An old-fashioned slide projector needs to create a 98-cm-high of a 2.0-cm-tall slide. The screen is 300 cm from the slide. What focal length does the lens need? Assume that it is a thin lens.

Verified step by step guidance

1. Step 1: Identify the given values. The height of the image (h') is 98 cm, the height of the object (h) is 2.0 cm, and the distance from the slide to the screen (image distance, d') is 300 cm. We need to find the focal length (f) of the lens.
2. Step 2: Use the magnification formula to relate the image height and object height: \( M = \frac{h'}{h} \). Substitute the given values: \( M = \frac{98}{2.0} \). This gives the magnification (M).
3. Step 3: Relate the magnification to the distances using the formula \( M = -\frac{d'}{d} \), where d is the object distance and d' is the image distance. Rearrange to solve for d: \( d = -\frac{d'}{M} \). Substitute the values for d' and M to find the object distance (d).
4. Step 4: Use the thin lens equation to relate the focal length, object distance, and image distance: \( \frac{1}{f} = \frac{1}{d} + \frac{1}{d'} \). Substitute the values for d and d' into this equation.
5. Step 5: Rearrange the thin lens equation to solve for the focal length (f): \( f = \frac{1}{\frac{1}{d} + \frac{1}{d'}} \). Simplify the expression to find the focal length.

Key Concepts

Magnification

Magnification is the ratio of the height of the image to the height of the object. In this case, it can be calculated using the formula M = h_image / h_object, where h_image is the height of the projected image and h_object is the height of the slide. Understanding magnification is crucial for determining how much larger the image will appear on the screen compared to the original slide.

Thin Lens Formula

The thin lens formula relates the object distance (d_o), image distance (d_i), and focal length (f) of a lens. It is expressed as 1/f = 1/d_o + 1/d_i. This formula is essential for calculating the focal length needed for the lens to project the image at the desired size and distance from the slide.

Lens Maker's Equation

The lens maker's equation provides a relationship between the focal length of a lens and its curvature and refractive index. While not directly needed for this problem, understanding how lens shape and material affect focal length can provide deeper insights into lens design and performance, especially in practical applications like projectors.