Let's break down your questions about lenses step by step. We'll start with the characteristics of the image formed by a concave lens and then move on to the convex lens scenario.
Characteristics of the Image Formed by a Concave Lens
When an object is placed at a distance of 40 cm from a concave lens with a focal length of 20 cm, the image formed has specific characteristics. Here are four key features:
- Virtual Image: The image produced by a concave lens is always virtual, meaning it cannot be projected onto a screen.
- Upright Orientation: The image is upright, which means it has the same orientation as the object.
- Reduced Size: The image is smaller than the object, indicating that the magnification is less than one.
- Location: The image is formed on the same side of the lens as the object, specifically between the focal point and the lens.
Calculating Image Characteristics
To understand why these characteristics hold, we can use the lens formula:
1/f = 1/v - 1/u
Where:
- f = focal length of the lens (negative for concave lenses)
- v = image distance from the lens
- u = object distance from the lens (negative in lens convention)
Plugging in the values:
f = -20 cm (since it's a concave lens) and u = -40 cm.
Now substituting into the lens formula:
1/(-20) = 1/v - 1/(-40)
Solving this gives:
1/v = 1/(-20) + 1/40
1/v = -2/40 + 1/40 = -1/40
Thus, v = -40 cm, confirming that the image is virtual, upright, and located at a distance of 40 cm on the same side as the object.
Finding Object Distance for a Convex Lens
Now, let's tackle the second part regarding the convex lens. If the size of the image is reduced to 1/3 of the object's size, we can use the magnification formula:
Magnification (m) = h'/h = -v/u
Here, h' is the height of the image, h is the height of the object, v is the image distance, and u is the object distance.
Given that the image size is 1/3 of the object size, we have:
m = 1/3
Substituting into the magnification formula:
1/3 = -v/u
This implies:
v = -u/3
Now, we can use the lens formula for the convex lens:
1/f = 1/v - 1/u
Given that the focal length (f) is 20 cm, we substitute:
1/20 = 1/(-u/3) - 1/u
To simplify, we can express everything in terms of u:
1/20 = -3/u - 1/u
Combining the fractions gives:
1/20 = -4/u
Cross-multiplying leads to:
u = -80 cm
This means the object is placed at a distance of 80 cm from the optical center of the convex lens. The negative sign indicates that the object is on the same side as the incoming light, which is standard in lens conventions.
Summary
In summary, the characteristics of the image formed by a concave lens include being virtual, upright, reduced in size, and located on the same side as the object. For the convex lens, if the image size is 1/3 of the object size, the object is positioned 80 cm from the lens. Understanding these principles helps in grasping how lenses manipulate light to form images.
