A thin lens separated from a mirror by the distance of 40 cm and have the focal lengths of 25 cm and -15 cm. An object is place 14 cm from the lens. Where is the image?
To determine the position of the image formed by a lens, we can use the lens formula:
1/f = 1/v - 1/u
Where:
f = focal length of the lens
v = distance of the image from the lens
u = distance of the object from the lens
In this case, since there is a combination of a lens and a mirror, we need to consider both focal lengths. However, the lens formula assumes that the light passes through the lens, so we'll treat the mirror as if the light rays are coming from the opposite side.
Given:
f1 = 25 cm (positive focal length for the lens)
f2 = -15 cm (negative focal length for the mirror)
u = -14 cm (negative because the object is placed on the same side as the mirror)
We can treat the distance between the lens and the mirror as the focal length of the virtual lens formed by their combination. The formula for combining lenses/mirrors in contact is:
1/f_total = 1/f1 + 1/f2
Substituting the values:
1/f_total = 1/25 + 1/(-15)
1/f_total = 3/75 - 5/75
1/f_total = -2/75
f_total = -75/2 = -37.5 cm
Now, we can use the lens formula with the combined focal length (f_total), distance of the object (u), and solve for the image distance (v):
1/f_total = 1/v - 1/u
Substituting the values:
1/-37.5 = 1/v - 1/(-14)
Simplifying:
-1/37.5 = 1/v + 1/14
Find a common denominator:
-2/75 = (14 + v)/14v
Cross-multiplying:
-14v = 2(14 + v)
Expanding:
-14v = 28 + 2v
Simplifying:
-14v - 2v = 28
-16v = 28
v = -28/16 = -7/4 = -1.75 cm
Therefore, the image is formed 1.75 cm on the same side of the lens as the object, which means it is a virtual image.