Question

An instructor wants to use a lens to project a real image of a light bulb onto a screen \(1.71 \mathrm{~m}\) from the bulb. In order to get the image to be twice as large as the bulb, what focal length lens will be needed?

Short answer

Answer: The focal length of the lens needed is 0.57 meters.

Step-by-step solution

Define the given variables and formula

First, let's define the given variables: - Object distance (from the lens), \(d_o\) - Image distance (from the lens), \(d_i\) - Focal length of the lens, \(f\) - Object size, \(h_o\) - Image size, \(h_i\) The problem states that the image size is twice the object size (\(h_i = 2h_o\)), and the screen (where the image is projected) is \(1.71 \mathrm{~m}\) from the bulb. To find the focal length of the lens, we will use the lens formula: $$\frac{1}{f} = \frac{1}{d_o}+\frac{1}{d_i}$$

Calculate the magnification

We know the magnification (\(M\)) is the ratio of the image size to the object size: $$M = \frac{h_i}{h_o} = 2$$ This implies that the ratio of the image distance to the object distance is also 2: $$\frac{d_i}{d_o} = M = 2$$

Solve for the object and image distances

Let the object distance be \(x\), then the image distance will be \(2x\). The total distance between the bulb and the screen is \(1.71 \mathrm{~m}\), so: $$x + 2x = 1.71$$ $$3x = 1.71$$ $$x = \frac{1.71}{3} = 0.57 \mathrm{~m}$$ Therefore, the object distance \(d_o = 0.57 \mathrm{~m}\) and the image distance \(d_i = 2(0.57) = 1.14 \mathrm{~m}\).

Use the lens formula to find the focal length

Now, we can apply the lens formula to find the focal length: $$\frac{1}{f} = \frac{1}{d_o}+\frac{1}{d_i} = \frac{1}{0.57}+\frac{1}{1.14}$$ $$\frac{1}{f} = \frac{2}{1.14}$$ $$f = \frac{1.14}{2} = 0.57\mathrm{~m}$$ Thus, the focal length of the lens needed to project the real image twice the size of the bulb is \(0.57 \mathrm{~m}\).