How far apart are an object and an image formed by a 75.0cm focal length converging lens if the image is 2.5x larger than the object and is real.
To find the distance between the object and the image formed by a converging lens, we can use the lens equation:
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
Given data:
f = 75.0 cm (converging lens focal length)
m = 2.5 (magnification)
di is real (which means it is positive)
We can start by finding the value of di. We know that the magnification (m) is given by the formula:
m = -di / do
Rearranging the formula, we get:
di = -m * do
Substituting the given values, we have:
di = -2.5 * do
Now, we can substitute this value of di into the lens equation:
1/f = 1/do + 1/(-2.5 * do)
Simplifying further:
1/f = (1 - 2.5)/(-2.5 * do)
Now, rearrange the equation to solve for do:
1/do = (1/f) + (1/(-2.5 * do))
Combine the terms on the right-hand side:
1/do = (1 - 0.4)/(-2.5 * do)
Simplify further:
1/do = (0.6)/(-2.5 * do)
Multiply both sides by (-2.5 * do):
-2.5 * do = 0.6
Divide both sides by -2.5:
do = 0.6 / -2.5
Solving this, we find:
do ≈ -0.24 m
Since the distance of the object cannot be negative, this negative value indicates that the object is located on the same side as the image (in front of the lens). Therefore, the absolute value of the object distance is:
|do| = 0.24 m
Now, we can find the distance of the image using the magnification formula:
m = -di / do
Substituting the values:
2.5 = -di / 0.24
Rearranging the equation:
di = -2.5 * 0.24
Solving this, we get:
di ≈ -0.6 m
Again, since di is positive, the absolute value of the image distance is:
|di| = 0.6 m
So, the object and the image formed by the lens are approximately 0.24 m (24 cm) and 0.6 m (60 cm) apart, respectively.