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Chapter 33: Problem 12

Several small drops of paint (less than \(1 \mathrm{~mm}\) in diameter) splatter on a painter's eyeglasses, which are approximately \(2 \mathrm{~cm}\) in front of the painter's eyes. Do the dots appear in what the painter sees? How do the dots affect what the painter sees?

Short Answer

Expert verified
Answer: Yes, the small paint drops on the painter's eyeglasses can affect their vision. The paint drops have an angular size of approximately 2.86°, which is significant enough to be noticeable and may obstruct a portion of the painter's field of view, potentially affecting their ability to paint precisely.

Step by step solution

01

Convert the paint drop size and the distance from the eye to meters

First, we need to convert the size of the paint drops and the distance from the eyes to the same unit, which is meters (m). The paint drops are less than 1mm in diameter, and eyeglasses are about 2cm in front of the painter's eyes. We can convert these values as follows: Paint drop diameter: \(1 mm = 0.001 m\) Distance from the eyes: \(2 cm = 0.02 m\)
02

Calculate the angular size of the paint drops

Next, we need to find the angular size (θ) of the paint drops as seen by the painter. We can use the formula for the small angle approximation, where: \( θ = \frac{d}{D}\), where d is the diameter of the paint drops and D is the distance from the eyes.
03

Plug in values and perform the calculation

Now we can plug in the values we obtained in step 1: \( θ = \frac{0.001}{0.02}\) \( θ = 0.05\) radian To convert the angle from radians to degrees, we can use the following formula: \( θ_{degrees} = \frac{θ_{radians}}{\pi} × 180\) \( θ_{degrees} = \frac{0.05}{\pi} × 180\) \( θ_{degrees} ≈ 2.86°\)
04

Interpret the results

The angular size of the paint drops is approximately 2.86°. Although the paint drops are small and close to the eyes, their angular size is significant enough to be noticed by the painter. This means that the small drops of paint will indeed appear in what the painter sees, and they will affect their vision. The drops can obstruct a portion of the painter's field of view, creating a nuisance and potentially affecting their ability to paint precisely. In summary, the small paint drops on the painter's eyeglasses will be noticeable to the painter and may affect their vision by obstructing a portion of their field of view.

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Most popular questions from this chapter

You have found in the lab an old microscope, which has lost its eyepiece. It still has its objective lens, and markings indicate that its focal length is \(7.00 \mathrm{~mm}\). You can put in a new eyepiece, which goes in \(20.0 \mathrm{~cm}\) from the objective. You need a magnification of about 200. Assume you want the comfortable viewing distance for the final image to be \(25.0 \mathrm{~cm}\). You find in a drawer eyepieces marked \(2.00-, 4.00-,\) and \(8.00-\mathrm{cm}\) focal length. Which is your best choice?

Galileo discovered the moons of Jupiter in the fall of \(1609 .\) He used a telescope of his own design that had an objective lens with a focal length of \(f_{o}=40.0\) inches and an eyepiece lens with a focal length of \(f_{e}=2.00\) inches. Calculate the magnifying power of Galileo's telescope.

You visit your eye doctor and discover that you require lenses having a diopter value of -8.4 . Are you nearsighted or farsighted? With uncorrected vision, how far away from your eyes must you hold a book to read clearly?

The distance from the lens (actually a combination of the cornea and the crystalline lens) to the retina at the back of the eye is \(2.0 \mathrm{~cm}\). If light is to focus on the retina, a) what is the focal length of the lens when viewing a distant object? b) what is the focal length of the lens when viewing an object \(25 \mathrm{~cm}\) away from the front of the eye?

An object is placed on the left of a converging lens at a distance that is less than the focal length of the lens. The image produced will be a) real and inverted. c) virtual and inverted. b) virtual and erect. d) real and erect.
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