To solve this problem, we will first find the image formed by the first lens (40 cm focal length) and then treat this image as the object for the second lens (20 cm focal length) to find the final image position and height.
First, we will use the lens equation for the first lens:
1/f = 1/do + 1/di,
where f is the focal length, do is the object distance, and di is the image distance.
For the first lens, f = 40 cm and do = 15 cm. Plug these values into the equation:
1/40 = 1/15 + 1/di
Now solve for di:
1/di = 1/40 - 1/15 = -1/60
di = -60 cm
The negative sign indicates that the image is an inverted virtual image, which is formed on the same side as the object. The distance between the first image and the first lens is 60 - 15 = 45 cm.
Now we will find the image height:
Image height = (Image distance / Object distance) × Object height
Image height = (45 / 15) × 2.0 = 6.0 cm
Now we will treat this image as an object for the second lens (20 cm focal length). For the second lens, the new object distance is:
do' = 45 cm - 10 cm = 35 cm (since the lenses are 10 cm apart)
Now use the lens equation for the second lens:
1/f' = 1/do' + 1/di',
where f' is the focal length of the second lens, do' is the new object distance, and di' is the final image distance.
For the second lens, f' = 20 cm and do' = 35 cm. Plug these values into the equation:
1/20 = 1/35 + 1/di'
Now solve for di':
1/di' = 1/20 - 1/35 = 3/140
di' = 140/3 = 46.67 cm
Next, we will find the magnification of the second lens:
Magnification = Image distance / Object distance
Magnification = 46.67 / 35 = 1.333
Now find the final image height by multiplying the height of the image formed by the first lens by the magnification of the second lens:
Final image height = Magnification × Image height
Final image height = 1.333 × 6 = 8.0 cm
So, the final image is formed at a distance of 46.67 cm on the other side of the second lens and has a height of 8.0 cm.
Two converging lenses with the focal lengths of 40 cm and 20 cm are 10 cm apart. a 2.0-cm-tall object is 15 cm in front of the 40-cm-focal-length lens. Calculate the image position and height.
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