A lens with a focal length of 25 cm is placed 45 cm in front of a lens with a focal length of 5.0 cm.
a) How far from the second lens is the final image of an object infinitely far from the first lens?
b) Is the image in part a in front of or behind the second lens?
To answer these questions, we can use the lens formula:
1/f = 1/di - 1/do
where:
- f is the focal length of the lens
- di is the image distance (distance between the lens and the image)
- do is the object distance (distance between the lens and the object)
Let's solve each part of the question step by step:
a) To find the final image distance, we need to calculate the image distance formed by the first lens (di1) and then use it as the object distance for the second lens (do2) to calculate the final image distance (di2).
Step 1: Find di1 using the lens formula of the first lens.
For the first lens: f1 = 25 cm, do1 = ∞ (object infinitely far)
1/25 = 1/di1 - 0
di1 = 25 cm
Step 2: Now, use di1 as the object distance for the second lens (do2) and find the final image distance (di2) using the lens formula of the second lens.
For the second lens: f2 = 5.0 cm, do2 = di1 = 25 cm
1/5.0 = 1/di2 - 1/25
Simplifying the equation:
1/5.0 = (25 - di2) / (25 * di2)
Multiplying both sides by 25 * di2:
(25 * di2) / 5.0 = 25 - di2
Simplifying further:
5 * di2 = 125 - 5 * di2
Rearranging the equation:
10 * di2 = 125
di2 = 12.5 cm
Therefore, the final image is located 12.5 cm in front of the second lens.
b) To determine if the final image is in front of or behind the lens, we need to check the sign of the image distance (di2).
In this case, di2 = 12.5 cm, which is a positive value. A positive image distance represents a real image formed on the opposite side of the lens from the object, which means the image is behind the lens.
So, the final image in part a) is behind the second lens.
To solve this problem, we can use the lens formula and apply the lensmaker's formula to find the distances and positions of the image formed by the system of lenses.
a) Let's start by finding the image formed by the first lens (Lens 1). We can use the lens formula:
1/f = 1/v - 1/u
Given:
Focal length of Lens 1 (f1) = 25 cm
Object distance from the first lens (u1) = infinity (as object is infinitely far)
Plugging in these values, we get:
1/25 = 1/v1 - 1/infinity
1/25 = 1/v1
Simplifying further, we find that v1 = 25 cm. This means that Lens 1 forms a real image 25 cm away from it.
Now, let's find the distance and position of the image formed by the second lens (Lens 2). The image formed by Lens 1 acts as an object for Lens 2. We can use the lensmaker's formula:
1/f = (n - 1)(1/R1 - 1/R2)
Given:
Focal length of Lens 2 (f2) = 5.0 cm
Object distance from Lens 2 (u2) = 45 cm
Plugging in these values, we get:
1/5 = (n - 1)(1/R1 - 1/R2)
1/5 = (1 - 1)(1/R1 - 1/R2)
1/5 = 0(1/R1 - 1/R2)
1/5 = 0
Simplifying further, we find that R1 and R2 do not impact the solution and are not needed.
Therefore, the image formed by Lens 2 will be at a distance of 5 cm from it. This is also the final image formed by the system of lenses.
b) The image formed by Lens 2 is 5 cm in front of the second lens.