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A pinhole camera is just a rectangular box with a tiny hole in one face. The film is on the face opposite this hole, and that is where the image is formed. The camera forms an image without a lens. (a) Make a clear ray diagram to show how a pinhole camera can form an image on the film without using a lens. (Hint: Put an object outside the hole, and then draw rays passing through the hole to the opposite side of the box.) (b) A certain pinhole camera is a box that is \(25 \mathrm{~cm}\) square and \(20.0 \mathrm{~cm}\) deep, with the hole in the middle of one of the \(25 \mathrm{~cm} \times 25 \mathrm{~cm}\) faces. If this camera is used to photograph a fierce chicken that is \(18 \mathrm{~cm}\) high and \(1.5 \mathrm{~m}\) in front of the camera, how large is the image of this bird on the film? What is the lateral magnification of this camera?

Step by step solution

01

Ray Diagram

Draw the object outside the camera and the film on the opposite side of the box. The pinhole would be the location where the light rays pass through. An inverted image is formed on the film as light rays passing through the pinhole diverges to the film.

02

Image Size Calculation

The size of the image can be calculated using similar triangles theory. The size (height) of the image can be given as \( h' = h \times \frac{d}{D}\), where d is the depth of the camera (20.0cm), D is the distance of the object from the camera (1.5m = 150cm) and h is the height of the object (18 cm). Substituting the given values, we get \( h' = 18 \times \frac{20.0}{150} = 2.4 \mathrm{~cm} \). Thus, the image of the bird on the film is 2.4cm high.

03

Lateral Magnification Calculation

The lateral magnification of the pinhole camera is defined as the ratio of the image size to the object size, which is same as the negative ratio of the image distance (inside the camera) to the object distance (outside the camera). Thus, \( M = -\frac{h'}{h} = -\frac{20.0}{150} = -\frac{1}{7.5} \). So, the lateral magnification of the camera is -1/7.5.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Ray Diagram

Drawing a ray diagram is an essential first step in understanding how a pinhole camera functions. A pinhole camera is essentially a light-tight box with a small hole on one side. To visualize how an image is formed without a lens, follow these simple steps:

• Place an object, such as a fierce chicken, outside the camera opposite the pinhole.
• Consider rays of light traveling from every point of the object. Only the rays that pass through the pinhole will reach the other side, where the film is located.
• Because the pinhole is small, light from the top of the object travels downward through the hole, and light from the bottom travels upward.
• This crossover causes an inverted image to form on the film.

By mapping the path of these rays, you can clearly see why the image appears upside down. The simplicity of the pinhole camera demonstrates the principles of light without complex optics or lenses.

Similar Triangles

Similar triangles help in comprehending how images in a pinhole camera are scaled versions of the objects they represent. When light rays from the object pass through the pinhole, triangles are formed between the object, pinhole, and image. These triangles are similar due to their corresponding angles being equal.Given the height of the object ($h")) and its distance from the pinhole (\(D\)), you can determine the image height (\(h'\)) and the camera depth (\(d\)) using the formula:\[ h' = h \frac{d}{D}\]

• Object height (h) = 18 cm.
• Object distance from pinhole (D) = 150 cm.
• Camera depth (d) = 20 cm.

Plug these values into the formula:\[ h' = 18 \times \frac{20.0}{150} = 2.4 cm\]This calculation shows that the image of the chicken is reduced to 2.4 cm in height. The concept of similar triangles allows us to rationalize how the dimensions of an image are proportionally smaller than the object.

Lateral Magnification

Lateral magnification is a measure of how much the image size is altered in comparison to the object size. In a pinhole camera, this magnification does not involve enlarging lenses as with typical cameras.The lateral magnification (\(M\)) for a pinhole camera is expressed as:\[ M = -\frac{h'}{h} = -\frac{d}{D}\]

• The negative sign indicates the inversion of the image.
• \(h'\) is the image height, and \(h\) is the object height.
• \(d\) is the depth of the camera, and \(D\) is the distance to the object.

Using the provided data:\[ M = -\frac{20.0}{150} = -\frac{1}{7.5}\]This implies that the image is scaled down by a factor of \(-1/7.5\) compared to the original object size. The negative signifies the image inversion, while the fraction denotes reduction, typical of pinhole cameras.