Question

A diverging lens has a focal length that has a magnitude of 37.0 cm. An object is placed 19.5 cm in front of this lens. Calculate the magnification.

Answer 1

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Answer 2

To calculate the magnification produced by a diverging lens, we can use the lens formula:

$\frac{1}{�}=\frac{1}{{�}_{�}}+\frac{1}{{�}_{�}}$f1=do1+di1

where:$�$f is the focal length of the lens,${�}_{�}$do is the object distance (distance of the object from the lens), and${�}_{�}$di is the image distance (distance of the image formed by the lens).

Since the focal length of the diverging lens is negative (for a diverging lens, the focal length is always negative), we will take it as -37.0 cm.

Given that${�}_{�}=-19.5$do=−19.5 cm (negative because the object is placed in front of the lens), we can solve for${�}_{�}$di using the lens formula:

$\frac{1}{-37}=\frac{1}{-19.5}+\frac{1}{{�}_{�}}$−371=−19.51+di1

Now, let's solve for${�}_{�}$di:

$\frac{-1}{37}=\frac{-1}{19.5}+\frac{1}{{�}_{�}}$37−1=19.5−1+di1

Multiply both sides by$-37\cdot 19.5\cdot {�}_{�}$−37⋅19.5⋅di to eliminate the fractions:

$-19.5\cdot {�}_{�}=-37\cdot {�}_{�}+37\cdot 19.5$−19.5⋅di=−37⋅di+37⋅19.5

Now, add$37\cdot {�}_{�}$37⋅di to both sides:

$17.5\cdot {�}_{�}=37\cdot 19.5$17.5⋅di=37⋅19.5

Finally, divide both sides by 17.5 to solve for${�}_{�}$di:

${�}_{�}=\frac{37\cdot 19.5}{17.5}$di=17.537⋅19.5

Now, we have the value for${�}_{�}$di, which is the image distance. To calculate the magnification (M) of the lens, we use the magnification formula:

$�=\frac{{�}_{�}}{{�}_{�}}$M=dodi

Plugging in the values:

$�=\frac{\frac{37\cdot 19.5}{17.5}}{-19.5}$M=−19.517.537⋅19.5

$�=\frac{37\cdot 19.5}{-17.5\cdot 19.5}$M=−17.5⋅19.537⋅19.5

$�=\frac{37}{-17.5}$M=−17.537

$�\approx -2.114$M≈−2.114

The magnification produced by the diverging lens is approximately -2.114. The negative sign indicates that the image is virtual and upright (opposite orientation compared to the object). The magnitude of the magnification tells us that the image is reduced in size by a factor of approximately 2.114.