An object is placed to the left of a lens, and a real image is formed to the right of the lens. The image is inverted relative to the object and is one-half the size of the object. The distance between the object and the image is 95 cm. (a) How far from the lens is the object? (b) What is the focal length of the lens?
Everything I have tried has been wrong, so any help is appreciated!
To solve this problem, we can use the lens formula, which relates the object distance (u), image distance (v), and focal length (f) of a lens.
The lens formula is given by:
1/f = 1/v - 1/u
From the given information, we know that:
- The image formed is real and inverted relative to the object.
- The distance between the object and the image is 95 cm.
- The size of the image is one-half the size of the object.
Let's solve these questions step by step:
(a) How far from the lens is the object?
To find the object distance (u), we first need to determine the image distance (v). From the given information, we know that the image is formed to the right of the lens (which means v is positive).
We also know that the image is one-half the size of the object and inverted. This indicates that the image is real and diminished. In this scenario, we can use the magnification formula:
magnification (m) = height of the image (h_i) / height of the object (h_o)
m = -1/2
(here, the negative sign indicates the inversion)
Now, we can use the lens formula:
1/f = 1/v - 1/u
Given that v = 95 cm, and m = -1/2, we substitute these values into the magnification formula:
-1/2 = h_i / h_o = v / u
Simplifying:
-1/2 = 95 / u
Cross-multiplying and solving for u:
-1 * u = 2 * 95
u = -190 cm
Since the object distance should be a positive quantity (u cannot be negative in this case), the object distance from the lens is 190 cm.
(b) What is the focal length of the lens?
To find the focal length (f), we can also use the lens formula:
1/f = 1/v - 1/u
Using the values we have determined:
1/f = 1/95 - 1/190
Simplifying:
1/f = (2 - 1)/(2 * 95)
1/f = 1/190
Cross-multiplying and solving for f:
f = 190 cm
Therefore, the focal length of the lens is 190 cm.