An image of the moon is focused onto a screen using a converging lens of focal length f= 35.3 cm. The diameter of the moon is 3.48×106 m, and its mean distance from the earth is 3.85×108 m. What is the diameter of the moon's image?
To find the diameter of the moon's image, we can use the thin lens formula which relates the object distance (distance of the moon from the lens), the image distance (distance of the image from the lens), and the focal length of the lens.
The thin lens formula is given by:
1/f = 1/di + 1/do
Where:
f = focal length of the lens
di = image distance (distance of the image from the lens)
do = object distance (distance of the moon from the lens)
First, let's convert the given values into meters:
f = 35.3 cm = 0.353 m
do = 3.85 x 10^8 m
Now, we can rearrange the thin lens formula to solve for the image distance:
1/di = 1/f - 1/do
Substituting the values, we have:
1/di = 1/0.353 - 1/(3.85 x 10^8)
Next, we can solve for di by taking the reciprocal of both sides:
di = 1 / (1/0.353 - 1/(3.85 x 10^8))
Using a calculator, we find that di is approximately equal to 0.00248 m.
Finally, we can calculate the diameter of the moon's image using the magnification formula:
Magnification (M) = -di/do
Substituting the values, we get:
M = -0.00248 m / (3.85 x 10^8 m)
The negative sign indicates that the image is inverted.
Using a calculator, we find that the magnification (M) is approximately -6.45 x 10^(-9).
To find the diameter of the image, multiply the magnification by the diameter of the moon (3.48 x 10^6 m):
Diameter of the image = |M| * Diameter of the moon
Diameter of the image = 6.45 x 10^(-9) * 3.48 x 10^6
Using a calculator, we find that the diameter of the moon's image is approximately 0.0224 meters or 22.4 millimeters.