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Chapter 24: Problem 14

Esperanza uses a 35 -mm camera with a standard lens of focal length $50.0 \mathrm{mm}\( to take a photo of her son Carlos, who is \)1.2 \mathrm{m}$ tall and standing \(3.0 \mathrm{m}\) away. (a) What must be the distance between the lens and the film to get a properly focused picture? (b) What is the magnification of the image? (c) What is the height of the image of Carlos on the film?

Short Answer

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Question: A 35-mm camera with a 50.0 mm focal length lens is used to take a picture of Carlos, who is 1.2 m tall and standing 3.0 m from the camera. Calculate (a) the distance between the lens and the film to get a properly focused picture, (b) the magnification of the image, and (c) the height of the image of Carlos on the film. Answer: (a) The distance between the lens and the film is approximately 52.6 mm, (b) the magnification of the image is 0.0175, and (c) the height of the image of Carlos on the film is 21 mm.

Step by step solution

01

Write down the lens formula

The lens formula is given by $$\frac{1}{f} = \frac{1}{d_o} + \frac{1}{d_i}$$ where \(f\) is the focal length, \(d_o\) is the object distance (distance between the object and lens) and \(d_i\) is the image distance (distance between the image produced and the lens).
02

Use given data and lens formula to calculate image distance

We are given the focal length \(f=50.0 \mathrm{mm}\), and the object distance \(d_o = 3.0 \mathrm{m}\). First, we need to convert object distance to mm. $$d_o = 3.0 \mathrm{m} \times 1000 \frac{\mathrm{mm}}{\mathrm{m}}= 3000 \mathrm{mm}$$ Now plug the given values into the lens formula: $$\frac{1}{50.0} = \frac{1}{3000} + \frac{1}{d_i}$$ Now solve for the image distance, \(d_i\).
03

Calculate image distance

To solve for \(d_i\), first subtract both sides using \(\frac{1}{3000}\): $$\frac{1}{50.0} - \frac{1}{3000} = \frac{1}{d_i}$$ Now, invert both sides to get \(d_i\): $$d_i = \frac{1}{\left(\frac{1}{50.0} - \frac{1}{3000}\right)} \approx 52.6 \: \mathrm{mm}$$ This is the distance between the lens and the film. So, the answer to (a) is 52.6 mm.
04

Write down formula for magnification

Magnification is given by $$M = \frac{h_i}{h_o} = \frac{d_i}{d_o}$$ where \(h_i\) is the height of the image, \(h_o\) is the height of the object, \(d_i\) is the image distance, and \(d_o\) is the object distance.
05

Calculate magnification

Using the given object distance and calculated image distance, compute the magnification: $$M = \frac{52.6}{3000} \approx 0.0175$$ The magnification of the image is 0.0175. So, the answer to (b) is 0.0175.
06

Calculate height of the image

Now, we can calculate the height of the image of Carlos on the film by using height of the object and magnification. $$h_i = M \times h_o$$ Carlos's height is given as \(1.2 \mathrm{m}\). Convert it to mm. $$h_o = 1.2 \mathrm{m} \times 1000 \frac{\mathrm{mm}}{\mathrm{m}}= 1200 \mathrm{mm}$$ Now, multiply the magnification we found in step 5 with the height of the object: $$h_i = 0.0175 \times 1200 \approx 21 \: \mathrm{mm}$$ The height of the image of Carlos on the film is 21 mm. So, the answer to (c) is 21 mm.

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