An object is 14 cm in front of a diverging lens that has a focal length of -7.9 cm. How far in front of the lens should the object be placed so that the size of its image is reduced by a factor of 1.9?
To solve this problem, we first need to understand the relationship between the object distance (o), image distance (i), and focal length (f) of a diverging lens.
The lens formula for a diverging lens is:
1/f = 1/o - 1/i
where:
- f is the focal length of the lens,
- o is the object distance (distance of the object from the lens), and
- i is the image distance (distance of the image from the lens).
In this particular problem, we already know the object distance (o) and focal length (f). The image distance (i) is the unknown we need to calculate.
The problem states that the size of the object's image should be reduced by a factor of 1.9. In simple terms, this means that the size of the image should be 1.9 times smaller than the size of the object.
Now we can set up the problem and solve for the image distance (i).
Given:
- Object distance (o) = 14 cm
- Focal length (f) = -7.9 cm
- Size reduction factor = 1.9
Using the lens formula, we have:
1/f = 1/o - 1/i
Plugging in the values we know:
1/-7.9 = 1/14 - 1/i
Now, we can rearrange the equation to solve for i:
1/i = 1/14 + 1/7.9
To simplify the calculation, we can find a common denominator:
1/i = (7.9 + 14) / (14 * 7.9)
1/i = 21.9 / 110.6
Now, we can take the reciprocal to find i:
i = 110.6 / 21.9
i ≈ 5.056 cm
Therefore, the image distance (i) should be approximately 5.056 cm.
To find the object distance (o') that would result in a 1.9 times smaller image, we can use the magnification formula:
magnification (m) = -i / o'
In this case, we want the image to be 1.9 times smaller, so the magnification is 1/1.9, which is approximately 0.526 (rounded to three decimal places).
Using the magnification formula:
0.526 = -5.056 / o'
Now, we can solve for o':
o' = -5.056 / 0.526
o' ≈ -9.607 cm
Therefore, the object should be placed approximately 9.607 cm in front of the lens to achieve a 1.9 times smaller image.