Two converging lenses are placed 31.0 cm apart. The focal length of the lens on the right is 22.0 cm, and the focal length of the lens on the left is 15.0 cm. An object is placed to the left of the 15.0 cm focal-length lens. A final image from both lenses is inverted and located halfway between the two lenses. How far to the left of the 15.0 cm focal-length lens is the original object?
To solve this problem, we can use the lens formula:
1/f = 1/v - 1/u
Where:
- f is the focal length of the lens
- v is the image distance
- u is the object distance
Let's denote the distance of the object from the left lens as x. Since the final image is located halfway between the two lenses, the image distance for the right lens will also be x.
For the left lens:
f₁ = 15.0 cm
v₁ = x
u₁ = -x (since object distance is to the left of the lens, it is negative)
For the right lens:
f₂ = 22.0 cm
v₂ = -x (since the image distance is also to the left of the right lens, it is negative)
u₂ = 31.0 cm - x (distance between the two lenses is 31.0 cm and the object distance for the right lens is the distance between the two lenses minus x)
Now we can set up the lens formula equation for each lens:
1/f₁ = 1/v₁ - 1/u₁
1/15.0 = 1/x - 1/-x
1/f₂ = 1/v₂ - 1/u₂
1/22.0 = 1/-x - 1/(31.0 - x)
Solving these two equations will give us the value of x, which represents the distance to the left of the 15.0 cm focal-length lens where the original object is located.
Let's solve these equations mathematically:
1/15.0 = 1/x - 1/-x
Multiplying both sides by 15.0x(-x):
-x = 15.0 - 15.0
x = 15.0 cm
1/22.0 = 1/-x - 1/(31.0 - x)
Multiplying both sides by 22.0x(31.0 - x):
31.0 - x = -22(31.0 - x)
31.0 - x = -682.0 + 22x
23x = 713.0
x = 30.99 cm (approximately)
Therefore, the original object is located approximately 30.99 cm to the left of the 15.0 cm focal-length lens.