A 1.0-cm-tall object is 10 cm in front of a converging lens that has a 30 cm focal length. Calculate the image position and height. Compare with your ray-tracing answers in part a.

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Start by identifying the given values: the object height \( h_o = 1.0 \, \text{cm} \), the object distance \( d_o = 10 \; \text{cm} \), and the focal length of the lens \( f = 30 \; \text{cm} \). The goal is to find the image distance \( d_i \) and the image height \( h_i \).

Use the lens equation to find the image distance \( d_i \): \( \frac{1}{f} = \frac{1}{d_o} + \frac{1}{d_i} \). Rearrange this equation to solve for \( \frac{1}{d_i} \): \( \frac{1}{d_i} = \frac{1}{f} - \frac{1}{d_o} \). Substitute the known values of \( f \) and \( d_o \) into the equation.

After calculating \( \frac{1}{d_i} \), take the reciprocal to find \( d_i \), which represents the image distance. Note whether the value of \( d_i \) is positive or negative, as this will indicate whether the image is real or virtual.

Next, calculate the magnification \( M \) using the formula \( M = -\frac{d_i}{d_o} \). Substitute the values of \( d_i \) and \( d_o \) to determine the magnification.

Finally, find the image height \( h_i \) using the relationship \( h_i = M \cdot h_o \). Multiply the magnification \( M \) by the object height \( h_o \) to determine the image height. Compare the results with your ray-tracing answers to verify consistency.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Lens Formula

The lens formula relates the object distance (u), image distance (v), and focal length (f) of a lens. It is given by the equation 1/f = 1/v + 1/u. This formula is essential for determining the position of the image formed by a lens, allowing us to calculate how far the image is from the lens based on the object's distance and the lens's focal length.

Magnification

Magnification is the ratio of the height of the image (h') to the height of the object (h), and it can also be expressed as the negative ratio of the image distance (v) to the object distance (u). The formula is given by magnification (M) = h'/h = -v/u. Understanding magnification is crucial for determining the size of the image relative to the object, which is necessary for solving the problem.

Ray Tracing

Ray tracing is a graphical method used to determine the position and characteristics of an image formed by a lens. By drawing rays from the object through the lens, one can visualize how the rays converge to form an image. This technique helps in understanding the behavior of light through lenses and provides a practical way to verify calculations made using the lens formula.

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