A small object is 25.0 cm from a diverging lens . A converging lens with a focal length of 11.6 cm is 30.0 cm to the right of the diverging lens. The two-lens system forms a real inverted image 17.0 cm to th right of the converging lens.
The Correct Answer and Explanation is :
To analyze the two-lens system described, we will break it down step by step, considering the diverging and converging lenses individually.
Step 1: Diverging Lens
For the diverging lens, we are given the object distance (denoted (d_o)) as 25.0 cm and the focal length (denoted (f_1)) as negative because it’s a diverging lens.
- The formula to find the image distance ((d_i)) for a lens is:
[
\frac{1}{f} = \frac{1}{d_o} + \frac{1}{d_i}
]
Using this formula for the diverging lens:
[
\frac{1}{f_1} = \frac{1}{d_o} + \frac{1}{d_i}
]
[
\frac{1}{-f_1} = \frac{1}{d_o} + \frac{1}{d_i}
]
Given (d_o = 25.0) cm and (f_1 = -11.6) cm (since it’s a diverging lens), we can solve for (d_i). Substituting the known values:
[
\frac{1}{-11.6} = \frac{1}{25} + \frac{1}{d_i}
]
Solving for (d_i), we get:
[
\frac{1}{d_i} = \frac{1}{-11.6} – \frac{1}{25}
]
[
d_i = -19.0 \, \text{cm}
]
This means the image formed by the diverging lens is virtual and located 19.0 cm to the left of the diverging lens.
Step 2: Converging Lens
The image formed by the diverging lens becomes the object for the converging lens. The object distance for the converging lens is the distance between the image of the diverging lens and the converging lens, which is:
[
d_o’ = 30.0 \, \text{cm} – 19.0 \, \text{cm} = 11.0 \, \text{cm}
]
We are also given that the focal length of the converging lens is (f_2 = 11.6 \, \text{cm}), and the image formed by the converging lens is real and located 17.0 cm to the right of the converging lens.
Now, applying the lens formula for the converging lens:
[
\frac{1}{f_2} = \frac{1}{d_o’} + \frac{1}{d_i’}
]
[
\frac{1}{11.6} = \frac{1}{11.0} + \frac{1}{d_i’}
]
Solving for (d_i’):
[
\frac{1}{d_i’} = \frac{1}{11.6} – \frac{1}{11.0}
]
[
d_i’ = 17.0 \, \text{cm}
]
Conclusion:
The image formed by the two-lens system is 17.0 cm to the right of the converging lens. The diverging lens creates a virtual image, which serves as the object for the converging lens. The converging lens then forms a real inverted image 17.0 cm to the right of its own focal point. This demonstrates how the combination of a diverging and a converging lens can create a real, inverted image with specific characteristics.