Two converging lenses are separated by 23.40 cm. The focal length of each lens is 12.20 cm. An object is placed 35.00 cm to the left of the lens that is on the left. Determine the final image distance relative to the lens on the right.
The image formed by the first lens acts as the object for the second lens. For the location of the first lens image,
1/di = 1/f - 1/do = 1/12.2 - 1/35 = 0.0534
do = 18.73 cm. That places it 4.67 cm in front of the second lens.
For the second lens,
1/4.67 + 1/di = 1/12.2
di = -7.57 cm
The final image is 7.57 cm to the left of the second lens, in the region between the lenses.
Oh, the wonderful world of lenses and distances! So, we have two converging lenses and an object placed to the left of the left lens. Let's dive in and find the final image distance relative to the lens on the right!
To solve this problem, we can use the lens formula: 1/f = 1/v - 1/u, where f is the focal length, v is the image distance, and u is the object distance.
For the lens on the left, we have:
f₁ = 12.20 cm
u₁ = -35.00 cm (negative because the object is to the left of the lens)
Let's plug these values into the lens formula and solve for v₁:
1/12.20 = 1/v₁ - 1/(-35.00)
After some calculations, we find v₁ to be approximately -10.99 cm. (Negative sign indicates a virtual image.)
Now, for the lens on the right:
f₂ = 12.20 cm
u₂ = v₁ - 23.40 cm (Since the lenses are separated by 23.40 cm, the object distance for the right lens is the image distance from the left lens minus the separation distance.)
Let's use the lens formula again and solve for v₂:
1/12.20 = 1/v₂ - 1/(-10.99 - 23.40)
After some more calculations, we find v₂ to be approximately -41.10 cm. (Again, negative sign represents a virtual image.)
So, the final image distance relative to the lens on the right is approximately -41.10 cm.
Don't worry, though! Even though it's negative, you can still laugh at how these lenses "negatively" juggle the light!
To determine the final image distance relative to the lens on the right, we can use the lens formula and apply it separately for each lens.
The lens formula is as follows:
1/f = 1/v - 1/u,
where:
- f is the focal length of the lens,
- v is the image distance (positive for images on the opposite side of the lens),
- u is the object distance (positive when objects are on the opposite side of the lens).
Let's calculate the image distance for the lens on the left:
Given:
- Focal length of the left lens, f1 = 12.20 cm,
- Object distance for the left lens, u1 = -35.00 cm (negative because the object is on the left side of the lens).
Applying the lens formula for the left lens, we have:
1/f1 = 1/v1 - 1/u1.
Substituting the given values, we get:
1/12.20 = 1/v1 - 1/(-35.00).
Simplifying:
1/12.20 = 1/v1 + 1/35.00.
Combining the two fractions on the right side:
1/12.20 = (1*35.00 + 1*v1) / (35.00 * v1).
As a next step, let's calculate the image distance for the right lens:
Given:
- Focal length of the right lens, f2 = 12.20 cm,
- Object distance for the right lens, u2 = final image distance relative to the lens on the right (unknown).
Applying the lens formula for the right lens, we have:
1/f2 = 1/v2 - 1/u2.
Substituting the given values, we get:
1/12.20 = 1/v2 - 1/u2.
Since the two lenses are separated by 23.40 cm, the image produced by the left lens at v1 becomes the object for the right lens. Therefore, we can write the equation as follows:
u2 = -23.40 - v1.
Now, we can substitute the value of u2 into the lens formula for the right lens to solve for v2:
1/12.20 = 1/v2 - 1/(-23.40 - v1).
Combining the two fractions on the right side:
1/12.20 = (1*(23.40 + v1) - 1*v2) / (v2 * (23.40 + v1)).
Now we have two equations: one for v1 and another for v2. We can solve them simultaneously to find the final image distance relative to the lens on the right.
To determine the final image distance relative to the lens on the right, we can use the lensmaker's formula and combine the formulas for two lenses in contact.
1. Begin by finding the object distance, u1, for the lens on the left.
- Given: Distance of the object from the left lens, D = -35.00 cm
- Since the object is to the left of the lens, the distance is negative.
- Therefore, u1 = -35.00 cm
2. Next, find the image distance, v1, using the lensmaker's formula for the left lens.
- Given: Focal length of each lens, f = 12.20 cm
- Given: Object distance for the left lens, u1 = -35.00 cm
- Using the lensmaker's formula: 1/v1 - 1/u1 = 1/f
- Substituting the values: 1/v1 - 1/(-35.00 cm) = 1/12.20 cm
- Simplifying the equation will give you the value of v1.
3. Once you have v1, which is the image distance for the left lens, you can consider it as the object distance, u2, for the right lens.
- u2 = v1 = "value of v1"
4. Now, we can find the image distance, v2, for the right lens using the lensmaker's formula.
- Given: Focal length of each lens, f = 12.20 cm
- Given: Object distance for the right lens, u2 = "value of v1"
- Using the lensmaker's formula: 1/v2 - 1/u2 = 1/f
- Substituting the values: 1/v2 - 1/"value of v1" = 1/12.20 cm
- Simplifying the equation will give you the value of v2.
The final image distance relative to the lens on the right is the value of v2.