A bi-convex lens with the focal length of 25 cm and a meniscus concave lens with the focal length of 10 cm, separated from each other as much as 70 cm. An object is located 100 cm from the bi-convex lens along the optical axis. Find out the location of ultimate image create by the combination of these two, using both algebra and roughly by ray tracing method. Also figure out the total magnification.
To find the location of the ultimate image created by the combination of the lenses, we can use the lens formula and the lens-maker's formula.
1. Algebraic Method:
Let's use the lens formula, which states:
1/f = 1/v - 1/u,
where f is the focal length of the lens, v is the image distance, and u is the object distance.
First, let's find the position of the bi-convex lens image:
Object distance (u1) = 100 cm
Focal length (f1) = 25 cm
1/f1 = 1/v1 - 1/u1
1/25 = 1/v1 - 1/100
Solving for v1 gives us:
1/v1 = 1/25 + 1/100
1/v1 = 5/100
v1 = 100/5 = 20 cm
Now, let's find the position of the meniscus concave lens image:
Focal length (f2) = -10 cm (negative because it's a concave lens)
Object distance (u2) = 70 cm - v1 (distance between the lenses)
1/f2 = 1/v2 - 1/u2
1/-10 = 1/v2 - 1/(70 - 20)
-1/10 = 1/v2 - 1/50
Solving for v2 gives us:
1/v2 = -1/10 + 1/50
1/v2 = -1/10 + 1/50 (common denominator 50)
1/v2 = -5/50 + 1/50
1/v2 = -4/50
v2 = -50/4 = -12.5 cm
Since v2 is negative, the image created by the meniscus concave lens is virtual and located 12.5 cm on the same side as the object.
Now, to find the location of the ultimate image, we add the distances from the lenses:
Ultimate image distance = v1 + v2
Ultimate image distance = 20 cm - 12.5 cm
Ultimate image distance = 7.5 cm
Therefore, the ultimate image is located 7.5 cm on the same side as the object.
2. Ray Tracing Method (roughly):
To get a rough idea of the image location, we can use the basic rules of ray tracing. Here's a simplified method:
- Draw a ray from the object parallel to the principal axis. It will pass through the focal point of the bi-convex lens (25 cm).
- Draw a ray from the object passing through the center of the bi-convex lens. This ray doesn't bend and continues straight.
- At the point where these rays intersect after passing through the bi-convex lens, draw a ray parallel to the principal axis. This ray will pass through the focal point of the meniscus concave lens (-10 cm).
- Extend these rays backward until they intersect. The point of intersection represents the location of the ultimate image.
By roughly measuring the distances on the diagram, we can estimate that the ultimate image is located around 7-8 cm from the original object, on the same side.
Total Magnification:
To find the total magnification, we can multiply the magnification of the bi-convex lens (M1) with the magnification of the meniscus concave lens (M2).
Magnification (M) = -v1/u1 = -20/100 = -1/5
Magnification (M1) = -v2/u2 = -(-12.5)/(70 - 20) = 12.5/50 = 1/4
Total Magnification (M_total) = M1*M2 = (-1/5)*(1/4) = -1/20
Therefore, the total magnification is -1/20, indicating that the image is reduced in size and inverted.