Problem 113 - Fundamentals of Physics

Question

A pinhole camera has the hole a distance \(12 \mathrm{~cm}\) from the film plane, which is a rectangle of height \(8.0 \mathrm{~cm}\) and width \(6.0 \mathrm{~cm} .\) How far from a painting of dimensions \(50 \mathrm{~cm}\) by \(50 \mathrm{~cm}\) should the camera be placed so as to get the largest complete image possible on the film plane?

Step-by-Step Solution

Answer

The camera should be placed 75 cm away from the painting.

Step 1: Understand the Problem

We need to find how far the camera should be from a square painting to get its largest complete image on the film plane. The film plane is a rectangle of size 8 cm by 6 cm, and the painting has dimensions of 50 cm by 50 cm. The hole is 12 cm from the film plane.

Step 2: Use the Similar Triangles Concept

In a pinhole camera, the image forms via similar triangles. The ratio of the dimensions of the object to the image corresponds to the ratio of their distances from the pinhole.

Step 3: Formulate the Similarity Equation

Let the distance from the painting to the pinhole be denoted as \(d\). From the similar triangles, the ratio of the painting's dimensions to the film's dimensions is equal to the ratio of the distances:\[ \frac{50}{8} = \frac{d}{12} \] (for height fitting) and \[ \frac{50}{6} = \frac{d}{12} \] (for width fitting).

Step 4: Solve the Equations

Solving \( \frac{50}{8} = \frac{d}{12} \), we get \(d = \frac{50 \times 12}{8} = 75 \) cm.Solving \( \frac{50}{6} = \frac{d}{12} \), we get \(d = \frac{50 \times 12}{6} = 100 \) cm.

Step 5: Determine the Correct Position

To have the largest complete image on the film plane, the limiting factor is the dimension that forms the larger ratio without exceeding. Thus, the smaller distance governs how far the camera should be placed. The smaller distance is 75 cm.

Key Concepts

Optics

Optics is the branch of physics that deals with the study of light and how it interacts with various materials. When discussing a pinhole camera, we delve into geometric optics, particularly the behavior of light rays. A pinhole camera works without a lens. Instead, light from a scene passes through a tiny hole to form an inverted image on the film plane. This phenomenon occurs because light travels in straight lines.

In the context of the exercise, optics plays a vital role in understanding how the image is formed. Since light rays converge at the pinhole, it creates an image on the film that is a scaled-down version of the original object (the painting). This light manipulation allows us to produce images by using simple properties of light and shapes.

Simulation of Triangles

The pinhole camera utilizes the principle of similar triangles to determine how far the object should be for the image to fit on the film. In geometric terms, two triangles are similar if their corresponding angles are equal and their corresponding sides are in proportion. This is a fundamental concept often used in optics to understand image formation via pinhole or lens systems.

For our pinhole camera, the triangles form between the object, the pinhole, and the image plane. One triangle is created by the height and distance of the painting, and the other by the height and distance on the film. Since the pinhole acts as a vertex for both triangles, we can use this similarity to create equations that help us solve how far the camera should be from the painting. Through the exercise formulas like \( \frac{50}{8} = \frac{d}{12} \), we understand how these comparable triangle dimensions determine the parameter \(d\), which signifies the optimal camera placement.

Image Formation

Image formation in a pinhole camera is a fascinating process governed by the laws of physics. Unlike traditional cameras that may employ a collection of lenses to focus light, the pinhole relies purely on the size of its opening and precise alignment. In this setup, the imagery occurs because only narrow beams of light pass through the tiny aperture, producing a coherent, albeit inverted, image on the opposite side – the film plane.

Each point of light from the painting transports through the pinhole to corresponding points on the film. This direct transfer creates a scaled representation of the scene outside. The key to capturing a complete image in the given problem is managing the distance from the painting so that its entire dimensions match constraints of the film. Achieving this requires solving for either the limiting size of the image – height or width, as shown in the similarity equations used.

Physics Education

Physics education offers rich insights into understanding principles like those demonstrated in a pinhole camera. Simple yet powerful, these devices show how foundational physics concepts can be used to create practical tools. The study of pinhole cameras covers an engaging intersection of geometry, optics, and photography. For students, it illustrates an example of physics' application in daily life and historical contexts.

By exploring this exercise, learners can contextualize abstract ideas through hands-on learning or virtual simulations. This enhances comprehension and nurtures curiosity about how devices such as cameras, although we might now see them as complex, originate from simple scientific concepts. In turn, continued study into such topics equips students with analytical skills vital for advanced physics explorations.