A camera with a telephoto lens of focal length 125mm takes a picture of a 1.8m tall woman standing 5.0m away.
(1). What must be the distance between the film and the lens to get a properly focused picture?
(2). What is the magnification of the image
(3). What is the size of the woman's image on the film?
To answer your questions step-by-step:
(1) To determine the distance between the film and the lens, we can use the lens formula:
1/f = 1/v - 1/u
Where:
f = focal length of the lens (125mm = 0.125m)
v = image distance (unknown)
u = object distance (5.0m)
Since we want a properly focused picture, the image should be formed at the focal length (v = f). Plugging in the values, we can solve for u:
1/0.125 = 1/f - 1/5.0
Simplifying the equation:
8 = 1/f - 1/5.0
1/f = 8 + 1/5.0
1/f = 8 + 0.2
1/f = 8.2
Now, we can calculate the distance between the film and the lens (v):
v = 1/(1/f)
v = f/1
v = 0.125m
Therefore, the distance between the film and the lens to get a properly focused picture is 0.125m.
(2) The magnification of the image can be calculated using the magnification formula:
m = -v/u
Where:
m = magnification (unknown)
v = image distance (0.125m)
u = object distance (5.0m)
Plugging in the values:
m = -0.125/5.0
m = -0.025
Therefore, the magnification of the image is -0.025.
(3) To determine the size of the woman's image on the film, we can use the magnification formula again:
m = size of the image/size of the object
Rearranging the formula:
size of the image = m * size of the object
Where:
m = magnification (-0.025)
size of the object = 1.8m (height of the woman)
Plugging in the values:
size of the image = -0.025 * 1.8
size of the image = -0.045m
Therefore, the size of the woman's image on the film is approximately -0.045m (negative because the image is inverted).
To answer these questions, we can use the thin lens equation and the magnification formula. Here's how:
(1) To determine the distance between the film and the lens for a properly focused picture, we need to use the thin lens equation:
1/f = 1/v - 1/u
Where:
- f is the focal length of the lens,
- v is the distance between the lens and the focused image (film), and
- u is the object distance (distance between the lens and the woman).
Rearranging the equation, we get:
1/v = 1/f + 1/u
Given values:
- f = 125 mm (or 0.125 m),
- u = 5.0 m,
- v is what we need to find.
Substituting the values into the equation, we have:
1/v = 1/0.125 + 1/5.0
Simplifying:
1/v = 8 + 0.2
1/v = 8.2
Taking the reciprocal of both sides:
v = 1/8.2
Therefore, the distance between the film and the lens for a properly focused picture is approximately 0.122 m (or 12.2 cm).
(2) To determine the magnification of the image, we can use the magnification formula:
magnification (m) = -v/u
Given values:
- u = 5.0 m,
- v = 0.122 m (from the previous calculation).
Substituting the values into the formula:
m = -0.122/5.0
Simplifying:
m ≈ -0.0244
Therefore, the magnification of the image is approximately -0.0244. The negative sign indicates that the image is formed inverted.
(3) To calculate the size of the woman's image on the film, we can use the magnification formula:
image size = magnification * object size
Given values:
- magnification (m) = -0.0244 (from the previous calculation),
- object size (height of the woman) = 1.8 m.
Substituting the values into the formula:
image size = -0.0244 * 1.8
Simplifying:
image size ≈ -0.04392 m
Therefore, the size of the woman's image on the film is approximately -0.04392 m, meaning the image is smaller than the actual size of the woman and is also inverted.