A photographer uses a camera lens with focal length of 50.0 mm to photograph a tree that is 4.60 m tall. She wishes to fill her 24.0-mm slide with the tree.
a) how far from the lens should the film be?
b) what is the magnification of the image?
a) To determine the distance from the lens to the film, we can use the thin lens formula:
1/f = 1/o + 1/i
Where:
f = focal length of the lens (50.0 mm = 0.050 m)
o = object distance (height of tree = 4.60 m)
i = image distance (distance from the lens to the film)
Since the slide is to be filled with the tree, the image distance should be the same as the height of the slide. Let's calculate it:
1/0.050 = 1/4.60 + 1/i
Simplifying the equation:
20 = 1/4.60 + 1/i
Multiply both sides by (4.60 * i):
20 * (4.60 * i) = (4.60 * i) / 4.60 + (4.60 * i) / i
92 * i = 1 + 4.60
92 * i = 5.60
i = 5.60 / 92
i ≈ 0.061 m
Therefore, the film should be located approximately 0.061 meters from the lens.
b) The magnification of the image can be calculated using the formula:
magnification = -i/o
Where:
magnification = magnification of the image
i = image distance (0.061 m)
o = object distance (4.60 m)
Substituting the values:
magnification = -0.061 / 4.60
magnification ≈ -0.013
The magnification of the image is approximately -0.013.
a) To determine the distance from the lens to the film, we can use the thin lens formula:
1/f = 1/o + 1/i
Where:
- f is the focal length of the lens,
- o is the object distance (distance from the lens to the tree),
- i is the image distance (distance from the lens to the film).
Given:
- f = 50.0 mm (or 0.050 m),
- o = 4.60 m.
Let's solve for i:
1/0.050 = 1/4.60 + 1/i
Simplifying, we get:
20 = (4.60 + 1/i)
Rearranging the equation:
1/i = 20 - 4.60
1/i = 15.40
Now, we can find the distance from the lens to the film (i):
i = 1 / 15.40
i ≈ 0.065 m (or 65 mm)
Therefore, the film should be approximately 0.065 meters (or 65 mm) away from the lens.
b) Now, let's calculate the magnification of the image. The magnification (M) can be determined using the equation:
M = -i / o
Given:
- o = 4.60 m,
- i = 0.065 m.
Plugging in the values:
M = -0.065 / 4.60
M ≈ -0.014
The magnification is approximately -0.014. Note that this negative sign indicates that the image is inverted.