In our provided solution, we assumed the eyepiece forms an image at infinity to relax the eye's accommodation.

Focal Length Calculation

Focal length is one of the critical determinants of how lenses behave when directing light. It is defined as the distance between the lens and the point where parallel rays of light converge, known as the focus.
In the calculation of a system using the lens formula, the focal length helps determine either the object distance or the image distance when the other values are known.
For a microscope, using lenses with a specific **Focal Length** - **Focal Length of Objective Lens**: This lens initiates the magnification process, and its focal length determines how much the image of the specimen is enlarged initially before further magnification by the eyepiece. - **Focal Length of Eyepiece Lens**: This lens magnifies the image relayed from the objective lens and often determines the overall field of view.
Using two lenses, each having a 12cm focal length in our calculation, required proper understanding of the interplay between these lenses to access how these distances affect overall magnification. The accuracy in setting focal lengths within an optical system directly correlates with achieving the correct magnification and minimizing errors such as the approximate magnification deviation discussed in the exercise.