A lens with f= 20.0 cm creates a
virtual image at -37.5 cm (in front
of the lens). What is the object
distance?
The Correct answer and Explanation is :
The correct answer is: 13.0 cm
To find the object distance for a lens that creates a virtual image, we can use the lens formula:
[
\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,
- (d_i) is the image distance.
Given:
- Focal length, (f = 20.0 \, \text{cm}) (positive for a converging lens),
- Image distance, (d_i = -37.5 \, \text{cm}) (negative because the image is virtual and located on the same side as the object).
We can rearrange the lens formula to solve for the object distance (d_o):
[
\frac{1}{d_o} = \frac{1}{f} – \frac{1}{d_i}
]
Substituting the given values into the equation:
[
\frac{1}{d_o} = \frac{1}{20.0} – \frac{1}{-37.5}
]
Now calculating the individual fractions:
[
\frac{1}{20.0} = 0.05
]
[
\frac{1}{-37.5} = -0.02667
]
Now substituting these values into the equation:
[
\frac{1}{d_o} = 0.05 + 0.02667
]
[
\frac{1}{d_o} = 0.07667
]
Taking the reciprocal gives:
[
d_o = \frac{1}{0.07667} \approx 13.0 \, \text{cm}
]
Thus, the object distance (d_o) is approximately 13.0 cm.
Explanation:
In optics, the relationship between object distance, image distance, and focal length is crucial for understanding how lenses function. The lens formula allows us to predict where an image will form based on the object’s position relative to the lens. A positive focal length indicates a converging lens, which brings parallel rays of light to a point.
A virtual image is formed when the object is located within the focal length of a converging lens. This situation leads to a negative image distance, meaning the image appears on the same side as the object.
By calculating the object distance, we determine that the object must be positioned approximately 13.0 cm from the lens to produce a virtual image at -37.5 cm. Understanding these relationships helps in various applications, from designing optical devices to predicting the behavior of light in different scenarios.