A lens has a focal length of +30 cm and a magnification of 6. How far apart are the object and the image?
1/s +1/s' = 1/f =1/30
m = s'/s = 6
so
1 / s + 1 / 6s = 1/30
7 / 6s = 1/30
6 s = 210
s = 35 cm source to lens
s' = 6 s = 210 cm image to lens
so 35 +210
To find the distance between the object and the image formed by the lens, we can use the lens formula:
1/f = 1/do + 1/di
Where:
f = focal length of the lens
do = distance of the object from the lens
di = distance of the image from the lens
In this case, the focal length (f) is given as +30 cm, which tells us that it is a convex lens, since the focal length of a convex lens is positive.
The magnification (m) is given as 6, which is the ratio of the height of the image to the height of the object:
m = hi/ho
Where:
hi = height of the image
ho = height of the object
Since the magnification is positive, it implies that the image formed by the lens is upright.
We can rearrange the magnification equation to solve for hi:
hi = m * ho
Let's assume that the distance of the object from the lens (do) is x cm. Therefore, the distance of the image from the lens (di) can be expressed as (di = -6x).
Now, substituting the above values into the lens formula:
1/f = 1/do + 1/di
1/30 = 1/x + 1/(-6x)
Now, we can solve this equation to find the value of x.
Multiply both sides by 30x(-6x) to eliminate the denominators:
(-6x) + 30 = 30(-6x)
Simplify the equation:
-6x + 30 = -180x
Combine like terms:
174x = 30
Divide both sides by 174:
x = 30/174
Simplify the fraction:
x ≈ 0.1724 cm
Therefore, the distance of the object from the lens (do) is approximately 0.1724 cm.
Now, we can calculate the distance of the image (di) using the equation (di = -6x):
di = -6 * x
di ≈ -6 * 0.1724
di ≈ -1.0344 cm
So, the object and the image are approximately 0.1724 cm apart from each other.