Textbook Question
A lightbulb is 3.0 m from a wall. What are the focal length and the position (measured from the bulb) of a lens that will form an on the wall that is twice the size of the lightbulb?
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33. Geometric Optics
Thin Lens And Lens Maker Equations
5:51 minutesProblem 30
Textbook Question
A converging lens with a focal length of 70.0 cm forms an image of a 3.20 cm tall real object that is to the left of the lens. The image is 4.50 cm tall and inverted. Where are the object and image located in relation to the lens? Is the image real or virtual?
Verified step by step guidance1
Step 1: Understand the problem. A converging lens is being used, and we are given the focal length \( f = 70.0 \; \text{cm} \), the height of the object \( h_o = 3.20 \; \text{cm} \), and the height of the image \( h_i = -4.50 \; \text{cm} \) (negative because the image is inverted). We need to find the object distance \( d_o \), the image distance \( d_i \), and determine whether the image is real or virtual.
Step 2: Use the magnification formula to relate the object and image distances. The magnification \( M \) is given by \( M = \frac{h_i}{h_o} = -\frac{d_i}{d_o} \). Substitute the known values for \( h_i \) and \( h_o \) to calculate the ratio \( \frac{d_i}{d_o} \).
Step 3: Use the lens equation to relate the focal length, object distance, and image distance. The lens equation is \( \frac{1}{f} = \frac{1}{d_o} + \frac{1}{d_i} \). Substitute the known value of \( f \) and the relationship between \( d_i \) and \( d_o \) from Step 2 into this equation to solve for \( d_o \).
Step 4: Once \( d_o \) is determined, use the magnification formula \( M = -\frac{d_i}{d_o} \) to calculate \( d_i \).
Step 5: Determine the nature of the image. Since the image is inverted and the lens is converging, the image is real. A real image is formed when light rays actually converge at the image location.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Converging Lens
A converging lens, or convex lens, is a transparent optical device that bends light rays inward to a focal point. The focal length is the distance from the lens to this point, where parallel rays of light converge. In this case, the focal length is 70.0 cm, which is crucial for determining the image location and characteristics when an object is placed in front of the lens.
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Magnification
Magnification is the ratio of the height of the image to the height of the object, indicating how much larger or smaller the image appears compared to the object. It can be calculated using the formula: magnification (M) = height of image (h') / height of object (h). In this scenario, the image is 4.50 cm tall, and the object is 3.20 cm tall, which helps determine the nature of the image formed by the lens.
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Real vs. Virtual Images
Real images are formed when light rays converge and can be projected onto a screen, while virtual images occur when light rays diverge and cannot be projected. In the context of a converging lens, if the object is placed outside the focal length, a real and inverted image is produced. Conversely, if the object is within the focal length, a virtual and upright image is formed. Understanding this distinction is essential for analyzing the image characteristics in the given problem.
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