How Is Angular Width Calculated in Lens Optics?

In summary, the question is asking for the angular width of a scene that is focused onto a flat screen using a converging lens with a 50.0 mm focal length. The solution involves calculating the height of the image as a function of its distance from the lens, then using that to find the distance and finally using the distance and width to calculate the angle. Assuming that the distance is equal to the focal length is incorrect.
  • #1
SamTsui86
31
0

Homework Statement



A converging lens with a 50.0 mm focal length is used to focus an image of a very distant scene onto a flat screen 37.0 mm wide. What is the angular width of the scene included in the image on the screen?

Homework Equations





The Attempt at a Solution



Basically I just did tan-1(37/50) = 36.5

What are I doing wrong?
 
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  • #2
Do you know how to calculate the height of the image as a function of it's distance form the lens? Use that to find get the distance and then use the distance and width to get the angle. Your problem is that you assumed that the distance was the focal length which isn't true.
 
  • #3


I would like to first clarify that the angular width of the image on the screen refers to the angular size of the object being viewed on the screen, not the physical size of the screen itself. This is an important distinction in optics.

To solve this problem, we can use the formula for angular magnification of a lens, which is given by M = -s'/s, where M is the magnification, s' is the image distance, and s is the object distance. In this case, the object distance is essentially infinity since the scene is very distant, so we can approximate s as infinity.

Using the thin lens equation, 1/s' + 1/s = 1/f, we can solve for s', the image distance. Plugging in the given values, we get s' = 83.3 mm.

Now, we can use the formula for angular magnification to find the angular width of the image on the screen, which is given by tan-1(s'/s). Plugging in our values, we get tan-1(83.3/∞) = 90 degrees. This means that the angular width of the scene included in the image on the screen is 90 degrees, which makes sense since the scene is very distant and thus appears to cover a wide angle.

In summary, to solve this problem, we need to use the thin lens equation and the formula for angular magnification. It's important to pay attention to the units and make sure they are consistent throughout the calculation. Also, it's important to understand the difference between angular size and physical size in optics.
 

1. What is a lens in physics?

A lens in physics is a transparent material that is curved in a way that causes light to bend as it passes through it. It can be either convex (thicker in the middle and thinner at the edges) or concave (thicker at the edges and thinner in the middle).

2. How does a lens work?

A lens works by refracting (bending) light as it passes through it. This bending of light allows the lens to focus or spread out light rays, depending on its curvature. It is this property that makes lenses useful in correcting vision and creating images in cameras and microscopes.

3. What is the difference between convex and concave lenses?

The main difference between convex and concave lenses is their shape and the way they bend light. Convex lenses bulge outwards and converge light rays, while concave lenses curve inward and diverge light rays. Convex lenses are used to create magnified, real images, while concave lenses are used to create smaller, virtual images.

4. How is the focal length of a lens determined?

The focal length of a lens is determined by the curvature of the lens and the medium through which light passes. It is the distance between the lens and the point where parallel light rays converge after passing through the lens. The longer the focal length, the more spread out the light rays will be, and the shorter the focal length, the more focused the light rays will be.

5. What are some real-world applications of lenses in physics?

Lenses have numerous real-world applications in physics. They are used in eyeglasses and contact lenses to correct vision, in cameras and telescopes to create images, in microscopes to magnify tiny objects, and in projectors to create enlarged images. Lenses are also used in optical instruments such as binoculars, magnifying glasses, and laser pointers. They are also used in everyday objects such as magnifying makeup mirrors and car headlights.

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