(II) In a film projector, the film acts as the object whose image is projected on a screen (Fig. 33–46). If a 105-mm-focal-length lens is to project an image on a screen 22.5 m away, how far from the lens should the film be? If the film is 24 mm wide, how wide will the picture be on the screen?

Verified step by step guidance

1. Step 1: Identify the lens equation to relate the focal length (f), object distance (d_o), and image distance (d_i). The lens equation is given as: $\frac{1}{f}=\frac{1}{d}$_o+1d_i. Here, f = 105 mm and d_i = 22.5 m (convert to mm: 22,500 mm).
2. Step 2: Rearrange the lens equation to solve for the object distance (d_o): $d$_o=11f-1d_i. Substitute the values of f and d_i into the equation.
3. Step 3: Calculate the magnification (M) of the lens, which is the ratio of the image height to the object height. The formula for magnification is: $M=\frac{d}{}$_id_o. Use the values of d_i and d_o obtained from the previous steps.
4. Step 4: Determine the width of the picture on the screen. The image width is given by: $\mathrm{Image}\mathrm{Width}=M\times \mathrm{Object}\mathrm{Width}$. Here, the object width is 24 mm, and M is the magnification calculated in Step 3.
5. Step 5: Verify the units and ensure all values are consistent (e.g., mm for distances and widths). Interpret the results to confirm the object distance and the image width on the screen.

Key Concepts

Lens Formula

The lens formula relates the object distance (u), image distance (v), and focal length (f) of a lens through the equation 1/f = 1/v + 1/u. This formula is essential for determining the position of the film relative to the lens in a projector setup, allowing us to calculate how far the film should be placed to achieve a clear image on the screen.

Magnification

Magnification in optics refers to the ratio of the height of the image to the height of the object, and it can also be expressed as the ratio of the image distance to the object distance (M = v/u). Understanding magnification is crucial for determining how the size of the projected image on the screen relates to the size of the film, which is necessary for calculating the width of the picture.

Image Formation

Image formation by a lens involves the process of light rays converging to create a visible image. In the context of a projector, the lens focuses light from the film onto the screen, and the characteristics of the image, such as size and clarity, depend on the distances involved and the properties of the lens. This concept is fundamental for understanding how the film's dimensions affect the projected image.