/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 20 A lightbulb is \(56 \mathrm{cm}\... [FREE SOLUTION] | 91影视

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Chapter 31: Problem 20

A lightbulb is \(56 \mathrm{cm}\) from a convex lens. Its image appears on a screen \(31 \mathrm{cm}\) from the lens, on the other side. Find (a) the lens's focal length and (b) how much the image is enlarged or reduced.

Short Answer

Expert verified
The focal length of the lens is 31 cm, and the image is reduced by a factor of 1.806 compared to the actual object.

Step by step solution

01

Apply Lens Formula

The lens formula is \(1/f = 1/v - 1/u\), where 'f' represents the lens's focal length, 'v' is the image distance, and 'u' is the object distance. Here, the object distance (the lightbulb) is \(u = -56 \, \mathrm{cm}\) (negative because the light is on the left side of the lens), and the image distance (the screen) is \(v = 31 \, \mathrm{cm}\) (positive because the image is formed on the right side of the lens). Substituting these values yields, \(1/f = 1/31 - 1/-56\).
02

Solve for Focal Length

Now, calculate the right-hand side of the formula to get the reciprocal of the lens's focal length. The result is \(1/f = 0.03226\). Therefore, the focal length of the lens 'f' can be found by taking the reciprocal of this value, which will yield \(1/0.03226 = 31.0 \, \mathrm{cm}\). So, the focal length of the lens is \(31 \, \mathrm{cm}\).
03

Apply Magnification Formula

The magnification formula is \( m = -v/u \), where 'm' represents the magnification. Substituting the values of image distance 'v' and object distance 'u' and solving the equation gives \(m = -31/-56 = 0.55357\). The magnification here is less than 1, which represents a reduction, and the absolute value reports how much the image is reduced.
04

Interpret Magnification Result

The magnification result shows an amount less than 1, meaning the image is reduced. To find out by how much, take the reciprocal of the magnification: 1/0.55357 = 1.806, which means that the image is just over 1.8 times smaller than the actual object. When magnification is less than 1, the image is smaller than the actual object, and when it is greater than 1, the image is larger than the object. Also, if the image is upright, the magnification is taken as positive, and if it is inverted, it is taken as negative.

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