A 1.00-cm-high object is placed 3.10 cm to the left of a converging lens of focal length 8.30 cm. A diverging lens of focal length −16.00 cm is 6.00 cm to the right of the converging lens. Find the position and height of the final image.
To find the position and height of the final image, we can use the lens formula and the lens-maker's formula. The lens formula relates the object distance (u), the image distance (v), and the focal length (f) of a lens:
1/f = 1/v - 1/u
Where u is positive for objects on the incident side of the lens, and v is positive for images formed on the opposite side of the lens.
The lens-maker's formula relates the focal length (f), the refractive indices of the lens medium (n), and the radii of curvature of the lens surfaces (R1 and R2):
1/f = (n - 1) * (1/R1 - 1/R2)
Using these formulas, we can go through the following steps to find the position and height of the final image:
Step 1: Calculate the image position formed by the converging lens.
- Given: Object distance (u) = -3.10 cm, Focal length (f) = 8.30 cm
- Apply the lens formula: 1/f = 1/v - 1/u
- Rearrange the formula to solve for v: 1/v = 1/f + 1/u
- Substitute the known values: 1/v = 1/8.30 + 1/(-3.10)
- Calculate 1/v: 1/v = 0.120 + (-0.323)
- Calculate v: v = 1 / (0.120 + (-0.323))
Step 2: Calculate the image position formed by the diverging lens.
- Given: Object distance (u) = 6.00 cm, Focal length (f) = -16.00 cm
- Apply the lens formula: 1/f = 1/v - 1/u
- Rearrange the formula to solve for v: 1/v = 1/f + 1/u
- Substitute the known values: 1/v = 1/-16.00 + 1/6.00
- Calculate 1/v: 1/v = (-0.063) + (0.167)
- Calculate v: v = 1 / (-0.063 + 0.167)
Step 3: Calculate the final image position.
- The final position of the image is the distance from the diverging lens.
- Subtract the image position formed by the diverging lens from the distance of the diverging lens to the converging lens.
Step 4: Calculate the final image height.
- Given: Object height = 1.00 cm
- Since the object height is the same as the image height formed by the converging lens, the final image height remains the same.
By following these steps and performing the necessary calculations, you will obtain the position and height of the final image.