Question
1. Find the distance to the image in the following lens problems and state whether the image is virtual or real AND whether it is upright or inverted. An object is placed 7.0 cm away from a converging lens with a focal length of 7.0 cm.
Answers
GPT-4o mini
To solve the problem, we will use the lens formula, which is given by:
\[
\frac{1}{f} = \frac{1}{d_o} + \frac{1}{d_i}
\]
where:
- \( f \) is the focal length of the lens,
- \( d_o \) is the object distance (measured from the lens to the object, positive for real objects),
- \( d_i \) is the image distance (measured from the lens to the image, positive for real images on the opposite side from the object and negative for virtual images on the same side as the object).
In this problem:
- The focal length \( f = 7.0 \, \text{cm} \) (positive, because it’s a converging lens),
- The object distance \( d_o = 7.0 \, \text{cm} \).
Now substitute the values into the lens formula:
\[
\frac{1}{7.0} = \frac{1}{7.0} + \frac{1}{d_i}
\]
Subtract \( \frac{1}{7.0} \) from both sides:
\[
\frac{1}{d_i} = \frac{1}{7.0} - \frac{1}{7.0} = 0
\]
Since \( \frac{1}{d_i} = 0 \), it follows that:
\[
d_i \rightarrow \infty
\]
This indicates that the image is formed at infinity.
### Analysis of the Image:
1. **Distance to the image**: \( d_i = \infty \) (the image is formed at infinity).
2. **Type of Image**: Since the object is at the focal point of a converging lens (\( d_o = f \)), the image is real.
3. **Orientation**: An image formed when the object is at a focal point of a converging lens is inverted.
### Summary:
- Distance to the image: \( d_i = \infty \)
- Type of image: Real
- Orientation: Inverted
\[
\frac{1}{f} = \frac{1}{d_o} + \frac{1}{d_i}
\]
where:
- \( f \) is the focal length of the lens,
- \( d_o \) is the object distance (measured from the lens to the object, positive for real objects),
- \( d_i \) is the image distance (measured from the lens to the image, positive for real images on the opposite side from the object and negative for virtual images on the same side as the object).
In this problem:
- The focal length \( f = 7.0 \, \text{cm} \) (positive, because it’s a converging lens),
- The object distance \( d_o = 7.0 \, \text{cm} \).
Now substitute the values into the lens formula:
\[
\frac{1}{7.0} = \frac{1}{7.0} + \frac{1}{d_i}
\]
Subtract \( \frac{1}{7.0} \) from both sides:
\[
\frac{1}{d_i} = \frac{1}{7.0} - \frac{1}{7.0} = 0
\]
Since \( \frac{1}{d_i} = 0 \), it follows that:
\[
d_i \rightarrow \infty
\]
This indicates that the image is formed at infinity.
### Analysis of the Image:
1. **Distance to the image**: \( d_i = \infty \) (the image is formed at infinity).
2. **Type of Image**: Since the object is at the focal point of a converging lens (\( d_o = f \)), the image is real.
3. **Orientation**: An image formed when the object is at a focal point of a converging lens is inverted.
### Summary:
- Distance to the image: \( d_i = \infty \)
- Type of image: Real
- Orientation: Inverted