Question
Describe the image created by a convex lens when the object is located between the focal point F and two times the focal point F. Use the lens and magnification equations to support your answer.
Answers
GPT-4o mini
When an object is placed between the focal point (F) and twice the focal point (2F) of a convex lens (also known as a converging lens), the image characteristics can be analyzed using the lens formula and magnification equations.
### Lens Formula
The lens formula for a convex lens 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 (the distance from the lens to the object).
- \( d_i \) is the image distance (the distance from the lens to the image).
### Magnification Equation
The magnification (\( M \)) produced by the lens is given by:
\[
M = -\frac{d_i}{d_o}
\]
### Analysis of the Image
1. **Object Location**: When the object is located between F and 2F, the object distance \( d_o \) is positive and less than \( 2f \), so \( d_o = f < d_o < 2f \).
2. **Image Distance Calculation**:
Since \( d_o < 2f \), we can calculate \( d_i \) using the lens formula. Rearranging gives:
\[
d_i = \frac{f \cdot d_o}{d_o - f}
\]
As \( d_o \) is less than \( 2f \), the term \( d_o - f \) is still positive, meaning that \( d_i \) will also be positive, indicating that the image is formed on the opposite side of the lens from the object.
3. **Image Characteristics**:
- **Location**: The image is located beyond 2F on the opposite side of the lens.
- **Size**: Since \( M = -\frac{d_i}{d_o} \) and \( d_i > d_o \) when \( d_o < 2f \), the absolute value of the magnification \( |M| > 1 \). This means that the image is magnified larger than the object.
- **Orientation**: The negative sign of the magnification indicates that the image is inverted.
### Summary of the Image Properties:
1. **Position**: The image is formed beyond 2F on the side opposite to the object.
2. **Size**: The image is larger than the object (magnified).
3. **Orientation**: The image is inverted compared to the object.
In conclusion, when an object is placed between the focal point and twice the focal point of a convex lens, the resulting image is formed beyond 2F, is inverted, and is larger than the object.
### Lens Formula
The lens formula for a convex lens 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 (the distance from the lens to the object).
- \( d_i \) is the image distance (the distance from the lens to the image).
### Magnification Equation
The magnification (\( M \)) produced by the lens is given by:
\[
M = -\frac{d_i}{d_o}
\]
### Analysis of the Image
1. **Object Location**: When the object is located between F and 2F, the object distance \( d_o \) is positive and less than \( 2f \), so \( d_o = f < d_o < 2f \).
2. **Image Distance Calculation**:
Since \( d_o < 2f \), we can calculate \( d_i \) using the lens formula. Rearranging gives:
\[
d_i = \frac{f \cdot d_o}{d_o - f}
\]
As \( d_o \) is less than \( 2f \), the term \( d_o - f \) is still positive, meaning that \( d_i \) will also be positive, indicating that the image is formed on the opposite side of the lens from the object.
3. **Image Characteristics**:
- **Location**: The image is located beyond 2F on the opposite side of the lens.
- **Size**: Since \( M = -\frac{d_i}{d_o} \) and \( d_i > d_o \) when \( d_o < 2f \), the absolute value of the magnification \( |M| > 1 \). This means that the image is magnified larger than the object.
- **Orientation**: The negative sign of the magnification indicates that the image is inverted.
### Summary of the Image Properties:
1. **Position**: The image is formed beyond 2F on the side opposite to the object.
2. **Size**: The image is larger than the object (magnified).
3. **Orientation**: The image is inverted compared to the object.
In conclusion, when an object is placed between the focal point and twice the focal point of a convex lens, the resulting image is formed beyond 2F, is inverted, and is larger than the object.